%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: tensor.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: XYATIP10 XYBTIP10 XYDASH10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips tensor.dvi -o tensor.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.05.02:1439 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet %%BeginFont: XYDASH10 %!PS-AdobeFont-1.1: XYDASH10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Aug 28 05:34:12 %%RevisionDate: 1997 Sep 18 10:23:31 % % XYDASH10: line segments for Xy-pic at 10 point % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYDASH10) readonly def /FamilyName (XYDASH) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch 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1ac6cf81a8eb952a043b1340a7fc883b28638897d15bb4c394d70df7df3d1312 2629b5a8c236d9e91fe466b264d6e5018581bb79e23dd527875cb97ef7962d44 2fb47682afcf9a3869ef323af9fec2d18e8a3613c10d546970216927e6740be8 0e272580db4cdc1b8fece17f94fc78640294a0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYBTIP10 %!PS-AdobeFont-1.1: XYBTIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:18 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYBTIP10) readonly def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBTIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 33 /d33 put dup 34 /d34 put dup 36 /d36 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 46 /d46 put dup 47 /d47 put dup 52 /d52 put dup 55 /d55 put dup 58 /d58 put dup 61 /d61 put dup 79 /d79 put dup 97 /d97 put dup 110 /d110 put dup 111 /d111 put dup 116 /d116 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 125 /d125 put dup 126 /d126 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092839 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139c8eef1237e57525cbc853d7cbe3c9a8b54c378e8af02257a8daa736c3d9ae fb18fd198a33681c334984d81e2d783d32adf54549f5bea0bf351b1016032908 81685bde8d44703654d97063c8ebfb896e029b2383f5754d467163ec07f3398a d88196c720fd98b9a2260de8d7d3aa6453f831ce18233cfbf6cb098bc3ca2cd1 495386a279ced386537228ec08f3b3e400cc040ab2e763b0cd93c9a2c5ee0436 f0a2f033ba5d3e4231aacc9b0aa820f7ad72a3cec593a1153ee5527693ad3bc8 eaef55ac2f52fdf27146c04dcc825181a275e632e75a94cb9b3d3f7d17c1c08b 83bbf5c681f864e234d10b0f7c64839aa1671931f39a001e4134030b91d9a473 6c7d5e101e04feb20a04907ab46ab24902c1844b018beefd9014c8b629674e57 f1f0d63ad79dfa8ce4d1fffabeb4315386d494a3ab66cc9f291a714ef0ee4f9f 1687f0ecbcd2acea0e98dd5f94dfd700e546599e58d1f25bc54ef6ec0f12b91e 6690287b7c527a51724cea71da655f2b2974633ba5484cc6c2300ba28dff89e2 0c37542986ec1e4613cc8a16521e5c2720d88fa18111a1083dcee82b011400a2 8b4124ab1a5bdde460e2589f2872b0436396df646b0eb176e75de9af54d7c4ab f628a596d7e1ead5815ec6bb58786913f0125dbe4a6328ea358185ce03fdf5b2 8d3e5cee90066a67f548590d69d197b1503ae8f993a1a7aa4248f0be3be623c0 fe29e1772ade5b00f22b228152d197d6b3e1ddf4dece5c7cbadfec3e7bb38696 91becf5079caf4c910b4a25a1b19e3c64eeb79e5223d56f7aae7411259fda2b2 2e2a4652323e63b97e76d22ca5fa800398fb0b4636201037f11794493ce10d4a 80fc85a0a26686c096d560a3a77cb2cf4fdcd105e98e7e88454412c6a107c5dc 7605eef7932c093879753c20b5397cdeba78d0a798f789304d63956f0f471a10 ad93e8ade240e9185f8028420ecc26e436fcff5421bcd39aeb8ac91800241d34 133cc54e557b68947337eb889fb494948d932519e0e19f3dd891220c04935f13 96f8196a907180e760be47484d144c1039ee7934d0f08397e9b640062ffd670c 1c7fc029acbe5f5c7982e68820e9140313eb390f785901879c614f5461f3e0e6 908f053b64f7f0ec48d567b4125934d21158a2ba50e361b6966e857922618b72 06d412a748706d7eac5fff41ee5c52f4142fce1ce9ba67a3c97ec186b187b80c 41b8ee73ba38e3bf2bc6741de19ef652832faba07c55e7bd7ae5fb49d86b1808 8bb5e80ed536552a211ce36d2961583aff648edd54f1de84ae741b98bbd57e37 577e55ca0f97c4ac1fed15732a0a072a176b0c5f14dbcee98f6d60fe3b3df180 2d93213c7eb7a37ff92ff4f97619596aa0771d0cbc82c5cf55b5be4f44fba75e 631ae71cbc9afbc42312cecdd86b7aec3152ddcf7fa1c6f5e051581597864746 2e975089ad928df03a4073bf06bfe1e262a551f24a12bca29408aacf7ed2e43a 3b7f7ff8e40af2ad2b131622058405cdf249568c2653b0be457fed5352e113fc 1107d6ed67a6ffeef5608109c13e5d5bc2432e65b82d3c3fe6d012e7a84e7730 13ae1201e1ad5ac9c2c11163be1d85f342956954f92651fcde7efbd58975fd01 37afbc60f68065b6b4af9856ef3a425a57a25b3ce8e5375aeccfc1146b78fb7e 6aa315ea438e2826d160cb12ae97fbd9c4704114f4269ae8ed6a114882c3fe34 9aa90531189c9e3b9fd1bb8f1772fdd681245ccfcb8b42165e018ad8af8f2440 954f0390129495317eab98a4d43da3e8b81c4269b19e4069c61c0ecb28a57873 610c578517a8e08f9762dce236f12bdb99718602cba86a56166ddcd42497ebf9 5b52ebedd9f09a6109ae30158fbe7d998aff28df96cb38f706e1b05b685c7df4 8410ed3d5558666c7c85dbae18e6e50330f7fd850d55e2c7a2277fa9fc2372b5 5e8102cd124121243810279c8d7773e43cc5233bdab4d5379ae5bf2f2ef82d78 9621ea5c345dba373f989f82a3f1d6edeb69dc7c9abb0de83996d732589a642c b63d80a8dd4420f3593380d56dabe56ffbf30f6d1bc7b8e3d8da03de5eb33aca 4b1dd4b2ae4b385c05b16d65d336149a68d790bfe240b23f6d38c19945037652 4b54fc9e0a60df4b23b9054d0c05b7081ed97195ef3fa26beecf5ea3fe6cae8b ebea88ed339f7e59494275fa923c8e6d1f6d34d82dd65764941e4578de48351c b2bdd01bcb8f95fbd8adc59d14b5212aa0f4243d1284c6c9325eb8331f11a031 c3c72668b913f9dc2b818ffa189229145614d11cc645beb56e006f2eddb395be f40407977ed4c2018231bdb1b3975e3471c917838f156ce50b6b7881cd5a924e f908a4246961874932ef7694a89836e8b2bdccf9d5e37465d72a051b28eacaa9 0c0499a2caf6f048e16a2d538f93db84d5aa1f5ca2bf5f59e688483eb6f7b76b 86fa39f5bb91ac2cb449490d02cacfae6eeda6cc9f81dd1f23f75dfaa147ed8a 8409b1df8fc6ab93b5e0faa71e6ccd142f1d922463b33e72adb48edb72841e71 6e736726f7502bb9acc792946494e722e21bf0fd127e18a5a841c4eb8e5da923 4b33a1ca992a972d9e9be9ab9ac03490a1ac894a14d0d968505d2e85e031c7ba 214a73d474109de5438f3d6c6fcf7d7c3328c23bb891d9a032b826499fef9f6a 6d2f4b931dac8a9e52f0cdc71237f90c0cc8d3ea2b77300e8eaf805a1c96a531 29116f4d980e6b5d2255be1b414827ec5f1c704957686a2a15eb97163c52aa2e 7ff983316875f3254b75360a5eb964fbbe2512639537c9d747ec797438cc6a1f c29c4ece2df509aca9f4432f2802404525ec3ccdd00c16d5e4b1d988e05a32d0 b3109c438bb4d96461ec7a01f31397d05613041ce4e7a058c76db266c85186ea c120fc8ded67831b7b6eaedc2697dbe21aa0be8c23376566c5c2368571454153 c6376da88f348eba5f988877fe62b7173a34031cf1a438f1d0681e252ba36789 ff08228ba054542b40c4d51023bbcc575acd9b5e48212cbdd008e48b5b455ae4 5e8f7abaac5079262c2f68dab05f1a2bfaf0c701cd9f6212786d56080b547abe de0a0c45b69b6735611e712287a2a2dd7c48bd41b61d5a521465d0056ad19950 f3b69224cceb5e44f4434014634678fad7c159c53fe7f1641358a043c397a25d dfcaaa727be1061288fd97407d8ca2e249e551e25cd28a4013136d80e56a9f03 7b1cf2a97c9d3e555c87c7442f038cda997d27c059816266af66e5af51445134 21ba39ffb81e96b689e4b1e8e6a31a64146f292793e8eef0390cb8cb8ef6ee21 95ffc8dc3707c1b5edd3e55958bafd54976526eaa15c355b4ff695b5dea821e4 78100344d5aa80c8968e88f8084a6c3d4f8498773f922ca08b95e293f39a66c0 7cf7a6471e41f1d213e63b2fc4ed95aaaff8523974de592f8781313988ee1fa1 da857e896e84708b791ce58ef30be9ab9d02ced5d8d6a72833dffeb44db429e3 a9c41e2c05da263b966233f2f52870aff1f5814a7460361ebc8146d9addcde2a 44347bd9eaeffcc9761fdc17a89ea5e8f445f95006e017e9276e079f74c8df0e c40a1c57cd36402eca43f744fa2586a1159bebcb5cdfb5cc7d85a50dcfa3a62d 3ee6c5bfb1867e002e8aa07a2acf3e66b535885f1e226169f48a0128d65481cf dcb8c0c147570b40a573595e69a63d1b7ea70ea2512bf2a3ec433d10ccbd63fb 692e642cb7397d926de8e29fa5320505e0023dc37be3ed06cc5d08b0efd62382 92c8a6fafb49304e2bb94b5a9603c34c7f30fe91bc74b9ba972013ac477ccfac 93541acea4d89fd121931f93bedede6bca4bb76959c2bae0655474de2364b0f2 3f0837b0dfb1ffd40fef2cad7cbd9f4e483227ec15a96a23badbf56d9c46a0b6 71771274545d4e81f830000c86c88934f4eda292bd7f13d93f158f446364831a 887ec603d4bb1308fed4f6effe728ed2460280640addf145df15c70d87970054 f86abf37ce50926a7d7f3a8dafb63ed484128a581d72a90c9ead1890f6551444 8fb113f29934bd2da3fc2a33a83bbcb5ea207c71cad63b05f82d1a55fc81280d 548451d254f5df3c97cd3bf42181bb306f365f419cf90ce7778ff5ffea51ea09 4fab9198f70afaac4dd6a703065f723b7d6e53db961c9699bfe7d79280e2f62f 4a05d7a1c0bbe55ae3be7e851f8bdc5206de3f6f1ea55f26248ca6fbdb061417 eb7689dd9d59d4a6d2e42d8a40b606ff7d4ff5fe0b6f6bf498528e0693b398f6 c74708bcec71b15d2cea5140321727a091677ac067f787001ea089276558c727 51a1e8644e499199d2f896154a8275d8b9b01aa32dec10a0f0196e4474e4b188 ab9a89e29a8edf5c449d7c25ea1091646d2c246599e5dcc4f0167f7ee1081ad8 c267205b3b256b96a3b54281236eb39a700fa43acf8517abc72a845bc110e1ec 153fc374710975248e3401bc8d1e66458fa2198d8885374f1feb57e7f77bf4ab 5edec4d089a223380cdd01d9ce39fdaf30ea1e3512db78a713a40159e3f7f578 fc119c61b32b73c6de5dbdf2899d6e9ccfd5c563887bc57122bc0dbac2091844 6b1ab98b5786c8f18477cb06b0df72dddc8c343deea9161c526b7210c5397fe9 faf8d92b0f886b1cda38491d477c1c8d082a8b542e505c6a8bc5bed72b14f1c3 9f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYATIP10 %!PS-AdobeFont-1.1: XYATIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:17 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYATIP10) readonly def /FamilyName (XYATIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYATIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /d14 put dup 15 /d15 put dup 16 /d16 put dup 33 /d33 put dup 34 /d34 put dup 36 /d36 put dup 38 /d38 put dup 39 /d39 put dup 40 /d40 put dup 41 /d41 put dup 42 /d42 put dup 47 /d47 put dup 52 /d52 put dup 55 /d55 put dup 58 /d58 put dup 61 /d61 put dup 79 /d79 put dup 97 /d97 put dup 102 /d102 put dup 111 /d111 put dup 116 /d116 put dup 118 /d118 put dup 119 /d119 put dup 120 /d120 put dup 125 /d125 put dup 126 /d126 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092838 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139d5cb0db6f50b26bd8ab173859c9c94db82970311d7eb0a02bee1be5f0d126 f9079e67107eda14b460e46b03b0422eb45a4f4afa382841c35b0bc0c8639b73 43c819afc69838ce781c2de7d22bf503ef6eec27c83cfd52a77bbc754b4f2050 55341700991f3ef4b5b5a54c21283034b38c8b6a6e65abccc94c0c26836806ec 4df8d8c64f595841395072f8a2d289b7fe5497bc0e061810b16a1e48653bb092 42ffa3ef0bba4e37a5aa4ddc31f3138aaf10998fd66f3817b77012eac677ed2d 7447908c771fbaba4dfcfeca5374a6b87e5657809a82f0ae068c5384c12f4653 2a82645a512212140d815e80ec76b3370c382e9f6d29ec6afc178a622249fda7 775d4f6a554f04748ed4210257fa6e376188a175db3c00f88421820b063a985c 07ad665eff7e2d32a27015c528227c2805aa8df134f4abad9958b841aa4263ec 9ec6d907308b0a5a51049de002cfe60ef35bf33fc863ba14ff361749554abd65 47426fbf3958ddb506ea3e303c932edec2896d3017a57913cde60cbdafeddbd3 cd5ef6dd49299783a92fe9deafb24e7f74a6ca6c0198b2fcda46b51445a2800f d1dc6a092b3192cfb314892e52753b5c7b94edd34c8213b032f8aab5d08753cb 65bd1a1225ba43194efa625ee5dc6eec6d0353d06ef3bef9c0d7df78fe189482 779c9276e83ccd71b50e87ba92cf092d65498e5b43cdc89436019e306c0d628e c7d1470ab322ca2d6adc3e9ff4196c0f7792ca0b20f741ad7bbd4391b0218511 14f0e97a44fd7d03e7003d1fe2c70d6266740dad7b07b3794dc53871887c8eba 2910d6fa654346318753cc752a7d25c1ef970e7dda8d8bbe249e9edf2c8c2ae6 abb93bab2dd8560466a7d08ce3e200d7cc488c6045871a8033ec1dc3a53e7056 3fe265f58c6ac98754d399b47b817b7aed7ebbbb503268e7324d6c0a0931978b 51a7d37187a99f234ae4c46b585383938e7cf59d2dfd20717031966eaa4317e7 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b(this)i(category)d(con)n(tains)i(a)g(sub)r (quotien)n(t)h(category)d(whic)n(h)i(is)h(equiv)-5 b(alen)n(t)25 b(to)g(the)456 923 y(MTC)20 b FL(C)721 893 y FM(in)n(t)804 923 y FQ(\()p FA(g)p FJ(;)14 b FH({)s FQ(\).)35 b(Com)n(bining)19 b(this)i(result)e(with)h(a)g(certain)f(dualit)n(y)h(b)r(et)n(w)n(een)f (the)i(categories)456 1023 y FL(O)522 1035 y FI(k)590 1023 y FQ(and)28 b FL(O)818 1035 y FE(\000)p FM(2)p FI(h)942 1019 y Fx(_)985 1035 y FE(\000)p FI(k)1078 1023 y FQ(,)g(Fink)n(elb)r (erg)f(sho)n(w)n(ed)f(that)i(this)h(sub)r(quotien)n(t)e(is)h(dual)g(to) f(the)h(category)456 1123 y FL(O)524 1092 y FM(in)n(t)522 1146 y FI(k)608 1123 y FQ(,)g(th)n(us)f(establishing)g(sim)n (ultaneously)g(Theorems)f(7.0.1)h(and)g(7.0.2.)605 1222 y(Finally)-7 b(,)22 b(the)e(third)g(pro)r(of)f(of)h(Theorem)f(7.0.1,)h (based)g(on)f(the)h(theory)g(of)f(v)n(ertex)g(op)r(erator)456 1322 y(algebras,)25 b(w)n(as)i(recen)n(tly)g(giv)n(en)f(b)n(y)i(Huang)f (and)g(Lep)r(o)n(wsky)g([)p FK(HL)p FQ(].)605 1421 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(\(whic)n(h)456 2119 y(to)r(ok)i(the)h(previous)f(140)f(pages)h(of)h (this)g(b)r(o)r(ok\))f(and)h(some)f(non-trivial)g(algebraic)f(geometry) 456 2218 y(used)i(in)h([)p FK(TUY)q FQ(],)g(also)e(not)i(an)f(easy)g (reading.)605 2318 y(The)22 b(pro)r(of)g(giv)n(en)g(in)g(this)h(c)n (hapter)e(is)i(based)e(on)h(the)h(mo)r(dular)f(functor)g(approac)n(h;)g (ho)n(w-)456 2418 y(ev)n(er,)i(our)f(pro)r(of)h(of)h(the)g(gluing)f (axiom)f(follo)n(ws)h(the)h(ideas)e(of)i([)p FK(BFM)p FQ(])f(rather)g(than)g([)p FK(TUY)q FQ(].)456 2517 y(This)g(pro)r(of)h (w)n(as)f(nev)n(er)f(published)j(b)r(efore;)f(ho)n(w)n(ev)n(er,)f(for)g (the)h(most)g(part)f(it)i(closely)d(follo)n(ws)456 2617 y(the)i(argumen)n(ts)e(in)i([)p FK(BFM)p FQ(],)h(so)e(all)g(the)i (credit)e(b)r(elongs)g(to)h(Beilinson,)g(F)-7 b(eigin,)26 b(and)e(Mazur.)456 2717 y(Mo)r(difying)k(their)g(argumen)n(ts)e(for)i (WZW)g(mo)r(del)g(w)n(as)f(rather)g(straigh)n(tforw)n(ard;)e(according) 456 2816 y(to)g(priv)-5 b(ate)26 b(comm)n(unications)f(from)h (Beilinson)f(and)h(F)-7 b(eigin,)26 b(they)h(in)n(tended)f(to)g (include)g(the)456 2916 y(pro)r(of)k(for)g(WZW)h(mo)r(del)g(in)g(the)g (\014nal)g(v)n(ersion)e(of)i(the)g(man)n(uscript.)46 b(Unfortunately)-7 b(,)31 b(it)h(is)456 3015 y(not)27 b(clear)g(when)g(\(and)h(if)6 b(\))29 b(suc)n(h)e(a)g(\014nal)h(v)n (ersion)e(app)r(ears,)g(so)h(w)n(e)g(include)h(this)g(pro)r(of)f(here.) 1115 3206 y FK(7.1.)46 b(Preliminaries)29 b(on)j(a\016ne)g(Lie)f (algebras)605 3356 y FQ(The)g(aim)g(of)f(this)i(subsection)e(is)h(just) g(to)g(\014x)g(the)g(notation,)g(w)n(e)g(refer)f(to)h(the)g(b)r(o)r(ok) g(of)456 3455 y(Kac)26 b([)p FK(K1)p FQ(])i(for)f(a)g(comprehensiv)n(e) f(treatmen)n(t.)605 3555 y(Let)40 b FA(g)g FQ(b)r(e)h(a)e(\014nite)i (dimensional)e(simple)i(Lie)f(algebra)e(o)n(v)n(er)g FH(C)15 b FQ(.)80 b(W)-7 b(e)41 b(\014x)f(a)f(Cartan)456 3655 y(subalgebra)22 b FA(h)i FL(\032)e FA(g)j FQ(and)f(let)h FL(h\001)p FJ(;)14 b FL(\001i)25 b FQ(b)r(e)g(an)f(in)n(v)-5 b(arian)n(t)23 b(bilinear)h(form)g(on)h FA(g)f FQ(normalized)f(so)h (that)456 3754 y FL(h)p FJ(\013;)14 b(\013)p FL(i)27 b FQ(=)f(2)j(for)g(long)g(ro)r(ots)g(of)g FA(g)p FQ(.)43 b(W)-7 b(e)30 b(will)f(use)h(the)g(same)f(notations)f(\(and)i (notions\))f(as)g(in)456 3854 y(Section)e(1.3.)605 3953 y(Let)j FA(g)p FQ(\(\()p FJ(t)p FQ(\)\))e FL(\021)f FA(g)20 b FL(\012)1203 3965 y Fz(C)1268 3953 y FH(C)15 b FQ(\()q(\()p FJ(t)p FQ(\)\))37 b(b)r(e)30 b(the)g FO(lo)l(op)k(algebr)l(a)j FQ(of)30 b FA(g)p FQ(.)44 b(Then)31 b(the)f FO(a\016ne)i(Lie)h(algebr)l (a)456 4053 y FQ(of)27 b FA(g)h FQ(is)1652 4191 y Fy(b)1654 4195 y FA(g)23 b FQ(=)g FA(g)p FQ(\(\()p FJ(t)p FQ(\)\))c FL(\010)f FH(C)d FJ(K)-1778 b FQ(\(7.1.1\))456 4337 y(with)28 b(comm)n(utation)f(relations)952 4479 y([)p FJ(a)19 b FL(\012)f FJ(f)t(;)c(b)k FL(\012)g FJ(g)s FQ(])k(=)h([)p FJ(a;)14 b(b)p FQ(])k FL(\012)g FJ(f)9 b(g)21 b FQ(+)d FL(h)p FJ(a;)c(b)p FL(i)g FQ(Res)2299 4491 y FM(0)2336 4479 y FQ(\()p FJ(d)-14 b(f)23 b(g)s FQ(\))p FJ(K)q(;)180 b FQ([)p FJ(K)q(;)2941 4475 y Fy(b)2943 4479 y FA(g)p FQ(])23 b(=)f(0)p FJ(:)456 4626 y FQ(F)-7 b(or)27 b(brevit)n(y)-7 b(,)27 b(w)n(e)g(often)h(use)f(the)h(notation)f FJ(x)p FQ([)p FJ(n)p FQ(])d(=)e FJ(x)d FL(\012)f FJ(t)2288 4596 y FI(n)2333 4626 y FJ(;)c(x)24 b FL(2)f FA(g)p FQ(.)605 4726 y(W)-7 b(e)29 b(let)869 4722 y Fy(b)871 4726 y FA(g)913 4696 y FM(+)993 4726 y FQ(=)c FJ(t)p FA(g)p FQ([[)p FJ(t)p FQ(]],)1328 4722 y Fy(b)1330 4726 y FA(g)1372 4696 y FE(\000)1453 4726 y FQ(=)g FJ(t)1573 4696 y FE(\000)p FM(1)1663 4726 y FA(g)p FQ([)p FJ(t)1758 4696 y FE(\000)p FM(1)1847 4726 y FQ(].)41 b(W)-7 b(e)29 b(ha)n(v)n(e)f(a)h(decomp)r (osition)f(of)2984 4722 y Fy(b)2986 4726 y FA(g)h FQ(in)n(to)g(subal-) 456 4826 y(gebras)1521 4964 y Fy(b)1523 4968 y FA(g)23 b FQ(=)1674 4964 y Fy(b)1676 4968 y FA(g)1718 4933 y FM(+)1791 4968 y FL(\010)18 b FA(g)h FL(\010)f FH(C)d FJ(K)30 b FL(\010)2253 4964 y Fy(b)2256 4968 y FA(g)2298 4933 y FE(\000)2353 4968 y FJ(:)605 5116 y FQ(W)-7 b(e)34 b(will)f(b)r(e)h(in)n(terested)f(in)1522 5112 y Fy(b)1524 5116 y FA(g)p FQ(-mo)r(dules)g(of)g(lev)n(el)g FJ(k)i FL(2)e FH(C)15 b FQ(,)41 b(i.e.,)35 b(mo)r(dules)e FJ(V)52 b FQ(suc)n(h)33 b(that)456 5216 y FJ(K)6 b FL(j)556 5228 y FI(V)636 5216 y FQ(=)22 b FJ(k)17 b FQ(id)852 5228 y FI(V)910 5216 y FQ(;)23 b(this)d(is)g(equiv)-5 b(alen)n(t)20 b(to)g(considering)f(mo)r(dules)h(o)n(v)n(er)f FJ(U)9 b FQ(\()2679 5212 y Fy(b)2681 5216 y FA(g)o FQ(\))2754 5228 y FI(k)2819 5216 y FQ(=)22 b FJ(U)2970 5212 y Fy(b)2972 5216 y FA(g)p FJ(=U)3120 5212 y Fy(b)3122 5216 y FA(g)o FQ(\()p FJ(K)9 b FL(\000)t FJ(k)s FQ(\).)p eop %%Page: 169 3 169 172 bop 1153 226 a FM(7.2.)29 b(REMINDERS)f(FR)n(OM)h(ALGEBRAIC)h (GEOMETR)-5 b(Y)575 b(169)456 425 y FQ(W)-7 b(e)41 b(will)g(denote)f(b) n(y)g FL(O)1256 437 y FI(k)1338 425 y FQ(the)h(category)e(of)1947 421 y Fy(b)1950 425 y FA(g)o FQ(-mo)r(dules)i(of)f(lev)n(el)h FJ(k)i FQ(whic)n(h)e(ha)n(v)n(e)e(w)n(eigh)n(t)456 525 y(decomp)r(osition)23 b(with)i(\014nite-dimensional)f(w)n(eigh)n(t)f (subspaces,)h(suc)n(h)g(that)h(the)g(action)e(of)3345 521 y Fy(b)3347 525 y FA(g)3389 495 y FM(+)456 624 y FQ(is)k(lo)r(cally)g(nilp)r(oten)n(t)h(and)g(the)g(action)f(of)g FA(g)h FQ(is)f(in)n(tegrable.)605 724 y(Of)33 b(sp)r(ecial)f(in)n (terest)g(for)g(us)g(are)g(t)n(w)n(o)g(classes)f(of)h(mo)r(dules)h (from)f FL(O)2821 736 y FI(k)2862 724 y FQ(:)47 b(W)-7 b(eyl)32 b(mo)r(dules)456 824 y(and)27 b(in)n(tegrable)f(mo)r(dules.)37 b(W)-7 b(eyl)28 b(mo)r(dule)g FJ(V)1930 794 y FI(k)1911 847 y(\025)1971 824 y FJ(;)14 b(\025)23 b FL(2)h FJ(P)2211 836 y FM(+)2266 824 y FQ(,)k(is)f(de\014ned)h(b)n(y)1545 975 y FJ(V)1612 941 y FI(k)1593 995 y(\025)1676 975 y FQ(=)22 b(Ind)1884 932 y Fu(b)1886 935 y Fk(g)1886 1004 y(g)p FE(\010)1972 1001 y Fu(b)1973 1004 y Fk(g)2008 988 y FF(+)2054 1004 y FE(\010)p Fz(C)t FI(K)2226 975 y FJ(V)2274 987 y FI(\025)2318 975 y FJ(;)-1885 b FQ(\(7.1.2\))456 1116 y(where)19 b FJ(V)736 1128 y FI(\025)799 1116 y FQ(is)h(the)f(irreducible)g(\014nite-dimensional)g FA(g)p FQ(-mo)r(dule)h(with)g(highest)f(w)n(eigh)n(t)g FJ(\025)p FQ(,)j(whic)n(h)456 1216 y(w)n(e)i(consider)f(as)h(a)h(mo)r(dule)f(o)n (v)n(er)f FA(g)12 b FL(\010)1654 1212 y Fy(b)1656 1216 y FA(g)1698 1186 y FM(+)1766 1216 y FL(\010)g FH(C)j FJ(K)36 b FQ(b)n(y)24 b(letting)2374 1212 y Fy(b)2376 1216 y FA(g)2418 1186 y FM(+)2498 1216 y FQ(act)g(as)g(0)h(and)f FJ(K)30 b FQ(act)25 b(as)e FJ(k)17 b FQ(id.)456 1316 y(The)27 b(W)-7 b(eyl)28 b(mo)r(dule)g(is)f(free)h(o)n(v)n(er)1547 1312 y Fy(b)1550 1316 y FA(g)1592 1285 y FE(\000)1648 1316 y FQ(.)605 1415 y(If)c FJ(k)35 b(=)-52 b FL(2)24 b FH(Q)5 b FQ(,)31 b(then)23 b(W)-7 b(eyl)23 b(mo)r(dules)g(are)g (irreducible)f(and)h(the)g(category)e FL(O)2885 1427 y FI(k)2949 1415 y FQ(is)i(semisimple.)456 1515 y(W)-7 b(e)24 b(will)g(b)r(e)g(mostly)g(in)n(terested)f(in)h(the)g(other)g (extreme)f(case)g FJ(k)j FL(2)d FH(Z)2640 1527 y FM(+)2689 1515 y FQ(.)35 b(In)24 b(this)h(case,)e(w)n(e)h(can)456 1614 y(also)j(consider)g(in)n(tegrable)g(highest-w)n(eigh)n(t)g(mo)r (dules.)39 b(W)-7 b(e)29 b(will)f(denote)h(b)n(y)f FL(O)2997 1584 y FM(in)n(t)2995 1638 y FI(k)3105 1614 y FL(\032)c(O)3260 1626 y FI(k)3329 1614 y FQ(the)456 1714 y(sub)r(category)h(of)h(in)n (tegrable)g(mo)r(dules,)h(i.e.,)g(suc)n(h)g(mo)r(dules)f(that)h(for)g (ev)n(ery)e(ro)r(ot)h FJ(\013)p FQ(,)i FJ(n)23 b FL(2)g FH(Z)o FQ(,)456 1814 y(the)34 b(action)f(of)g FJ(e)999 1826 y FI(\013)1046 1814 y FQ([)p FJ(n)p FQ(])h(is)g(lo)r(cally)f(nilp) r(oten)n(t.)55 b(It)34 b(is)g(kno)n(wn)f(that)h FL(O)2649 1784 y FM(in)n(t)2647 1837 y FI(k)2767 1814 y FQ(is)f(semisimple)h (with)456 1916 y(simple)27 b(ob)5 b(jects)27 b FJ(L)1055 1886 y FI(k)1055 1939 y(\025)1099 1916 y FJ(;)14 b(\025)23 b FL(2)g FJ(P)1350 1886 y FI(k)1338 1936 y FM(+)1394 1916 y FQ(,)k(where)h FJ(P)1750 1886 y FI(k)1738 1936 y FM(+)1821 1916 y FQ(is)f(the)h FO(p)l(ositive)j(Weyl)g(alc)l(ove)1418 2062 y FJ(P)1483 2027 y FI(k)1471 2082 y FM(+)1550 2062 y FQ(=)23 b FL(f)p FJ(\025)g FL(2)g FJ(P)1882 2074 y FM(+)1960 2062 y FL(j)h FQ(\()p FJ(\025;)14 b(\022)2165 2027 y FE(_)2214 2062 y FQ(\))24 b FL(\024)e FJ(k)s FL(g)p FJ(;)-2012 b FQ(\(7.1.3\))456 2210 y(see)31 b([)p FK(K1)p FQ(].)51 b(\(Note)32 b(that)h FJ(P)1324 2179 y FI(k)1312 2230 y FM(+)1399 2210 y FQ(is)f(the)h(same)f(set)g(whic)n(h)g(w)n(e)g (denoted)g(b)n(y)g FJ(C)38 b FQ(in)33 b(Section)f(3.3.\))456 2316 y(The)25 b(mo)r(dules)g FJ(L)1004 2286 y FI(k)1004 2339 y(\025)1072 2316 y FQ(are)f(irreducible)h(and)g(can)f(b)r(e)i (describ)r(ed)f(as)f(the)i(quotien)n(t)f FJ(L)3023 2286 y FI(k)3023 2339 y(\025)3089 2316 y FQ(=)e FJ(V)3243 2286 y FI(k)3225 2339 y(\025)3284 2316 y FJ(=)-5 b(Z)3378 2328 y FI(\025)3421 2316 y FQ(,)456 2418 y(where)29 b FJ(Z)755 2430 y FI(\025)828 2418 y FQ(is)h(the)g(unique)h(maximal)e (prop)r(er)g(submo)r(dule)h(of)g FJ(V)2525 2388 y FI(k)2506 2442 y(\025)2566 2418 y FQ(.)44 b(It)30 b(is)g(kno)n(wn)f(that)h FJ(Z)3315 2430 y FI(\025)3389 2418 y FQ(is)456 2518 y(generated)c(b)n (y)h(one)h(v)n(ector:)35 b FJ(Z)1440 2530 y FI(\025)1506 2518 y FQ(=)23 b FJ(U)1658 2514 y Fy(b)1660 2518 y FA(g)p FQ(\()p FJ(e)1773 2530 y FI(\022)1810 2518 y FQ([)p FL(\000)p FQ(1]\))1995 2488 y FI(a)p FM(+1)2119 2518 y FJ(v)2159 2530 y FI(\025;k)2259 2518 y FQ(,)28 b(where)f FJ(a)c FQ(=)f FJ(k)g FL(\000)c FQ(\()p FJ(\025;)c(\022)3010 2488 y FE(_)3059 2518 y FQ(\).)605 2620 y(It)24 b(is)f(useful)i(to)e (note)h(that)f(b)r(oth)h FJ(V)1717 2590 y FI(k)1698 2644 y(\025)1782 2620 y FQ(and)f FJ(L)1996 2590 y FI(k)1996 2644 y(\025)2063 2620 y FQ(ha)n(v)n(e)f(a)i(natural)e FH(Z)2664 2632 y FE(\000)2714 2620 y FQ(-grading)g(\(sometimes)456 2720 y(called)f(the)g FO(homo)l(gene)l(ous)k(gr)l(ading)p FQ(\),)f(de\014ned)e(b)n(y)f(deg)15 b FJ(v)2225 2732 y FI(\025;k)2348 2720 y FQ(=)22 b(0)p FJ(;)14 b FQ(deg)g FJ(a)p FQ([)p FJ(n)p FQ(])23 b(=)g FJ(n;)14 b(a)23 b FL(2)g FA(g)p FJ(;)14 b(n)23 b FL(2)g FH(Z)o FQ(.)456 2822 y(It)28 b(is)f(easy)g(to)h(see)f(that)h(homogeneous)e(comp)r(onen) n(ts)i(of)f FJ(V)2355 2792 y FI(k)2336 2846 y(\025)2424 2822 y FQ(\(and,)h(in)g(fact,)g(an)n(y)f(mo)r(dule)h(in)456 2922 y(the)g(category)d FL(O)999 2934 y FI(k)1040 2922 y FQ(\))j(are)f(\014nite-dimensional.)605 3022 y(Finally)-7 b(,)23 b(w)n(e)f(will)g(de\014ne)g(the)g(dualit)n(y)g(in)g(the)g (category)e FL(O)2437 3034 y FI(k)2500 3022 y FQ(b)n(y)i FJ(D)r(V)42 b FQ(=)22 b(\()p FJ(V)2957 2992 y FE(\003)2996 3022 y FQ(\))3028 2992 y FI(\\)3059 3022 y FQ(,)h(where)e FJ(V)3406 2992 y FE(\003)456 3121 y FQ(is)29 b(the)h(restricted)f(dual) g(to)h FJ(V)19 b FQ(,)30 b(i.e.)42 b(the)30 b(direct)g(sum)f(of)h(the)g (dual)f(spaces)g(to)g(homogeneous)456 3221 y(comp)r(onen)n(ts)40 b(of)g FJ(V)19 b FQ(,)44 b(and)d FJ(\\)f FQ(is)h(de\014ned)g(as)f (follo)n(ws:)62 b(for)40 b(a)2476 3217 y Fy(b)2478 3221 y FA(g)g FQ(mo)r(dule)h FJ(W)12 b FQ(,)44 b(the)d(mo)r(dule)456 3321 y FJ(W)546 3290 y FI(\\)616 3321 y FQ(coincides)e(with)i FJ(W)51 b FQ(as)39 b(a)h(v)n(ector)e(space,)k(and)e(the)g(action)f(of) 2721 3317 y Fy(b)2724 3321 y FA(g)g FQ(is)h(t)n(wisted)g(b)n(y)f(the) 456 3420 y(automorphism)1272 3555 y FJ(\\)9 b FQ(:)28 b FJ(x)p FQ([)p FJ(n)p FQ(])23 b FL(7!)g FQ(\()p FL(\000)p FQ(1\))1807 3521 y FI(n)1852 3555 y FJ(x)p FQ([)p FL(\000)p FJ(n)p FQ(])p FJ(;)180 b(K)29 b FL(7!)23 b(\000)p FJ(K)q(:)-2173 b FQ(\(7.1.4\))456 3690 y(It)23 b(is)g(easy)e(to)i(see)g(that)g FJ(D)i FQ(is)d(an)h(an)n(ti-automorphism)e(of)i(the)g(category)e FL(O)2819 3702 y FI(k)2883 3690 y FQ(whic)n(h)i(preserv)n(es)456 3790 y FL(O)524 3760 y FM(in)n(t)522 3813 y FI(k)608 3790 y FQ(.)48 b(In)31 b(particular,)g(for)g(an)g(in)n(tegrable)f(mo)r (dule)h FJ(L)2189 3760 y FI(k)2189 3813 y(\025)2232 3790 y FQ(,)h FJ(D)r(L)2415 3760 y FI(k)2415 3813 y(\025)2490 3790 y FQ(is)f(also)f(an)h(irreducible)g(in)n(te-)456 3889 y(grable)f(mo)r(dule,)j(whose)d(top)i(homogeneous)d(comp)r(onen)n (t)j(is)f FJ(V)2524 3859 y FE(\003)2505 3913 y FI(\025)2562 3889 y FQ(.)49 b(It)31 b(is)h(\(non-canonically\))456 3992 y(isomorphic)26 b(to)h FJ(L)1034 3962 y FI(k)1034 4018 y FE(\000)p FI(w)1134 4026 y FF(0)1166 4018 y FM(\()p FI(\025)p FM(\))1262 3992 y FQ(.)1101 4169 y FK(7.2.)47 b(Reminders)29 b(from)h(algebraic)i(geometry)605 4319 y FQ(In)k(this)g(section)g(w)n(e)f(brie\015y)g(list)i(some)e(facts)g (from)h(algebraic)e(geometry)g(whic)n(h)i(will)456 4419 y(b)r(e)g(used)f(b)r(elo)n(w.)61 b(All)36 b(of)g(them)g(are)f(quite)h (standard,)h(so)e(a)g(reader)f(who)i(has)f(ev)n(en)g(basic)456 4518 y(kno)n(wledge)26 b(of)h(algebraic)f(geometry)g(o)n(v)n(er)g FH(C)49 b FQ(can)27 b(safely)g(skip)h(this)g(section.)605 4618 y(All)33 b(v)-5 b(arieties)32 b(considered)f(in)i(this)f(section)g (are)g(considered)f(with)i(analytic)f(top)r(ology;)456 4717 y(as)h(b)r(efore,)h(w)n(e)g(use)f(the)h(w)n(ords)e(\\manifold")h (and)g(\\non-singular)e(v)-5 b(ariet)n(y")33 b(as)g(synon)n(yms.)456 4817 y(F)-7 b(or)22 b(a)h(v)-5 b(ariet)n(y)23 b FJ(S)5 b FQ(,)24 b(w)n(e)f(denote)g(b)n(y)g FL(O)1597 4829 y FI(S)1669 4817 y FQ(the)h(structure)f(sheaf)g(of)g FJ(S)28 b FQ(\(i.e.,)d(the)f(sheaf)f(of)g(analytic)456 4917 y(functions)35 b(on)f FJ(S)5 b FQ(\).)58 b(W)-7 b(e)35 b(assume)f(that)h(the)g(reader) e(is)i(familiar)f(with)h(the)g(notion)f(of)h(a)f FL(O)r FQ(-)456 5016 y(mo)r(dule)d(and)g(a)g(coheren)n(t)f FL(O)r FQ(-mo)r(dule.)48 b(As)32 b(usual,)f(for)g(a)g(p)r(oin)n(t)g FJ(s)e FL(2)h FJ(S)36 b FQ(w)n(e)30 b(de\014ne)i(b)n(y)f FL(O)3349 5028 y FI(S;s)456 5116 y FQ(the)k(lo)r(cal)g(ring)g(at)g FJ(s)p FQ(,)i(i.e.)61 b(the)35 b(ring)g(of)g(germs)g(of)g(analytic)g (functions)g(at)g FJ(s)p FQ(,)j(and)d(b)n(y)g FJ(m)3409 5128 y FI(s)456 5216 y FQ(the)28 b(maximal)g(ideal)g(of)h(this)f(ring,) g(whic)n(h)h(consists)e(of)i(functions)f(v)-5 b(anishing)28 b(at)h FJ(s)p FQ(.)39 b(W)-7 b(e)29 b(also)p eop %%Page: 170 4 170 173 bop 456 226 a FM(170)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(denote)h(b)n(y)862 404 y Fy(b)842 425 y FL(O)908 437 y FI(S;s)1033 425 y FQ(the)h(completion)f(of)g(the)h(lo)r (cal)e(ring)h(with)h(resp)r(ect)f(to)g(top)r(ology)f(giv)n(en)h(b)n(y) 456 525 y(the)34 b(p)r(o)n(w)n(ers)e(of)i(the)g(maximal)f(ideal.)55 b(In)34 b(particular,)g(if)g(dim)14 b FJ(S)38 b FQ(=)33 b(1,)i FJ(s)e FL(2)h FJ(S)k FQ(is)c(a)f(regular)456 624 y(p)r(oin)n(t,)d(and)g FJ(t)g FQ(is)g(a)g FO(lo)l(c)l(al)j(p)l(ar)l (ameter)g(at)e FJ(s)p FQ(,)g(i.e.,)g(an)f(analytic)f(function)i(in)f(a) g(neigh)n(b)r(orho)r(o)r(d)456 730 y(of)d FJ(s)h FQ(suc)n(h)f(that)h FJ(t)p FQ(\()p FJ(s)p FQ(\))23 b(=)g(0)p FJ(;)14 b FQ(\()p FJ(dt)p FQ(\))1444 742 y FI(s)1503 730 y FL(6)p FQ(=)22 b(0,)28 b(then)1892 709 y Fy(b)1872 730 y FL(O)1938 742 y FI(S;s)2056 730 y FL(')23 b FH(C)15 b FQ([[)p FJ(t)p FQ(]].)605 829 y(F)-7 b(or)34 b(an)h FL(O)950 841 y FI(S)998 829 y FQ(-mo)r(dule)g FL(F)43 b FQ(w)n(e)35 b(de\014ne)g(its)g(\014b)r (er)g(at)g(p)r(oin)n(t)g FJ(s)g FL(2)h FJ(S)k FQ(to)35 b(b)r(e)g FL(F)3006 841 y FI(s)3041 829 y FJ(=m)3156 841 y FI(s)3191 829 y FL(F)3251 841 y FI(s)3286 829 y FQ(.)59 b(In)456 929 y(particular,)40 b(if)f FL(F)46 b FQ(is)39 b(the)f(sheaf)h(of)f(sections)g(of)g(a)h(v)n(ector)e(bundle) i FJ(F)12 b FQ(,)41 b(then)e(in)g(this)g(w)n(a)n(y)456 1029 y(one)31 b(reco)n(v)n(ers)e(the)k(\014b)r(ers)f(of)g FJ(F)12 b FQ(.)50 b(W)-7 b(e)32 b(will)g(sa)n(y)f(that)h(an)g FL(O)r FQ(-mo)r(dule)h FL(F)40 b FQ(is)31 b FO(lisse)i FQ(if)g(it)f(is)g(the)456 1128 y(sheaf)d(of)h(sections)f(of)g(a)g (\014nite-dimensional)h(v)n(ector)e(bundle.)44 b(Note)29 b(that)h(ev)n(ery)f(lisse)g(sheaf)456 1228 y(is)e(coheren)n(t,)g(but)h (con)n(v)n(erse)d(is)j(not)g(true.)605 1328 y(In)g(general,)f(for)g(an) h(op)r(en)g(subset)g FJ(U)k FL(\032)24 b FJ(S)32 b FQ(and)c(a)g(sheaf)f FL(F)36 b FQ(on)28 b FJ(S)5 b FQ(,)28 b(w)n(e)g(denote)f(b)n(y)h FL(F)8 b FQ(\()p FJ(U)h FQ(\))456 1427 y(the)31 b(v)n(ector)f(space)g (of)h(sections)g(of)g FL(F)39 b FQ(o)n(v)n(er)29 b FJ(U)9 b FQ(.)47 b(Ho)n(w)n(ev)n(er,)30 b(in)h(the)h(case)e(when)h FJ(U)38 b FQ(=)28 b FJ(C)f FL(n)20 b FJ(D)r FQ(,)456 1527 y(where)38 b FJ(C)45 b FQ(is)38 b(compact)h(and)f FJ(D)j FQ(is)d(a)h(divisor,)h(and)f FL(F)8 b FQ(|an)38 b FL(O)r FQ(-mo)r(dule)h(o)n(v)n(er)e FJ(C)6 b FQ(,)42 b(w)n(e)c(will)456 1626 y(denote)e(b)n(y)g FL(F)8 b FQ(\()p FJ(C)31 b FL(\000)24 b FJ(D)r FQ(\))37 b(the)g(space)f(of)h FO(mer)l(omorphic)h FQ(sections)e(of)h FL(F)44 b FQ(o)n(v)n(er)35 b FJ(C)43 b FQ(whic)n(h)36 b(are)456 1726 y(regular)25 b(outside)j(of)f FJ(D)r FQ(.)37 b(W)-7 b(e)28 b(hop)r(e)g(it)g(w)n (on't)f(cause)g(confusion.)605 1826 y(W)-7 b(e)23 b(will)g(use)g(the)g (follo)n(wing)f(w)n(ell)h(kno)n(wn)f(facts)h(ab)r(out)g(complex)f(curv) n(es.)34 b(As)23 b(b)r(efore,)h(all)456 1925 y(the)d(curv)n(es)f(are)f (assumed)i(to)f(b)r(e)i(compact)e(and)h(non-singular)e(\(unless)i(sp)r (eci\014ed)g(otherwise\),)456 2025 y(but)28 b(not)f(necesarily)g (connected.)605 2176 y FP(Theorem)32 b FQ(7.2.1)e(\(Riemann-Ro)r(c)n (h\))p FP(.)41 b FO(L)l(et)f FJ(C)47 b FO(b)l(e)41 b(a)g(c)l(onne)l (cte)l(d)g(c)l(omplex)g(curve,)j(and)456 2276 y FJ(p)498 2288 y FM(1)535 2276 y FJ(;)14 b(:)g(:)g(:)f(;)h(p)761 2288 y FI(n)806 2276 y FJ(;)g(q)s FO(|distinct)28 b(p)l(oints)h(of)f FJ(C)35 b FQ(\()p FJ(n)23 b FL(\025)g FQ(0\))p FO(.)38 b(L)l(et)27 b(us)h(\014x)f(the)h(princip)l(al)i(p)l(arts)e(of)h(L)l (aur)l(ent)456 2376 y(exp)l(ansions)k FQ(\()p FJ(f)9 b FQ(\))989 2388 y FI(i)1045 2376 y FL(2)29 b FH(C)15 b FQ(\()q(\()p FJ(t)1278 2388 y FI(i)1312 2376 y FQ(\)\))p FJ(=)p FH(C)g FQ([[)p FJ(t)p FQ(]])39 b FO(ne)l(ar)32 b FJ(p)1865 2388 y FI(i)1893 2376 y FO(.)48 b(Then)33 b(ther)l(e)g(exists)g(a)g(function)g FJ(f)k FL(2)29 b(O)r FQ(\()p FJ(C)f FL(\000)456 2475 y(f)p FJ(p)540 2487 y FM(1)576 2475 y FJ(;)14 b(:)g(:)g(:)g(;)g(p)803 2487 y FI(n)848 2475 y FJ(;)g(q)s FL(g)p FQ(\))29 b FO(which)i(has)g(given)f (princip)l(al)i(p)l(arts)e(of)h(L)l(aur)l(ent)d(exp)l(ansion)j(at)e FJ(p)3107 2487 y FI(i)3164 2475 y FO(and)i(has)456 2575 y(a)j(p)l(ole)g(at)g FJ(q)s FO(.)50 b(Mor)l(e)l(over,)36 b(the)e(or)l(der)g(of)h(p)l(ole)f(at)g FJ(q)j FO(c)l(an)c(b)l(e)h(b)l (ounde)l(d)g(by)g(a)g(c)l(onstant)f(which)456 2675 y(only)d(dep)l(ends) h(on)f(the)f(or)l(der)i(of)g(p)l(oles)f(at)g FJ(p)1864 2687 y FI(i)1921 2675 y FO(and)g(the)g(genus)f(of)i(the)f(curve)g FJ(C)6 b FO(.)605 2826 y FQ(This)39 b(theorem)g(can)f(b)r(e)i (generalized)d(to)i(curv)n(e)f(whic)n(h)h(ma)n(y)f(ha)n(v)n(e)g (ordinary)g(double)456 2926 y(p)r(oin)n(t)31 b(singularities)f(and)h (ma)n(y)g(b)r(e)h(disconnected.)48 b(In)31 b(this)h(case,)f(w)n(e)g(ha) n(v)n(e)f(to)i(allo)n(w)e(p)r(oles)456 3025 y(at)f(a)h(collection)f(of) h(p)r(oin)n(ts)g FJ(q)1388 3037 y FM(1)1425 3025 y FJ(;)14 b(:)g(:)g(:)g(;)g(q)1647 3037 y FI(m)1740 3025 y FQ(suc)n(h)30 b(that)g(on)f(ev)n(ery)g(comp)r(onen)n(t)h(of)g FJ(C)36 b FQ(there)30 b(is)g(at)456 3125 y(least)d(one)g(of)h(the)g(p)r(oin)n (ts)f FJ(q)1326 3137 y FI(i)1354 3125 y FQ(.)605 3276 y FP(Theorem)32 b FQ(7.2.2)p FP(.)40 b FO(L)l(et)34 b FJ(C)41 b FO(b)l(e)35 b(a)g(c)l(omplex)h(curve)41 b FQ(\()p FO(p)l(ossibly)c(disc)l(onne)l(cte)l(d)e(and)h(singu-)456 3376 y(lar)9 b FQ(\))p FO(.)42 b(L)l(et)30 b FJ(q)e FL(2)d FJ(C)37 b FO(b)l(e)31 b(a)g(r)l(e)l(gular)g(p)l(oint,)g(and)h FJ(t)p FO(|a)f(lo)l(c)l(al)g(p)l(ar)l(ameter)h(at)e FJ(q)s FO(.)42 b(Then)31 b(the)g(ve)l(ctor)456 3476 y(sp)l(ac)l(e)1490 3614 y FH(C)15 b FQ(\(\()p FJ(t)p FQ(\)\))q FJ(=)p FH(C)20 b FQ([[)p FJ(t)p FQ(]])25 b(+)18 b FL(O)r FQ(\()p FJ(C)25 b FL(\000)18 b FJ(q)s FQ(\))p FJ(;)456 3752 y FO(is)35 b(\014nite)g(dimensional.)58 b(Mor)l(e)l(over,)39 b(ther)l(e)c(exists)g FJ(N)42 b FL(2)34 b FH(Z)2387 3764 y FM(+)2471 3752 y FO(which)j(only)f(dep)l(ends)h(on)e(the)456 3852 y(top)l(olo)l(gy)c(of) f FJ(C)36 b FO(such)30 b(that)1245 3990 y FL(O)r FQ(\()p FJ(C)25 b FL(\000)18 b FJ(q)s FQ(\))h(+)f FH(C)d FQ([[)p FJ(t)p FQ(]])29 b FL(\033)23 b FJ(t)2009 3955 y FE(\000)p FI(N)2124 3990 y FH(C)14 b FQ([)q FJ(t)2231 3955 y FE(\000)p FM(1)2326 3990 y FQ(])k(+)g FH(C)d FQ([[)p FJ(t)p FQ(]])p FJ(:)1256 4169 y FK(7.3.)47 b(Conformal)30 b(blo)s(c)m(ks:)42 b(de\014nition)605 4319 y FQ(In)34 b(this)g(section,)h(w)n(e)f(will)g (de\014ne)g(the)g(v)n(ector)e(spaces)h(of)h(coin)n(v)-5 b(arian)n(ts;)35 b(later)e(w)n(e)g(will)456 4419 y(sho)n(w)c(that)h (these)h(v)n(ector)e(spaces)g(satisfy)h(the)g(axioms)f(of)i(a)e(mo)r (dular)h(functor.)45 b(The)30 b(basic)456 4518 y(references)c(for)h (this)h(section)f(are)g([)p FK(TUY)q FQ(],)h([)p FK(Be)o FQ(])g(\(with)h(minor)e(c)n(hanges\).)605 4618 y(Fix)40 b(a)f(compact)g(nonsingular)f(complex)i(curv)n(e)e FJ(C)46 b FQ(\(not)40 b(necessarily)e(connected\),)43 b(a)456 4717 y(\014nite)28 b(dimensional)f(simple)h(Lie)f(algebra)f FA(g)p FQ(,)h(and)h(a)f(p)r(ositiv)n(e)g(in)n(teger)g FJ(k)s FQ(.)605 4817 y(Let)k FJ(p)799 4829 y FM(1)836 4817 y FJ(;)14 b(:)g(:)g(:)g(;)g(p)1063 4829 y FI(n)1138 4817 y FQ(b)r(e)31 b(distinct)h(p)r(oin)n(ts)e(on)h FJ(C)37 b FQ(with)31 b(lo)r(cal)f(co)r(ordinates)f FJ(t)2890 4829 y FM(1)2928 4817 y FJ(;)14 b(:)g(:)g(:)f(;)h(t)3142 4829 y FI(n)3218 4817 y FQ(\(recall)456 4917 y(that)28 b(a)g(lo)r(cal)f(co)r(ordinate)g(at)h(a)g(p)r(oin)n(t)g FJ(p)g FQ(is)g(a)g(holomorphic)f(function)h FJ(t)h FQ(in)f(a)g(neigh)n (b)r(orho)r(o)r(d)456 5016 y(of)e FJ(p)g FQ(suc)n(h)g(that)g FJ(t)p FQ(\()p FJ(p)p FQ(\))e(=)e(0)p FJ(;)14 b FQ(\()p FJ(dt)p FQ(\))1444 5028 y FI(p)1506 5016 y FL(6)p FQ(=)23 b(0\).)36 b(W)-7 b(e)27 b(will)f(alw)n(a)n(ys)f(assume)g(that)i(on)f (ev)n(ery)f(connected)456 5116 y(comp)r(onen)n(t)e(of)g FJ(C)30 b FQ(there)23 b(is)g(at)g(least)g(one)g(p)r(oin)n(t.)35 b(Let)24 b FJ(V)2217 5128 y FM(1)2255 5116 y FJ(;)14 b(:)g(:)g(:)f(;)h(V)2487 5128 y FI(n)2556 5116 y FL(2)23 b(O)2700 5128 y FI(k)2765 5116 y FQ(b)r(e)g(some)3075 5112 y Fy(b)3077 5116 y FA(g)p FQ(-mo)r(dules)456 5216 y(asso)r(ciated)j(to)h(these)h(p)r(oin)n(ts.)p eop %%Page: 171 5 171 174 bop 1298 226 a FM(7.3.)29 b(CONF)n(ORMAL)h(BLOCKS:)f (DEFINITION)720 b(171)605 425 y FQ(W)-7 b(e)28 b(will)g(use)f(the)h (notations)1611 560 y FJ(~)-43 b(p)23 b FQ(=)g(\()p FJ(p)1837 572 y FM(1)1874 560 y FJ(;)14 b(:)g(:)g(:)f(;)h(p)2100 572 y FI(n)2145 560 y FQ(\))p FJ(;)1610 685 y(V)42 b FQ(=)23 b FJ(V)1836 697 y FM(1)1892 685 y FL(\012)18 b FJ(:)c(:)g(:)k FL(\012)g FJ(V)2221 697 y FI(n)2267 685 y FJ(:)456 833 y FQ(In)27 b(particular,)g(if)h FJ(V)1090 845 y FI(i)1141 833 y FQ(=)23 b FJ(L)1286 803 y FI(k)1286 856 y(\025)1325 864 y FG(i)1383 833 y FQ(are)j(in)n(tegrable)h(mo)r (dules,)g(w)n(e)g(will)h(use)g(the)g(notation)1190 968 y FJ(~)1193 990 y(\025)23 b FQ(=)g(\()p FJ(\025)1432 1002 y FM(1)1470 990 y FJ(;)14 b(:)g(:)g(:)g(;)g(\025)1703 1002 y FI(n)1748 990 y FQ(\))p FJ(;)97 b(L)1957 955 y FI(k)1955 1010 y(~)1957 1026 y(\025)2023 990 y FQ(=)23 b FJ(L)2168 955 y FI(k)2168 1010 y(\025)2207 1018 y FF(1)2262 990 y FL(\012)18 b(\001)c(\001)g(\001)k(\012)g FJ(L)2600 955 y FI(k)2600 1010 y(\025)2639 1018 y FG(n)2684 990 y FJ(:)605 1127 y FQ(Consider)27 b(the)h(Lie)f(algebra)1472 1262 y FA(g)p FQ(\()p FJ(C)e FL(\000)18 b FJ(~)-42 b(p)o FQ(\))24 b(=)e FA(g)d FL(\012)2023 1274 y Fz(C)2087 1262 y FL(O)r FQ(\()p FJ(C)25 b FL(\000)19 b FJ(~)-43 b(p)p FQ(\))-1972 b(\(7.3.1\))456 1397 y(of)40 b FA(g)p FQ(-v)-5 b(alued)39 b(functions)h(on)g FJ(C)47 b FQ(whic)n(h)40 b(are)f(regular)f(outside)i(the)g(p)r(oin)n(ts)g FJ(p)2962 1409 y FM(1)2999 1397 y FJ(;)14 b(:)g(:)g(:)g(;)g(p)3226 1409 y FI(n)3311 1397 y FQ(and)456 1497 y(meromorphic)26 b(at)h(these)h(p)r(oin)n(ts.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(Lie)i (algebra)d(homomorphisms)1562 1632 y FJ(\015)1605 1644 y FI(i)1642 1632 y FQ(:)j FA(g)p FQ(\()p FJ(C)d FL(\000)18 b FJ(~)-42 b(p)o FQ(\))24 b FL(!)f FA(g)p FQ(\(\()p FJ(t)p FQ(\)\))456 1767 y(giv)n(en)33 b(b)n(y)h(Lauren)n(t)f(expansion)g (around)g(the)h(p)r(oin)n(t)g FJ(p)2214 1779 y FI(i)2276 1767 y FQ(in)g(the)g(lo)r(cal)g(co)r(ordinate)f FJ(t)3176 1779 y FI(i)3203 1767 y FQ(.)57 b(This)456 1867 y(do)r(es)33 b(not)g(giv)n(e)g(a)g(Lie)g(algebra)f(homomorphism)g FA(g)p FQ(\()p FJ(C)d FL(\000)23 b FJ(~)-43 b(p)p FQ(\))33 b FL(!)2558 1863 y Fy(b)2560 1867 y FA(g)g FQ(b)r(ecause)g(of)h(the)f (cen)n(tral)456 1966 y(term)27 b(in)h(de\014nition)g(of)1212 1962 y Fy(b)1215 1966 y FA(g)o FQ(.)37 b(Ho)n(w)n(ev)n(er,)26 b(b)n(y)h(the)h(Residue)g(Theorem,)1031 2101 y FJ(~)-39 b(\015)28 b FQ(=)22 b FJ(\015)1235 2113 y FM(1)1291 2101 y FL(\010)c(\001)c(\001)g(\001)k(\010)g FJ(\015)1615 2113 y FI(n)1670 2101 y FQ(:)27 b FA(g)p FQ(\()p FJ(C)e FL(\000)19 b FJ(~)-43 b(p)p FQ(\))23 b FL(!)g FA(g)p FQ(\(\()p FJ(t)p FQ(\)\))d FL(\010)e(\001)c(\001)g(\001)k(\010)g FA(g)p FQ(\(\()p FJ(t)p FQ(\)\))456 2237 y(can)27 b(b)r(e)h(lifted)g (to)g(a)f(homomorphism)602 2533 y FJ(~)-39 b(\015)14 b FQ(:)28 b FA(g)p FQ(\()p FJ(C)d FL(\000)18 b FJ(~)-42 b(p)o FQ(\))24 b FL(!)f FJ(U)9 b FQ(\()1253 2529 y Fy(b)1255 2533 y FA(g)p FQ(\))1329 2545 y FI(k)1388 2533 y FL(\012)18 b(\001)c(\001)g(\001)19 b(\012)f FJ(U)9 b FQ(\()1766 2529 y Fy(b)1768 2533 y FA(g)o FQ(\))1841 2545 y FI(k)1883 2533 y FJ(;)93 b(~)-39 b(\015)5 b FQ(\()p FJ(x)p FQ(\))24 b(=)2312 2429 y FI(n)2273 2454 y Fy(X)2279 2631 y FI(i)p FM(=1)2407 2533 y FQ(1)17 b FL(\012)h(\001)c(\001)g(\001)19 b(\012)f FJ(\015)2791 2545 y FI(i)2818 2533 y FQ(\()p FJ(x)p FQ(\))i FL(\012)e(\001)c(\001)g(\001)k(\012)g FQ(1)p FJ(:)456 2745 y FQ(In)27 b(particular,)g FA(g)p FQ(\()p FJ(C)e FL(\000)18 b FJ(~)-42 b(p)o FQ(\))28 b(acts)f(on)h FJ(V)19 b FQ(.)605 2895 y FP(Definition)32 b FQ(7.3.1)p FP(.)40 b FQ(The)32 b(space)g(of)g FO(c)l(onformal)j(blo)l(cks)40 b FQ(is)33 b(the)f(v)n(ector)f(space)h(of)g(coin-)456 2995 y(v)-5 b(arian)n(ts)1267 3130 y FJ(\034)9 b FQ(\()p FJ(C)q(;)17 b(~)-45 b(p)r(;)14 b(V)k FQ(\))24 b(:=)e FJ(V)1802 3145 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)q FM(\))2055 3130 y FQ(=)22 b FJ(V)9 b(=)p FA(g)p FQ(\()p FJ(C)25 b FL(\000)19 b FJ(~)-42 b(p)o FQ(\))p FJ(V)5 b(:)605 3297 y FQ(W)-7 b(e)26 b(will)g(write)f FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)r(;)1361 3281 y(~)1366 3297 y(t)o(;)14 b(V)19 b FQ(\))26 b(when)g(w)n(e)f(need)h (to)f(sho)n(w)g(the)h(dep)r(endence)g(on)f(the)h(c)n(hoice)456 3400 y(of)h(lo)r(cal)g(parameters)1171 3385 y FJ(~)1176 3400 y(t)c FQ(=)g(\()p FJ(t)1379 3412 y FM(1)1416 3400 y FJ(;)14 b(:)g(:)g(:)g(;)g(t)1631 3412 y FI(n)1676 3400 y FQ(\).)605 3500 y(It)23 b(is)f(easy)g(to)g(see)g(that)h(the)g (construction)e(ab)r(o)n(v)n(e)g(also)h(mak)n(es)f(p)r(erfect)i(sense)f (if)h(w)n(e)f(allo)n(w)456 3605 y FJ(t)486 3617 y FI(i)543 3605 y FQ(b)r(e)30 b FO(formal)h FQ(lo)r(cal)e(parameters)f(at)h FJ(p)1697 3617 y FI(i)1725 3605 y FQ(,)h(i.e.,)g FJ(t)1967 3617 y FI(i)2021 3605 y FL(2)2123 3584 y Fy(b)2103 3605 y FL(O)2169 3617 y FI(p)2203 3625 y FG(i)2234 3605 y FJ(;)14 b FQ(\()p FJ(dt)2376 3617 y FI(i)2404 3605 y FQ(\))2436 3617 y FI(p)2470 3625 y FG(i)2527 3605 y FL(6)p FQ(=)26 b(0.)43 b(Note)29 b(that)h(once)f FJ(t)3331 3617 y FI(i)3389 3605 y FQ(is)456 3718 y(c)n(hosen,)d(one)i(has)1065 3697 y Fy(b)1045 3718 y FL(O)1111 3730 y FI(p)1145 3738 y FG(i)1198 3718 y FQ(=)23 b FH(C)15 b FQ([[)p FJ(t)1416 3730 y FI(i)1450 3718 y FQ(]].)605 3871 y FP(Lemma)31 b FQ(7.3.2)f(\(Beauville)e([)p FK(Be)o FQ(]\))p FP(.)42 b FO(L)l(et)23 b FJ(~)-43 b(p)p FO(,)25 b FJ(V)42 b FO(b)l(e)23 b(as)g(ab)l(ove,)j(and)e(let)f FJ(q)j FL(2)d FJ(C)10 b FL(\000)5 b FJ(~)-43 b(p)p FO(,)25 b FJ(\025)e FL(2)h FJ(P)3376 3841 y FI(k)3364 3891 y FM(+)3419 3871 y FO(.)456 3977 y(As)33 b(b)l(efor)l(e,)k(let)d FJ(V)1024 3989 y FI(\025)1102 3977 y FO(b)l(e)g(the)g(c)l(orr)l(esp)l(onding)h (\014nite-dimensional)g FA(g)p FO(-mo)l(dule,)h(and)f(let)f FJ(V)3297 3947 y FI(k)3279 4001 y(\025)3372 3977 y FO(b)l(e)456 4079 y(the)c(Weyl)g(mo)l(dule)g(over)1269 4075 y Fy(b)1272 4079 y FA(g)o FO(.)39 b(Then)30 b(the)g(inclusion)h FJ(V)2134 4091 y FI(\025)2201 4079 y FJ(,)-14 b FL(!)23 b FJ(V)2383 4049 y FI(k)2364 4103 y(\025)2453 4079 y FO(induc)l(es)30 b(an)g(isomorphism)965 4224 y FQ(\()p FJ(V)38 b FL(\012)18 b FJ(V)1214 4236 y FI(\025)1258 4224 y FQ(\))1290 4239 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FM(\))1570 4177 y FE(\030)1542 4224 y FL(\000)c(!)23 b FQ(\()p FJ(V)37 b FL(\012)18 b FJ(V)1941 4190 y FI(k)1922 4245 y(\025)1982 4224 y FQ(\))2014 4239 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FE(\000)p FI(q)r FM(\))2351 4224 y FQ(=)22 b FJ(\034)9 b FQ(\()p FJ(C)q(;)17 b(~)-45 b(p)20 b FL([)f FJ(q)s(;)14 b(V)37 b FL(\012)18 b FJ(V)3060 4190 y FI(k)3041 4245 y(\025)3101 4224 y FQ(\))p FJ(;)-2700 b FQ(\(7.3.2\))456 4364 y FO(wher)l(e)22 b FA(g)p FQ(\()p FJ(C)6 b FL(\000)q FJ(~)-43 b(p)p FQ(\))22 b FO(acts)g(on)f FJ(V)1304 4376 y FI(\025)1370 4364 y FO(via)h(the)g(evaluation)h(map)f FJ(a)p FL(\012)p FJ(f)32 b FL(7!)23 b FJ(f)9 b FQ(\()p FJ(q)s FQ(\))p FJ(a)p FO(,)24 b FJ(a)f FL(2)g FA(g)p FO(,)g FJ(f)32 b FL(2)23 b(O)r FQ(\()p FJ(C)6 b FL(\000)q FJ(~)-43 b(p)r FQ(\))p FO(.)605 4514 y FP(Pr)n(oof.)41 b FQ(Since)33 b(the)g(natural)f(em)n(b)r(edding)h FJ(V)41 b FL(\012)21 b FJ(V)2247 4526 y FI(\025)2323 4514 y FJ(,)-14 b FL(!)32 b FJ(V)41 b FL(\012)21 b FJ(V)2689 4484 y FI(k)2670 4537 y(\025)2763 4514 y FQ(is)33 b(clearly)f FA(g)p FQ(\()p FJ(C)c FL(\000)22 b FJ(~)-43 b(p)p FQ(\))456 4614 y(equiv)-5 b(arian)n(t,)31 b(it)g(induces)g(a)g(map)g(from)g(the)h(left)f(hand)g (side)h(of)37 b(\(7.3.2\))31 b(to)g(the)g(righ)n(t)g(hand)456 4713 y(side.)605 4813 y(By)d(the)g(Riemann{Ro)r(c)n(h)f(form)n(ula,)h (there)f(exists)h(a)g(function)g FJ(z)j FQ(on)d FJ(C)34 b FQ(regular)27 b(outside)456 4912 y FJ(~)-42 b(p)18 b FL([)h FJ(q)30 b FQ(and)e(ha)n(ving)e(a)h(simple)h(p)r(ole)g(at)f (the)h(p)r(oin)n(t)g FJ(q)s FQ(.)37 b(Then)1253 5109 y FL(O)r FQ(\()p FJ(C)25 b FL(\000)19 b FJ(~)-43 b(p)18 b FL(\000)g FJ(q)s FQ(\))24 b(=)e FL(O)r FQ(\()p FJ(C)k FL(\000)18 b FJ(~)-42 b(p)p FQ(\))18 b FL(\010)2319 5006 y FE(1)2289 5030 y Fy(M)2298 5207 y FI(i)p FM(=1)2429 5109 y FH(C)d FJ(z)2526 5075 y FE(\000)p FI(i)2610 5109 y FJ(;)p eop %%Page: 172 6 172 175 bop 456 226 a FM(172)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(therefore)e FA(g)p FQ(\()p FJ(C)f FL(\000)19 b FJ(~)-43 b(p)18 b FL(\000)g FJ(q)s FQ(\))24 b FL(')e FA(g)p FQ(\()p FJ(C)j FL(\000)19 b FJ(~)-43 b(p)p FQ(\))19 b FL(\010)1786 421 y Fy(b)1788 425 y FA(g)1830 395 y FE(\000)1886 425 y FQ(.)605 525 y(By)27 b(de\014nition,)g FJ(V)1193 495 y FI(k)1174 548 y(\025)1261 525 y FQ(is)g(a)g(free)g FJ(U)9 b FQ(\()1668 521 y Fy(b)1670 525 y FA(g)1712 495 y FE(\000)1768 525 y FQ(\)-mo)r(dule)27 b(isomorphic)f(to)h FJ(U)9 b FQ(\()2737 521 y Fy(b)2739 525 y FA(g)2781 495 y FE(\000)2837 525 y FQ(\))p FJ(V)2917 537 y FI(\025)2961 525 y FQ(;)28 b(hence,)f FJ(V)3313 537 y FI(\025)3380 525 y FL(')456 627 y FQ(\()p FJ(V)555 597 y FI(k)536 651 y(\025)596 627 y FQ(\))627 640 y Fu(b)628 643 y Fk(g)663 627 y Fx(\000)8 b FQ(.)64 b(Then)37 b(\(7.3.2\))f(follo)n(ws)g(b)n(y)g (tensoring)g(with)h FJ(V)56 b FQ(and)36 b(taking)g(coin)n(v)-5 b(arian)n(ts)35 b(with)456 727 y(resp)r(ect)27 b(to)g FA(g)p FQ(\()p FJ(C)e FL(\000)19 b FJ(~)-43 b(p)p FQ(\).)p 3384 727 4 57 v 3388 674 50 4 v 3388 727 V 3437 727 4 57 v 605 896 a FP(Lemma)31 b FQ(7.3.3)p FP(.)40 b FO(L)l(et)d FJ(C)44 b FO(b)l(e)38 b(c)l(onne)l(cte)l(d,)j(and)d(let)g FJ(V)2271 908 y FI(i)2336 896 y FO(b)l(e)g(quotients)g(of)g(Weyl)h(mo)l (dules)p FQ(:)456 996 y FJ(V)504 1008 y FI(i)571 996 y FQ(=)g FJ(V)742 965 y FI(k)723 1019 y(\025)762 1027 y FG(i)793 996 y FJ(=I)871 1008 y FI(i)937 996 y FQ(\()p FO(the)g(ide)l(als)h FJ(I)1390 1008 y FI(i)1457 996 y FO(may)f(b)l(e)g(zer)l(o,)j(maximal,)g(or)d(anything)g(in)g(b)l(etwe)l (en)6 b FQ(\))p FO(.)66 b(As-)456 1106 y(sume)30 b(also)i(that)f(at)f (le)l(ast)h(one)g(of)h FJ(V)1608 1118 y FI(i)1667 1106 y FO(is)f(inte)l(gr)l(able,)h(i.e.,)h(e)l(qual)e(to)g FJ(L)2696 1076 y FI(k)2696 1130 y(\025)2735 1138 y FG(i)2765 1106 y FO(.)42 b(Then)32 b(the)f(natur)l(al)456 1217 y(surje)l(ction)d FJ(V)42 b FQ(=)23 b FJ(V)1063 1229 y FM(1)1115 1217 y FL(\012)15 b(\001)f(\001)g(\001)h(\012)f FJ(V)1434 1229 y FI(n)1503 1217 y Fw(\020)23 b FJ(L)1666 1187 y FI(k)1666 1240 y(\025)1705 1248 y FF(1)1756 1217 y FL(\012)15 b(\001)f(\001)g(\001)h(\012)f FJ(L)2084 1187 y FI(k)2084 1240 y(\025)2123 1248 y FG(n)2191 1217 y FQ(=)23 b FJ(L)2336 1187 y FI(k)2334 1242 y(~)2336 1257 y(\025)2407 1217 y FO(gives)29 b(rise)g(to)f(an)g(isomorphism)1491 1392 y FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)14 b(V)19 b FQ(\))1896 1345 y FE(\030)1868 1392 y FL(\000)-40 b(!)23 b FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)r(;)14 b(L)2311 1358 y FI(k)2309 1413 y(~)2311 1428 y(\025)2353 1392 y FQ(\))p FJ(:)-1952 b FQ(\(7.3.3\))605 1552 y FP(Pr)n(oof.)41 b FQ(It)28 b(su\016ces)f(to)g(pro)n(v)n(e)f(that)553 1799 y(\()p FJ(L)642 1765 y FI(k)642 1820 y(\025)681 1828 y FF(1)737 1799 y FL(\012)18 b FJ(V)868 1811 y FM(2)924 1799 y FL(\012)g(\001)c(\001)g(\001)k(\012)g FJ(V)1253 1811 y FI(n)p FE(\000)p FM(1)1402 1799 y FL(\012)g FJ(V)1552 1765 y FI(k)1533 1820 y(\025)1572 1828 y FG(n)1618 1799 y FQ(\))1650 1814 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FM(\))1902 1799 y FQ(=)23 b(\()p FJ(L)2079 1765 y FI(k)2079 1820 y(\025)2118 1828 y FF(1)2173 1799 y FL(\012)18 b FJ(V)2304 1811 y FM(2)2360 1799 y FL(\012)g(\001)c(\001)g (\001)k(\012)g FJ(V)2689 1811 y FI(n)p FE(\000)p FM(1)2838 1799 y FL(\012)h FJ(L)2979 1765 y FI(k)2979 1820 y(\025)3018 1828 y FG(n)3062 1799 y FQ(\))3094 1814 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FM(\))3323 1799 y FJ(:)456 1961 y FQ(Let)35 b FJ(Z)41 b FQ(=)34 b FL(f)p FJ(v)k FL(2)e FJ(V)1087 1930 y FI(k)1068 1984 y(\025)1107 1992 y FG(n)1188 1961 y FL(j)f FJ(L)1303 1973 y FI(\025)1342 1981 y FF(1)1402 1961 y FL(\012)23 b(\001)14 b(\001)g(\001)23 b(\012)g FJ(V)1746 1973 y FI(n)p FE(\000)p FM(1)1900 1961 y FL(\012)g FJ(v)38 b FL(\032)d FQ(Im)14 b FA(g)p FQ(\()p FJ(C)30 b FL(\000)24 b FJ(~)-43 b(p)p FQ(\))p FL(g)p FQ(.)58 b(Ob)n(viously)-7 b(,)36 b(this)f(is)g(a)456 2071 y(submo)r(dule)26 b(in)g FJ(V)1034 2041 y FI(k)1015 2095 y(\025)1054 2103 y FG(n)1100 2071 y FQ(;)g(our)f(goal)g(is)h(to)g (pro)n(v)n(e)e(that)i FJ(V)2118 2041 y FI(k)2099 2095 y(\025)2138 2103 y FG(n)2184 2071 y FJ(=)-5 b(Z)31 b FQ(is)26 b(in)n(tegrable.)35 b(This)26 b(is)g(equiv)-5 b(alen)n(t)456 2182 y(to)27 b(the)i(follo)n(wing)e(statemen)n(t:)37 b(for)28 b(ev)n(ery)f(ro)r(ot)g FJ(\013)h FQ(and)g FJ(v)f FL(2)d FJ(V)2452 2152 y FI(k)2433 2205 y(\025)2472 2213 y FG(n)2518 2182 y FQ(,)k(one)g(has)f(\()p FJ(e)2941 2194 y FI(\013)2988 2182 y FQ([)p FL(\000)p FQ(1]\))3173 2152 y FI(N)3236 2182 y FJ(v)g FL(2)d FJ(Z)456 2284 y FQ(for)30 b FJ(N)38 b FL(\035)30 b FQ(0)h(\(in)g(fact,)i(it)f (su\016ces)f(to)g(c)n(hec)n(k)f(this)i(for)f FJ(\013)e FQ(=)g FJ(\022)r FQ(\).)49 b(W)-7 b(e)31 b(lea)n(v)n(e)f(it)i(to)f(the) h(reader)456 2384 y(to)h(c)n(hec)n(k)f(that)i(if)f(w)n(e)g(c)n(ho)r (ose)f FJ(f)41 b FL(2)33 b FH(C)15 b FQ(\(\()q FJ(t)p FQ(\)\))39 b(suc)n(h)33 b(that)h FJ(f)42 b FQ(has)32 b(\014rst)h(order)f(p)r(ole)i(at)f(0,)h(then)456 2483 y(the)c(ab)r(o)n(v)n(e)f(condition)g(is)h(equiv)-5 b(alen)n(t)30 b(to)g(\()p FJ(e)1863 2495 y FI(\013)1910 2483 y FJ(f)9 b FQ(\))1992 2453 y FI(N)2055 2483 y FJ(v)30 b FL(2)d FJ(Z)36 b FQ(for)30 b FJ(N)35 b FL(\035)27 b FQ(0)j(\(in)g(other)g(w)n (ords,)f(the)456 2583 y(notion)e(of)g(an)g(in)n(tegrable)g(mo)r(dule)h (do)r(es)f(not)g(dep)r(end)h(on)g(the)g(c)n(hoice)e(of)i(lo)r(cal)f (parameter\).)605 2683 y(No)n(w)c(let)g FJ(f)32 b FL(2)23 b(O)r FQ(\()p FJ(C)17 b FL(\000)10 b FJ(p)1349 2695 y FM(1)1394 2683 y FL(\000)g FJ(p)1511 2695 y FI(n)1556 2683 y FQ(\))23 b(b)r(e)h(a)e(function)i(whic)n(h)f(has)g(a)f(\014rst)h (order)f(p)r(ole)h(at)g FJ(p)3238 2695 y FI(n)3283 2683 y FQ(.)36 b(By)456 2782 y(the)28 b(Riemann{Ro)r(c)n(h)f(theorem,)h(suc) n(h)g(a)g(function)g(exists)g(if)h(w)n(e)e(allo)n(w)g(it)i(to)f(ha)n(v) n(e)f(a)g(p)r(ole)i(of)456 2882 y(su\016cien)n(tly)19 b(high)h(order)f(at)g FJ(p)1385 2894 y FM(1)1422 2882 y FQ(.)35 b(Since)20 b FJ(L)1746 2852 y FI(k)1746 2905 y(\025)1785 2913 y FF(1)1841 2882 y FQ(is)g(in)n(tegrable,)g(and)g FJ(f)28 b FQ(is)20 b(regular)e(at)i FJ(p)3028 2894 y FM(2)3065 2882 y FJ(;)14 b(:)g(:)g(:)f(;)h(p)3291 2894 y FI(n)p FE(\000)p FM(1)3421 2882 y FQ(,)456 2993 y(w)n(e)25 b(easily)f(see)h(that)h(action)f(of)g FJ(e)1490 3005 y FI(\013)1537 2993 y FJ(f)34 b FQ(on)25 b FJ(L)1782 2962 y FI(k)1782 3016 y(\025)1821 3024 y FF(1)1872 2993 y FL(\012)13 b(\001)h(\001)g(\001)g(\012)g FJ(V)2188 3005 y FI(n)p FE(\000)p FM(1)2344 2993 y FQ(is)25 b(lo)r(cally)g(nilp)r (oten)n(t.)37 b(Therefore,)456 3103 y(for)26 b(an)n(y)g FJ(v)778 3115 y FM(1)839 3103 y FL(2)d FJ(L)974 3073 y FI(k)974 3127 y(\025)1013 3135 y FF(1)1050 3103 y FJ(;)14 b(:)g(:)g(:)f(;)h(v)1274 3115 y FI(n)1343 3103 y FL(2)23 b FJ(V)1488 3073 y FI(k)1469 3127 y(\025)1508 3135 y FG(n)1554 3103 y FQ(,)k(one)g(has)f FJ(v)1943 3115 y FM(1)1998 3103 y FL(\012)17 b(\001)d(\001)g(\001)j(\012)g FJ(v)2316 3115 y FI(n)p FE(\000)p FM(1)2464 3103 y FL(\012)g FQ(\()p FJ(e)2617 3115 y FI(\013)2664 3103 y FJ(f)9 b FQ(\))2746 3073 y FI(N)2809 3103 y FJ(v)2849 3115 y FI(n)2917 3103 y FL(2)24 b FQ(Im)14 b FA(g)p FQ(\()p FJ(C)23 b FL(\000)18 b FJ(~)-43 b(p)p FQ(\).)456 3213 y(But)27 b(this)h(exactly)f(means)g(that)h(\()p FJ(e)1573 3225 y FI(\013)1621 3213 y FJ(f)9 b FQ(\))1703 3183 y FI(N)1765 3213 y FJ(v)1805 3225 y FI(n)1874 3213 y FL(2)23 b FJ(Z)34 b FQ(for)27 b FJ(N)32 b FL(\035)23 b FQ(0.)p 3384 3213 V 3388 3160 50 4 v 3388 3213 V 3437 3213 4 57 v 605 3382 a(This)i(theorem)g(can)g(b)r(e)h(rewritten)f(in)h(more)e(in)n(v)-5 b(arian)n(t)25 b(terms.)36 b(F)-7 b(or)24 b(a)h(mo)r(dule)h FJ(V)42 b FL(2)23 b(O)3380 3394 y FI(k)3421 3382 y FQ(,)456 3482 y(denote)36 b(b)n(y)g FJ(V)922 3452 y FM(in)n(t)1042 3482 y FQ(its)h(maximal)e(in)n(tegrable)g(quotien)n(t)h(\(it)h(is)f (easy)f(to)h(see)g(that)g(it)h(is)f(w)n(ell-)456 3582 y(de\014ned\).)h(Then)28 b(the)g(previous)e(lemma)i(immediately)g (implies)f(the)h(follo)n(wing)f(corollary)-7 b(.)605 3739 y FP(Cor)n(ollar)h(y)33 b FQ(7.3.4)p FP(.)39 b FO(L)l(et)30 b FJ(V)1519 3751 y FI(i)1570 3739 y FL(2)23 b(O)1716 3709 y FI(I)1714 3763 y(k)1755 3739 y FJ(N)9 b(T)j FO(,)29 b(and)h(at)g(le)l(ast)g(one)g(of)g FJ(V)2698 3751 y FI(i)2756 3739 y FO(is)g(inte)l(gr)l(able.)40 b(Then)1057 3893 y FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)14 b(V)1359 3905 y FM(1)1415 3893 y FL(\012)k(\001)c(\001)g(\001)k(\012)g FJ(V)1744 3905 y FI(n)1790 3893 y FQ(\))23 b(=)g FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)14 b(V)2254 3859 y FM(in)n(t)2235 3913 y(1)2356 3893 y FL(\012)k(\001)c(\001)g(\001)k (\012)g FJ(V)2704 3859 y FM(in)n(t)2685 3913 y FI(n)2788 3893 y FQ(\))p FJ(:)605 4062 y FP(Cor)n(ollar)-6 b(y)33 b FQ(7.3.5)p FP(.)39 b FO(L)l(et)d FJ(V)53 b FQ(=)34 b FJ(V)1725 4074 y FM(1)1777 4062 y FL(\001)14 b(\001)g(\001)23 b(\012)f FN(V)2028 4074 y Fj(n)2063 4062 y FJ(;)14 b FN(V)2144 4074 y Fj(i)2213 4062 y FL(2)35 b FN(O)2347 4032 y FM(in)n(t)2347 4083 y Fj(k)2430 4062 y FO(.)58 b(Then)37 b(the)f(emb)l(e)l(dding)h FH(C)56 b FQ(=)456 4162 y FJ(V)504 4174 y FM(0)564 4162 y FJ(,)-14 b FL(!)24 b FJ(L)737 4132 y FI(k)737 4182 y FM(0)807 4162 y FO(induc)l(es)30 b(an)f(isomorphism)1364 4314 y FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)r(;)14 b(V)k FQ(\))24 b FL(')e FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)20 b FL([)f FJ(q)s(;)14 b(V)37 b FL(\012)18 b FJ(L)2440 4280 y FI(k)2440 4335 y FM(0)2480 4314 y FQ(\))p FJ(:)-2079 b FQ(\(7.3.4\))605 4472 y FP(Pr)n(oof.)41 b FQ(This)27 b(follo)n(ws)g(from)g(Lemmas)g (7.3.2)g(and)g(7.3.3:)1036 4620 y(\()p FJ(V)37 b FL(\012)18 b FJ(L)1293 4586 y FI(k)1293 4641 y FM(0)1334 4620 y FQ(\))1366 4635 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FE(\000)p FI(q)r FM(\))1702 4620 y FL(')23 b FQ(\()p FJ(V)38 b FL(\012)18 b FJ(V)2058 4586 y FI(k)2039 4641 y FM(0)2098 4620 y FQ(\))2130 4635 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FE(\000)p FI(q)r FM(\))2467 4620 y FL(')23 b FQ(\()p FJ(V)37 b FL(\012)18 b FH(C)d FQ(\))2842 4635 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)5 b FM(\))3076 4620 y FJ(:)p 3384 4769 V 3388 4716 50 4 v 3388 4769 V 3437 4769 4 57 v 456 4938 a FQ(Ha)n(ving)26 b(pro)n(v)n(ed)g(these)i(results,)f(w)n(e)g(can)g(pro)n(v)n(e)f(no)n(w) h(the)h(follo)n(wing)f(prop)r(osition.)605 5095 y FP(Pr)n(oposition)k FQ(7.3.6)p FP(.)40 b FO(If)21 b FJ(V)42 b FQ(=)23 b FJ(V)1682 5107 y FM(1)1734 5095 y FL(\001)14 b(\001)g(\001\012)p FN(V)1940 5107 y Fj(n)1974 5095 y FJ(;)g FN(V)2055 5107 y Fj(i)2113 5095 y FL(2)23 b FN(O)2235 5065 y FM(in)n(t)2235 5116 y Fj(k)2319 5095 y FO(,)g(then)e(the)h(sp)l(ac)l(es)g(of)g(c)l (oinvariants)456 5195 y FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)14 b(V)19 b FQ(\))30 b FO(ar)l(e)g(\014nite)f (dimensional.)p eop %%Page: 173 7 173 176 bop 1298 226 a FM(7.3.)29 b(CONF)n(ORMAL)h(BLOCKS:)f (DEFINITION)720 b(173)605 425 y FP(Pr)n(oof.)41 b FQ(W)-7 b(e)34 b(ma)n(y)f(assume)g(that)i FJ(C)40 b FQ(is)34 b(connected.)55 b(Com)n(bining)33 b(Lemma)h(7.3.2)f(and)456 525 y(7.3.3,)c(w)n(e)h(see)g(that)g(it)g(su\016ces)g(to)g(pro)n(v)n(e)f (the)h(statemen)n(t)g(for)g FJ(n)d FQ(=)g(1)p FJ(;)14 b(V)2784 537 y FM(1)2848 525 y FQ(=)27 b FJ(L)2997 495 y FI(k)2997 548 y(\025)3040 525 y FQ(.)44 b(It)30 b(follo)n(ws)456 626 y(from)h(Theorem)g(7.2.2)f(that)1394 622 y Fy(b)1397 626 y FA(g)1439 596 y FM(+)1515 626 y FQ(+)20 b FA(g)p FQ(\()p FJ(C)28 b FL(\000)20 b FJ(p)p FQ(\))30 b FL(\033)2042 622 y Fy(b)2044 626 y FA(g)2086 596 y FM(+)2162 626 y FQ(+)21 b FJ(t)2278 596 y FE(\000)p FI(N)2391 622 y Fy(b)2393 626 y FA(g)2435 596 y FE(\000)2522 626 y FQ(for)31 b FJ(N)39 b FL(\035)29 b FQ(0.)49 b(Therefore,)31 b(it)456 726 y(su\016ces)c(to)g(pro)n(v)n(e)f(that)i(the)g(v)n(ector)e(space) 1586 881 y FJ(W)1664 893 y FI(N)1750 881 y FQ(=)d FJ(L)1895 847 y FI(k)1895 902 y(\025)1938 881 y FJ(=t)2010 847 y FE(\000)p FI(N)2122 877 y Fy(b)2124 881 y FA(g)2166 847 y FE(\000)2222 881 y FJ(V)2270 893 y FI(\025)456 1037 y FQ(is)k(\014nite-dimensional.)605 1136 y(T)-7 b(o)28 b(pro)n(v)n(e)g(this,)h(note)g(that)g(one)f(has)h(a)f(w)n (ell-de\014ned)h(action)f(of)2680 1132 y Fy(b)2682 1136 y FA(g)2724 1106 y FE(\024)p FM(0)2838 1136 y FQ(=)c FA(g)p FQ([)p FJ(t)3022 1106 y FE(\000)p FM(1)3111 1136 y FQ(])29 b(on)g FJ(W)3358 1148 y FI(N)3421 1136 y FQ(,)456 1236 y(whic)n(h)c(factors)g(through)g(the)h(\014nite-dimensional)f (quotien)n(t)h FA(a)d FQ(=)2556 1232 y Fy(b)2558 1236 y FA(g)2600 1206 y FE(\024)p FM(0)2689 1236 y FJ(=t)2761 1206 y FE(\000)p FI(N)2873 1232 y Fy(b)2875 1236 y FA(g)2917 1206 y FE(\024)p FM(0)3006 1236 y FQ(.)37 b(Ob)n(viously)-7 b(,)456 1336 y FJ(W)534 1348 y FI(N)631 1336 y FQ(=)34 b(\()p FJ(U)9 b FA(a)p FQ(\))p FJ(v)942 1348 y FI(\025;k)1042 1336 y FQ(.)57 b(On)34 b(the)h(other)f(hand,)i FA(a)e FQ(is)g(generated)f(b)n(y)h FJ(e)2592 1348 y FI(\013)2639 1336 y FJ(;)14 b(f)2717 1348 y FI(\013)2764 1336 y FQ(,)36 b FJ(e)2862 1348 y FI(\013)2909 1336 y FJ(t)2939 1305 y FE(\000)p FM(1)3028 1336 y FQ(,)g(and)e(all)h(of)456 1435 y(these)24 b(generators)e(act)i(nilp)r(oten)n(tly)h(on)f FJ(W)1808 1447 y FI(N)1871 1435 y FQ(.)36 b(Th)n(us,)25 b(all)f(w)n(e)g(need)h(is)f(to)g(pro)n(v)n(e)f(the)i(follo)n(wing)456 1535 y(lemma.)605 1696 y FP(Lemma)31 b FQ(7.3.7)p FP(.)40 b FO(If)f FA(a)21 b FO(is)h(a)f(\014nite-dimensional)i(Lie)f(algebr)l (a)g(with)g(gener)l(ators)g FJ(x)3148 1708 y FM(1)3185 1696 y FJ(;)14 b(:)g(:)g(:)g(;)g(x)3417 1708 y FI(n)3463 1696 y FO(,)456 1796 y(and)25 b FJ(W)37 b FO(is)26 b(a)g(cyclic)g FA(a)p FO(-mo)l(dule)f(such)g(that)h(the)f(action)h(of)g FJ(x)2326 1808 y FI(i)2379 1796 y FO(in)f FJ(W)38 b FO(is)25 b(lo)l(c)l(al)t(ly)i(nilp)l(otent,)g(then)456 1895 y FJ(W)41 b FO(is)30 b(\014nite-dimensional.)605 2056 y FQ(T)-7 b(o)43 b(pro)n(v)n(e)e(this)j(lemma,)j(w)n(e)c(pass)f(from)h (the)h(mo)r(dule)f FJ(W)56 b FQ(o)n(v)n(er)41 b FJ(U)9 b FA(a)43 b FQ(to)g(the)g(corre-)456 2156 y(sp)r(onding)36 b(graded)f(mo)r(dule)h FJ(Gr)17 b(W)48 b FQ(o)n(v)n(er)35 b FJ(Gr)r FQ(\()p FJ(U)9 b FA(a)p FQ(\))38 b(=)f FJ(S)5 b FQ(\()p FA(a)p FQ(\).)63 b(Consider)36 b(the)g(v)-5 b(ariet)n(y)36 b FJ(S)42 b FQ(=)456 2256 y FJ(S)5 b(upp)p FQ(\()p FJ(Gr)16 b(W)c FQ(\))38 b FL(\032)f FA(a)1098 2226 y FE(\003)1136 2256 y FQ(.)63 b(Then)37 b(it)g(follo)n(ws)e(from)i (the)f(nilp)r(otency)h(condition)f(that)h FJ(x)3202 2268 y FI(i)3230 2256 y FQ(,)i(con-)456 2355 y(sidered)f(as)g(a)g(function)i (on)e FA(a)1447 2325 y FE(\003)1485 2355 y FQ(,)k(v)-5 b(anishes)38 b(on)g FJ(S)5 b FQ(.)71 b(By)38 b(Gabb)r(er's)g(in)n (tegrabilit)n(y)g(theorem)456 2455 y([)p FK(Gab)p FQ(],)27 b(if)g FJ(x;)14 b(y)29 b FQ(v)-5 b(anish)26 b(on)g FJ(S)5 b FQ(,)26 b(then)h([)p FJ(x;)14 b(y)s FQ(])26 b(also)g(v)-5 b(anishes.)35 b(Therefore,)26 b FJ(S)i FQ(=)22 b FL(f)p FQ(0)p FL(g)p FQ(.)35 b(But)27 b(ev)n(ery)456 2555 y(\014nitely)k (generated)f(mo)r(dule)h(o)n(v)n(er)e(the)i(p)r(olynomial)g(ring,)g (whic)n(h)g(has)f(a)g(\014nite)i(supp)r(ort,)g(is)456 2654 y(\014nite-dimensional.)k(This)28 b(pro)n(v)n(es)d(the)j(lemma,)g (and)f(th)n(us,)h(the)g(prop)r(osition.)p 3384 2654 4 57 v 3388 2601 50 4 v 3388 2654 V 3437 2654 4 57 v 605 2833 a(As)g(an)f(illustration,)g(consider)g(the)h(simplest)f(case)g FJ(C)j FQ(=)22 b FH(P)2485 2803 y FM(1)2521 2833 y FQ(.)605 2994 y FP(Pr)n(oposition)31 b FQ(7.3.8)p FP(.)40 b FO(L)l(et)29 b FJ(C)g FQ(=)23 b FH(P)1749 2964 y FM(1)1785 2994 y FO(,)30 b FJ(p)1882 3006 y FM(1)1919 2994 y FJ(;)14 b(:)g(:)g(:)g(;)g (p)2146 3006 y FI(n)2190 2994 y FO(|distinct)30 b(p)l(oints)g(on)g FJ(C)6 b FO(.)605 3094 y FQ(\(i\))24 b FO(L)l(et)g FJ(V)920 3064 y FI(k)900 3119 y(~)902 3134 y(\025)984 3094 y FQ(=)f FJ(V)1139 3064 y FI(k)1120 3117 y(\025)1159 3125 y FF(1)1202 3094 y FL(\012)5 b FJ(:)14 b(:)g(:)t FL(\012)5 b FJ(V)1510 3064 y FI(k)1491 3117 y(\025)1530 3125 y FG(n)1576 3094 y FO(,)25 b(and)f FJ(V)1827 3107 y FI(~)1829 3122 y(\025)1896 3094 y FQ(=)f FJ(V)2032 3106 y FI(\025)2071 3114 y FF(1)2114 3094 y FL(\012)5 b FJ(:)14 b(:)g(:)t FL(\012)5 b FJ(V)2403 3106 y FI(\025)2442 3114 y FG(n)2488 3094 y FO(.)37 b(Then)24 b(the)g(homomorphism)1590 3272 y FQ(\()p FJ(V)1668 3285 y FI(~)1670 3301 y(\025)1715 3272 y FQ(\))1747 3284 y Fk(g)1809 3272 y FL(!)f FJ(\034)9 b FQ(\()p FJ(C)q(;)17 b(~)-45 b(p)r(;)14 b(V)2236 3238 y FI(k)2216 3293 y(~)2218 3309 y(\025)2277 3272 y FQ(\))456 3443 y FO(obtaine)l(d)30 b(by)g(r)l(estricting)f(the)g(natur)l(al)g(map)h FJ(V)1959 3413 y FI(k)1938 3468 y(~)1940 3483 y(\025)2023 3443 y FL(!)23 b FJ(V)2196 3413 y FI(k)2175 3468 y(~)2177 3483 y(\025)2236 3443 y FJ(=)p FA(g)p FQ(\()p FJ(C)g FL(\000)18 b FJ(~)-43 b(p)p FQ(\))p FJ(V)2657 3413 y FI(k)2636 3468 y(~)2638 3483 y(\025)2698 3443 y FO(,)30 b(is)f(an)h(isomorphism.)605 3559 y FQ(\(ii\))j FO(L)l(et)f FJ(z)k FO(b)l(e)c(a)h(glob)l(al)h(c)l(o)l(or)l(dinate)f(on)g FH(P)1954 3529 y FM(1)1990 3559 y FO(;)h(assume)e(that)h FJ(z)t FQ(\()p FJ(p)2632 3571 y FI(i)2659 3559 y FQ(\))f FO(is)h(\014nite.)46 b(De\014ne)32 b(the)456 3659 y(endomorphism)g FJ(T)20 b FQ(:)28 b FJ(V)1179 3672 y FI(~)1181 3687 y(\025)1248 3659 y FL(!)23 b FJ(V)1400 3672 y FI(~)1402 3687 y(\025)1475 3659 y FO(by)986 3888 y FJ(T)12 b FQ(\()p FJ(v)1119 3900 y FM(1)1174 3888 y FL(\012)18 b(\001)c(\001)g(\001)k(\012)g FJ(v)1495 3900 y FI(n)1541 3888 y FQ(\))23 b(=)1723 3784 y FI(n)1684 3809 y Fy(X)1690 3986 y FI(i)p FM(=1)1818 3888 y FJ(v)1858 3900 y FM(1)1913 3888 y FL(\012)c(\001)14 b(\001)g(\001)k(\012)g FJ(z)t FQ(\()p FJ(p)2312 3900 y FI(i)2339 3888 y FQ(\))c FJ(e)2424 3900 y FI(\022)2461 3888 y FJ(v)2501 3900 y FI(i)2547 3888 y FL(\012)k(\001)c(\001)g(\001) 19 b(\012)f FJ(v)2869 3900 y FI(n)456 4116 y FO(Then)30 b(one)g(has)h(an)e(isomorphism)1221 4271 y FQ(\()p FJ(V)1301 4283 y FI(\025)1340 4291 y FF(1)1396 4271 y FL(\012)18 b FJ(:)c(:)g(:)19 b FL(\012)f FJ(V)1726 4283 y FI(\025)1765 4291 y FG(n)1810 4271 y FQ(\))1842 4288 y Fk(g)p FE(\010)p Fz(C)5 b FI(T)2014 4271 y FG(k)q FF(+1)37 b FL(')23 b FJ(\034)9 b FQ(\()p FH(P)2371 4236 y FM(1)2408 4271 y FJ(;)14 b(~)-42 b(p)o(;)14 b(L)2580 4236 y FI(k)2578 4291 y(~)2580 4307 y(\025)2623 4271 y FQ(\))p FJ(:)605 4439 y FP(Pr)n(oof.)41 b FQ(P)n(art)33 b(\(i\))i(is)f(pro)n(v)n(ed)f (in)i(the)g(same)e(w)n(a)n(y)h(as)f(Lemma)h(7.3.2,)h(if)g(w)n(e)f(also) g(note)456 4539 y(that)26 b(for)f(one)h(p)r(oin)n(t,)h FA(g)p FQ(\()p FH(P)1275 4508 y FM(1)1326 4539 y FL(\000)15 b FJ(p)p FQ(\))23 b(=)f FA(g)15 b FL(\010)1725 4535 y Fy(b)1727 4539 y FA(g)1769 4508 y FE(\000)1825 4539 y FQ(.)37 b(As)26 b(for)f(part)h(\(ii\),)h(it)f(can)g(b)r(e)g(deduced)g (from)g(the)456 4638 y(fact)h(that)h FJ(L)856 4608 y FI(k)856 4662 y(\025)922 4638 y FQ(=)23 b FJ(V)1077 4608 y FI(k)1058 4662 y(\025)1118 4638 y FJ(=U)1224 4634 y Fy(b)1226 4638 y FA(g)o FQ(\()p FJ(e)1338 4650 y FI(\022)1375 4638 y FQ([)p FL(\000)p FQ(1]\))1560 4608 y FI(a)p FM(+1)1684 4638 y FJ(v)1724 4650 y FI(\025;k)1824 4638 y FQ(.)p 3384 4638 V 3388 4585 50 4 v 3388 4638 V 3437 4638 4 57 v 605 4817 a(Let)d(us)f(relate)g(this)h(description)e(with)i(the)g (one)f(usually)g(giv)n(en)g(in)h(the)f(ph)n(ysics)g(literature.)456 4917 y(As)39 b(b)r(efore,)i(let)f FJ(C)48 b FQ(=)42 b FH(P)1274 4887 y FM(1)1349 4917 y FQ(with)e(global)d(co)r(ordinate)h FJ(z)t 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eop %%Page: 174 8 174 177 bop 456 226 a FM(174)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(the)41 b(de\014nition)g(of)f(coin)n(v)-5 b(arian)n(ts)39 b FJ(V)1641 395 y FI(k)1622 449 y(\025)1661 457 y FG(i)1733 425 y FQ(b)n(y)i FJ(V)1910 437 y FI(\025)1949 445 y FG(i)2021 425 y FQ(and)f(the)h(algebra)e FA(g)p FQ(\()p FH(P)2781 395 y FM(1)2844 425 y FL(\000)27 b(f)p FQ(0)p FJ(;)14 b(z)3096 437 y FI(i)3122 425 y FJ(;)g FL(1g)p FQ(\))41 b(b)n(y)456 533 y FA(g)p FQ(\()p FH(P)582 503 y FM(1)636 533 y FL(\000)18 b(f)p FQ(0)p FJ(;)c FL(1g)p FQ(\))22 b(=)h FA(g)p FQ([)p FJ(z)t(;)14 b(z)1295 503 y FE(\000)p FM(1)1382 533 y FQ(].)37 b(Th)n(us)456 702 y(\(7.3.5\))82 b FJ(\034)9 b FQ(\()p FH(P)903 668 y FM(1)940 702 y FJ(;)14 b FQ(0)p FJ(;)g(z)1095 714 y FM(1)1131 702 y FJ(;)g(:)g(:)g(:)g(;)g(z)1355 714 y FI(n)1399 702 y FJ(;)g FL(1)p FJ(;)g(V)1604 714 y FM(0)1642 702 y FJ(;)g(:)g(:)g(:)f (;)h(V)1874 714 y FE(1)1945 702 y FQ(\))824 857 y(=)23 b(\()p FJ(V)992 869 y FM(0)1048 857 y FL(\012)18 b FJ(V)1179 869 y FI(\025)1218 877 y FF(1)1274 857 y FL(\012)g FJ(:)c(:)g(:)k FL(\012)g FJ(V)1603 869 y FI(\025)1642 877 y FG(n)1706 857 y FL(\012)h FJ(V)1838 869 y FE(1)1908 857 y FQ(\))p FJ(=)1982 789 y Fy(\000)2020 857 y FQ(\()p FJ(x)p FQ([)p FJ(n)p FQ(]\))2227 869 y FM(0)2284 857 y FQ(+)2367 778 y Fy(X)2501 857 y FJ(z)2544 822 y FI(n)2540 877 y(i)2588 857 y FJ(x)2635 869 y FI(i)2682 857 y FQ(+)f(\()p FL(\000)p FQ(1\))2936 822 y FI(n)2980 857 y FQ(\()p FJ(x)p FQ([)p FL(\000)p FJ(n)p FQ(]\))3252 869 y FE(1)3323 789 y Fy(\001)456 1012 y FQ(where)31 b FJ(n)e FL(2)h FH(Z)o FJ(;)14 b(x)24 b FL(2)30 b FA(g)p FQ(,)i(and)f(notation)g FJ(x)1763 1024 y FI(i)1823 1012 y FQ(means)g FJ(x)h FQ(acting)f(on)g FJ(V)2580 1024 y FI(\025)2619 1032 y FG(i)2650 1012 y FQ(,)i(etc.)48 b(W)-7 b(e)32 b(can)f(pass)g(to)456 1111 y(the)d(dual)f(space)g FJ(\034)1050 1081 y FE(\003)1116 1111 y FQ(whic)n(h)h(will)g(b)r(e)g(a)f(subspace)g(in)595 1366 y(Hom)768 1378 y Fz(C)814 1366 y FQ(\()p FJ(V)894 1378 y FM(0)951 1366 y FL(\012)18 b FJ(V)1082 1378 y FI(\025)1121 1386 y FF(1)1176 1366 y FL(\012)g FJ(:)c(:)g(:)19 b FL(\012)f FJ(V)1506 1378 y FI(\025)1545 1386 y FG(n)1609 1366 y FL(\012)g FJ(V)1740 1378 y FE(1)1811 1366 y FJ(;)c FH(C)h FQ(\))29 b(=)23 b(Hom)2224 1378 y Fz(C)2270 1366 y FQ(\()p FJ(V)2350 1378 y FM(0)2406 1366 y FL(\012)18 b FJ(V)2537 1378 y FI(\025)2576 1386 y FF(1)2632 1366 y FL(\012)g FJ(:)c(:)g(:)k FL(\012)g FJ(V)2961 1378 y FI(\025)3000 1386 y FG(n)3046 1366 y FJ(;)3099 1348 y FH([)3083 1366 y FJ(D)r(V)3202 1378 y FE(1)3273 1366 y FQ(\))456 1535 y(where)697 1514 y Fy(c)693 1535 y FJ(W)38 b FQ(is)25 b(the)g(completion)g(of)h(a)e FJ(W)35 b FL(2)24 b(O)1869 1547 y FI(k)1935 1535 y FQ(with)i(resp)r(ect)f(to)g(the)h (homogeneous)d(grading.)456 1635 y(Rewriting)k(the)h(coin)n(v)-5 b(ariance)25 b(condition,)j(w)n(e)f(get)815 1790 y FJ(\034)860 1756 y FE(\003)922 1790 y FQ(=)c FL(f)p FQ(\010)f(:)h FJ(V)1228 1802 y FM(0)1285 1790 y FL(\012)18 b FJ(V)1416 1802 y FI(\025)1455 1810 y FF(1)1510 1790 y FL(\012)g FJ(:)c(:)g(:)19 b FL(\012)f FJ(V)1840 1802 y FI(\025)1879 1810 y FG(n)1947 1790 y FL(!)2070 1772 y FH([)2054 1790 y FJ(D)r(V)2173 1802 y FE(1)2266 1790 y FL(j)24 b FQ(\010\()p FJ(x)p FQ([)p FJ(n)p FQ(])19 b(+)2650 1711 y Fy(X)2783 1790 y FJ(z)2826 1756 y FI(n)2822 1810 y(i)2871 1790 y FJ(x)2918 1802 y FI(i)2946 1790 y FQ(\))k(=)g FJ(x)p FQ([)p FJ(n)p FQ(]\010)p FL(g)922 1955 y FQ(=)g(Hom)1183 1971 y Fk(g)p FM([)p FI(t;t)1307 1955 y Fx(\000)p FF(1)1384 1971 y FM(])1407 1955 y FQ(\()p FJ(V)1487 1967 y FM(0)1543 1955 y FL(\012)18 b FJ(V)1674 1967 y FI(\025)1713 1975 y FF(1)1751 1955 y FQ(\()p FJ(z)1822 1967 y FM(1)1859 1955 y FQ(\))g FL(\012)g FJ(:)c(:)g(:)g(V)2151 1967 y FI(\025)2190 1975 y FG(n)2236 1955 y FQ(\()p FJ(z)2307 1967 y FI(n)2352 1955 y FQ(\))p FJ(;)2438 1937 y FH([)2421 1955 y FJ(D)r(V)2540 1967 y FE(1)2611 1955 y FQ(\))p FJ(;)456 1867 y FQ(\(7.3.6\))456 2104 y(where,)27 b(as)g(b)r(efore,)g FJ(V)19 b FQ(\()p FJ(z)t FQ(\))27 b(is)h(the)g(ev)-5 b(aluation)27 b(represen)n(tation.)605 2203 y(F)-7 b(or)31 b(the)h(case)f FA(g)f FQ(=)g FA(sl)1312 2215 y FM(2)1349 2203 y FJ(;)14 b(n)30 b FQ(=)f(1)j(the)g(dimensions)f(of)h(these)g (spaces)f(\(whic)n(h,)i(as)e(w)n(e)g(will)456 2303 y(sho)n(w)d(b)r(elo) n(w,)h(pla)n(y)g(the)h(role)e(of)h(m)n(ultiplicit)n(y)h(co)r(e\016cien) n(ts)f FJ(N)2454 2273 y FI(k)2445 2325 y(ij)2532 2303 y FQ(for)g(the)h(mo)r(dular)e(category)456 2413 y FL(O)524 2383 y FM(in)n(t)522 2437 y FI(k)608 2413 y FQ(\))c(w)n(ere)e (calculated)h(in)h([)p FK(TK)p FQ(];)h(their)e(answ)n(er)f(agrees)g (with)i(the)f(form)n(ula)g(for)g FJ(U)3082 2425 y FI(q)3118 2413 y FQ(\()p FA(sl)3210 2425 y FM(2)3248 2413 y FQ(\))p FJ(;)14 b(q)26 b FQ(=)456 2519 y FJ(e)495 2489 y FI(\031)r FM(i)p FI(=)p FM(\()p FI(k)q FM(+2\))792 2519 y FQ(giv)n(en)h(in)h (\(3.3.24\))o(|as)f(exp)r(ected)h(from)f(Theorem)g(7.0.2.)605 2674 y FP(Remark)32 b FQ(7.3.9)p FP(.)39 b FQ(It)26 b(is)f(a)g(natural) g(question)g(to)g(generalize)f(the)h(de\014nition)h(of)f(coin)n(v)-5 b(ari-)456 2773 y(an)n(ts,)26 b(whic)n(h)g(can)h(b)r(e)g(view)n(ed)f (as)g(Lie)g(algebra)f(homology)g(in)i(degree)e(zero)h FJ(H)2928 2785 y FM(0)2965 2773 y FQ(\()p FA(g)p FQ(\()p FJ(C)d FL(\000)17 b FJ(~)-43 b(p)p FQ(\))p FJ(;)14 b(V)19 b FQ(\))456 2873 y(and)f(consider)g(all)g(homology)f(spaces)g FJ(H)1710 2885 y FE(\003)1749 2873 y FQ(\()p FA(g)p FQ(\()p FJ(C)6 b FL(\000)q FJ(~)-43 b(p)p FQ(\))p FJ(;)14 b(V)19 b FQ(\).)35 b(T)-7 b(o)18 b(the)h(b)r(est)g(of)f(out)h(kno)n(wledge,)g (this)456 2972 y(approac)n(h)k(w)n(as)i(\014rst)g(suggested)g(b)n(y)g (B.)h(F)-7 b(eigin.)36 b(One)26 b(of)f(the)h(\014rst)g(results)f(in)h (this)g(direction,)456 3072 y(pro)n(v)n(ed)i(in)j([)p FK(T)-8 b(el)o FQ(],)31 b(is)f(the)h(v)-5 b(anishing)30 b(theorem:)41 b(if)31 b FJ(V)2145 3084 y FI(i)2203 3072 y FQ(are)e(W)-7 b(eyl)31 b(mo)r(dules,)g(then)f(all)g(higher)456 3172 y(homology)25 b(v)-5 b(anish.)36 b(In)27 b(particular,)e(this)i (theorem)f(allo)n(ws)g(one)g(to)g(calculate)g(dimensions)h(of)456 3271 y(the)e(v)n(ector)e(spaces)g(of)i(coin)n(v)-5 b(arian)n(ts)23 b FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)14 b(L)1953 3241 y FI(k)1951 3296 y(~)1953 3312 y(\025)1996 3271 y FQ(\),)25 b(b)n(y)g(writing)f(for)g(eac)n(h)g(of)g FJ(L)2928 3241 y FI(k)2928 3295 y(\025)2967 3303 y FG(i)3022 3271 y FQ(a)h(resolution)456 3377 y(consisting)34 b(of)g(W)-7 b(eyl)36 b(mo)r(dules,)g(and)f(then)h(using)e(the)h(fact)g(that)h(for)e (the)h(W)-7 b(eyl)35 b(mo)r(dules,)456 3477 y(dimension)d(of)h(the)g (space)g(of)f(coin)n(v)-5 b(arian)n(ts)31 b(is)i(kno)n(wn)f(\(see)h (Lemma)f(7.3.8\).)52 b(This)33 b(answ)n(er)456 3577 y(coincides)f(with) i(the)f(dimension)g(of)g(the)h(spaces)e(of)h(homomorphisms)f(in)h(the)g (category)f(of)456 3676 y(represen)n(tations)25 b(of)j(quan)n(tum)f (group)g(at)g(ro)r(ot)g(of)h(unit)n(y)f(\(see)h(Prop)r(osition)e (3.3.23\).)605 3776 y(The)18 b(meaning)g(of)h(the)f(higher)g(homology)f (spaces)g(\(\\higher)h(conformal)f(blo)r(c)n(ks"\))g FJ(H)3245 3788 y FI(i)3273 3776 y FQ(\()p FA(g)p FQ(\()p FJ(C)6 b FL(\000)456 3876 y FJ(~)-42 b(p)o FQ(\))p FJ(;)14 b(V)19 b FQ(\))24 b(when)f FJ(V)949 3888 y FI(i)1001 3876 y FQ(are)f(in)n(tegrable)g(and)h(the)g(role)f(they)i(pla)n(y)e(in) h(conformal)f(\014eld)i(theory)e(is)h(still)456 3975 y(unclear.)1527 4169 y FK(7.4.)47 b(Flat)32 b(connection)605 4319 y FQ(In)k(the)g(previous)f(section,)j(w)n(e)d(ha)n(v)n(e)g (de\014ned)h(and)g(studied)g(some)f(prop)r(erties)g(of)h(the)456 4419 y(v)n(ector)31 b(spaces)h(of)h(coin)n(v)-5 b(arian)n(ts)31 b(for)i(a)f(giv)n(en)g(curv)n(e)g FJ(C)39 b FQ(with)34 b(mark)n(ed)e(p)r(oin)n(ts)h(and)f(c)n(hosen)456 4518 y(lo)r(cal)40 b(parameters)f(at)i(these)h(p)r(oin)n(ts.)77 b(No)n(w,)44 b(let)d(us)g(study)h(what)f(happ)r(ens)g(with)g(these)456 4618 y(spaces)31 b(when)i(w)n(e)f(c)n(hange)f(the)i(lo)r(cal)f (parameters,)f(or)h(mo)n(v)n(e)f(the)i(p)r(oin)n(ts.)51 b(Let)33 b(us)f(assume)456 4717 y(that)g(w)n(e)f(ha)n(v)n(e)f(a)i(smo)r (oth)f(family)h(of)f(p)r(oin)n(ted)h(curv)n(es)f FJ(C)2310 4729 y FI(s)2346 4717 y FJ(;)14 b(s)29 b FL(2)i FJ(S)36 b FQ(o)n(v)n(er)30 b(a)h(smo)r(oth)h(base)f FJ(S)5 b FQ(.)456 4817 y(As)30 b(men)n(tioned)g(ab)r(o)n(v)n(e,)f(it)i(means)f (that)g(w)n(e)g(ha)n(v)n(e)f(a)g(smo)r(oth)h(manifold)g FJ(C)2864 4829 y FI(S)2943 4817 y FQ(with)h(a)e(prop)r(er)456 4917 y(\015at)h(smo)r(oth)f(morphism)h FJ(\031)g FQ(:)d FJ(C)1478 4929 y FI(S)1553 4917 y FL(!)g FJ(S)35 b FQ(suc)n(h)29 b(that)h(eac)n(h)g(\014b)r(er)g FJ(C)2563 4929 y FI(s)2625 4917 y FQ(=)d FJ(\031)2767 4887 y FE(\000)p FM(1)2856 4917 y FQ(\()p FJ(s)p FQ(\))k(is)e(a)h(complex)456 5016 y(curv)n(e;)f(w)n(e)g(also)f(ha)n(v)n(e)g FJ(n)h FQ(non-in)n (tersecting)f(sections)g FJ(p)2229 5028 y FI(i)2282 5016 y FQ(:)e FJ(S)31 b FL(!)25 b FJ(C)2580 5028 y FI(S)2629 5016 y FQ(,)30 b(and)f(lo)r(cal)f(parameters)456 5116 y FJ(t)486 5128 y FI(i)513 5116 y FQ(,)e(whic)n(h)g(are)e(functions)i (in)f(a)g(neigh)n(b)r(orho)r(o)r(d)g(of)g FJ(p)2105 5128 y FI(i)2133 5116 y FQ(\()p FJ(S)5 b FQ(\))23 b FL(\032)g FJ(C)2423 5128 y FI(S)2496 5116 y FQ(suc)n(h)j(that)f FJ(p)2901 5128 y FI(i)2929 5116 y FQ(\()p FJ(S)5 b FQ(\))25 b(is)h(the)g(zero)456 5216 y(lo)r(cus)g(of)g FJ(t)788 5228 y FI(i)816 5216 y FQ(,)h(and)f FJ(dt)1099 5228 y FI(i)1150 5216 y FL(6)p FQ(=)d(0)j(on)g FJ(p)1462 5228 y FI(i)1489 5216 y FQ(\()p FJ(S)5 b FQ(\).)37 b(Suc)n(h)27 b(a)f(data)g(de\014nes)g(on)g(eac)n(h)g(\014b)r(er)g(a)h(structure)e (of)i(a)p eop %%Page: 175 9 175 178 bop 1563 226 a FM(7.4.)29 b(FLA)-5 b(T)29 b(CONNECTION)986 b(175)456 425 y FQ(p)r(oin)n(ted)26 b(complex)f(curv)n(e,)h(with)g(a)g (lo)r(cal)f(parameter)g(at)g(eac)n(h)h(puncture;)g(as)g(b)r(efore,)g(w) n(e)f(will)456 525 y(assume)d(tah)n(t)i(on)f(eac)n(h)g(connected)g (comp)r(onen)n(t)g(of)h FJ(C)2159 537 y FI(s)2218 525 y FQ(there)f(is)h(at)f(least)g(one)g(mark)n(ed)f(p)r(oin)n(t.)456 624 y(Similarly)d(to)g(the)h(construction)f(of)h(the)g(previous)e (section,)j(it)f(is)g(con)n(v)n(enien)n(t)e(to)i(allo)n(w)e FJ(t)3218 636 y FI(i)3266 624 y FQ(to)h(b)r(e)456 730 y(formal)26 b(parameter,)h(i.e.)37 b(an)27 b(elemen)n(t)h(of)f(the)h (completed)g(lo)r(cal)f(ring)2723 709 y Fy(b)2703 730 y FL(O)2769 745 y FI(C)2817 753 y FG(S)2858 745 y FI(;p)2912 753 y FG(i)2938 745 y FM(\()p FI(S)s FM(\))3062 730 y FL(')22 b(O)3215 742 y FI(S)3263 730 y FQ([[)p FJ(t)3339 742 y FI(i)3367 730 y FQ(]].)605 829 y(W)-7 b(e)38 b(will)g(denote)f(b) n(y)h(\002)1393 841 y FI(S)1478 829 y FQ(the)g(sheaf)f(of)h(v)n(ector)e (\014elds)h(on)h FJ(S)5 b FQ(.)66 b(W)-7 b(e)38 b(will)g(also)f(denote) 456 929 y(b)n(y)i FL(O)r FQ(\()p FJ(C)742 941 y FI(S)818 929 y FL(\000)26 b FJ(~)-42 b(p)p FQ(\()p FJ(S)5 b FQ(\)\))40 b(the)g(sheaf)g(on)f FJ(S)44 b FQ(whose)c(sections)f(o)n(v)n(er)f FJ(U)52 b FL(\032)42 b FJ(S)j FQ(are)39 b(b)n(y)g(de\014nition)456 1029 y(meromorphic)22 b(functions)j(o)n(v)n(er)d FJ(\031)1534 999 y FE(\000)p FM(1)1623 1029 y FQ(\()p FJ(U)9 b FQ(\))24 b FL(\032)e FJ(C)1923 1041 y FI(S)1996 1029 y FQ(whic)n(h)i(are)f (regular)f(outside)i(of)g FJ(p)3059 1041 y FI(i)3087 1029 y FQ(\()p FJ(S)5 b FQ(\);)25 b(when)456 1128 y FJ(S)i FQ(=)c FL(f)p FJ(point)p FL(g)p FQ(,)i(this)i(coincides)f(with)h(the)f (de\014nition)h(in)g(the)f(previous)g(section.)36 b(In)26 b(a)g(similar)456 1228 y(w)n(a)n(y)-7 b(,)26 b(w)n(e)h(de\014ne)h FA(g)p FQ(\()p FJ(C)1135 1240 y FI(S)1202 1228 y FL(\000)19 b FJ(~)-43 b(p)p FQ(\()p FJ(S)5 b FQ(\)\))p FJ(;)14 b FQ(\002\()p FJ(C)1672 1240 y FI(S)1739 1228 y FL(\000)19 b FJ(~)-43 b(p)p FQ(\()p FJ(S)5 b FQ(\)\)|all)28 b(of)f(them)i(are)d (shea)n(v)n(es)g(on)h FJ(S)5 b FQ(.)605 1328 y(Throughout)29 b(this)i(section,)f(let)h(us)f(\014x)g(a)g(family)g FJ(C)2276 1340 y FI(S)2355 1328 y FQ(as)f(ab)r(o)n(v)n(e,)h(c)n(ho)r(ose)f(in)n (tegrable)3373 1324 y Fy(b)3375 1328 y FA(g)p FQ(-)456 1427 y(mo)r(dules)i FJ(V)833 1439 y FM(1)871 1427 y FJ(;)14 b(:)g(:)g(:)g(;)g(V)1104 1439 y FI(n)1179 1427 y FL(2)30 b(O)1332 1397 y FM(in)n(t)1330 1451 y FI(k)1417 1427 y FQ(,)i(and)g(let)g FJ(V)49 b FQ(=)29 b FJ(V)2001 1439 y FM(1)2060 1427 y FL(\012)21 b FJ(V)2194 1439 y FI(n)2240 1427 y FQ(.)49 b(Then)32 b(for)f(ev)n(ery)f(p)r(oin)n(t)i FJ(s)e FL(2)g FJ(S)37 b FQ(w)n(e)456 1527 y(can)27 b(de\014ne)h(the)g (v)n(ector)e(space)h(of)g(coin)n(v)-5 b(arian)n(ts)1246 1678 y FJ(\034)1282 1690 y FI(s)1341 1678 y FQ(=)23 b FJ(\034)9 b FQ(\()p FJ(C)1565 1690 y FI(s)1602 1678 y FJ(;)14 b(~)-42 b(p)o FQ(\()p FJ(s)p FQ(\))p FJ(;)14 b(V)20 b FQ(\))j(=)f FJ(V)9 b(=)p FA(g)p FQ(\()p FJ(C)2262 1690 y FI(s)2317 1678 y FL(\000)19 b FJ(~)-43 b(p)p FQ(\()p FJ(s)p FQ(\)\))p FJ(V)5 b(:)-2197 b FQ(\(7.4.1\))456 1829 y(The)27 b(main)h(goal)e(of)i(this)g(section)f(is)g(to)h(pro)n(v)n (e)e(the)i(follo)n(wing)e(theorem.)605 1987 y FP(Theorem)32 b FQ(7.4.1)p FP(.)40 b FO(Under)h(the)g(ab)l(ove)h(assumptions,)j(the)c (ve)l(ctor)h(sp)l(ac)l(es)f FJ(\034)3113 1999 y FI(s)3190 1987 y FO(form)h(a)456 2087 y(ve)l(ctor)34 b(bund)t(le)h FJ(\034)1011 2099 y FI(S)1093 2087 y FO(over)g FJ(S)k FO(which)c(c)l(arries)g(a)g(natur)l(al)f(pr)l(oje)l(ctively)i(\015at)e (c)l(onne)l(ction.)51 b(The)456 2187 y(assignment)31 b FJ(S)f FL(7!)c FJ(\034)1114 2199 y FI(S)1194 2187 y FO(is)32 b(functorial)g(in)f FJ(S)5 b FQ(:)42 b FO(for)32 b(every)g(map)g FJ( )12 b FQ(:)29 b FJ(S)2602 2157 y FE(0)2650 2187 y FL(!)d FJ(S)36 b FO(and)c(a)g(family)h FJ(C)3396 2199 y FI(S)456 2286 y FO(over)38 b FJ(S)43 b FO(as)38 b(b)l(efor)l(e,)j(ther)l(e)d(is)g(a)g(c)l(anonic)l(al)h (isomorphism)h FJ(\034)2427 2298 y FI(S)2471 2282 y Fx(0)2535 2286 y FQ(=)d FJ( )2694 2256 y FE(\003)2733 2286 y FQ(\()p FJ(\034)2801 2298 y FI(S)2849 2286 y FQ(\))p FO(,)k(wher)l(e)d FJ(C)3248 2298 y FI(S)3292 2282 y Fx(0)3357 2286 y FQ(:=)456 2386 y FJ( )513 2356 y FE(\003)551 2386 y FQ(\()p FJ(C)642 2398 y FI(S)691 2386 y FQ(\))p FO(.)605 2545 y FQ(W)-7 b(e)34 b(remind)h(that)f(a)f(connection)h(is)g(called)g(pro)5 b(jectiv)n(ely)32 b(\015at)i(if)h([)p FL(r)2845 2557 y FI(X)2908 2545 y FJ(;)14 b FL(r)3014 2557 y FI(Y)3072 2545 y FQ(])23 b FL(\000)f(r)3274 2560 y FM([)p FI(X)q(;Y)14 b FM(])456 2644 y FQ(is)28 b(an)h(op)r(erator)e(of)i(m)n(ultiplication) g(b)n(y)g(a)f(function)i(for)e(an)n(y)g(t)n(w)n(o)g(v)n(ector)g (\014elds)h FJ(X)r(;)14 b(Y)47 b FQ(on)29 b FJ(S)5 b FQ(.)456 2744 y(The)28 b(failure)g(of)h(the)g(connection)f(to)g(b)r(e)h (\015at)f(is,)h(of)g(course,)e(related)h(with)h(the)g(cen)n(tral)f (term)456 2844 y(in)i(the)g(de\014nition)g(of)1166 2840 y Fy(b)1168 2844 y FA(g)p FQ(:)41 b(for)29 b FJ(k)h FQ(=)c(0,)k(the)h (connection)e(is)h(\015at)g(\(but)g(of)g(little)h(in)n(terest,)f(since) 456 2943 y(the)e(only)f(in)n(tegrable)f(mo)r(dule)i(of)g(lev)n(el)f(0)g (is)g FJ(L)1952 2913 y FI(k)1952 2964 y FM(0)2016 2943 y FQ(=)22 b FH(C)15 b FQ(\))q(.)43 b(W)-7 b(e)28 b(will)f(discuss)h (this)f(later.)605 3043 y(The)35 b(remaining)f(part)h(of)g(this)g (section)g(is)g(dev)n(oted)f(to)h(the)h(construction)e(of)h(the)g (\015at)456 3142 y(connection)22 b(and)h(the)g(pro)r(of)g(of)g(the)g (theorem.)35 b(F)-7 b(or)23 b(simplicit)n(y)-7 b(,)24 b(w)n(e)e(will)i(assume)e(that)h FJ(n)g FQ(=)g(1;)456 3242 y(the)f(general)f(case)g(can)g(b)r(e)i(treated)e(similarly)-7 b(.)34 b(Our)22 b(exp)r(osition)f(follo)n(ws)g([)p FK(BFM)p FQ(])h(\(somewhat)456 3342 y(simpli\014ed\).)605 3500 y FP(Lemma)31 b FQ(7.4.2)p FP(.)40 b FO(The)24 b(ve)l(ctor)f(sp)l(ac)l (es)h FJ(\034)1834 3512 y FI(s)1893 3500 y FO(form)g(a)f FL(O)2218 3512 y FI(S)2266 3500 y FO(-c)l(oher)l(ent)g(she)l(af)i(over) e FJ(S)5 b FO(,)25 b(i.e.,)h(ther)l(e)456 3600 y(exists)34 b(a)h(c)l(oher)l(ent)g(she)l(af)h FJ(\034)1349 3612 y FI(S)1432 3600 y FO(such)e(that)h FJ(\034)1836 3612 y FI(s)1904 3600 y FQ(=)d FJ(\034)2037 3612 y FI(S)2085 3600 y FJ(=I)2163 3612 y FI(s)2198 3600 y FJ(\034)2234 3612 y FI(S)2283 3600 y FO(,)k FJ(I)2380 3612 y FI(s)2451 3600 y FO(b)l(eing)f(the)g(ide)l(al)h(of)f(functions)456 3700 y(vanishing)c(at)e FJ(s)p FO(.)605 3858 y FP(Pr)n(oof.)41 b FQ(Let)19 b FJ(V)1121 3870 y FI(S)1192 3858 y FQ(=)k FL(O)1346 3870 y FI(S)1395 3858 y FL(\012)q FJ(V)37 b FQ(\(usual)19 b(algebraic)e(v)n(ector)g(pro)r(duct,)k(no)e (completions\);)i(this)456 3958 y(is)h(an)g FL(O)710 3970 y FI(S)758 3958 y FQ(-mo)r(dule,)h(whic)n(h)f(carries)f(an)h FL(O)1764 3970 y FI(S)1812 3958 y FQ(-linear)f(action)h(of)g(the)h FL(O)2602 3970 y FI(S)2650 3958 y FQ(-mo)r(dule)f FA(g)p FQ(\()p FJ(C)3098 3970 y FI(S)3155 3958 y FL(\000)8 b FJ(p)p FQ(\()p FJ(S)d FQ(\)\).)456 4058 y(De\014ne)28 b(the)g(sheaf)1485 4208 y FJ(\034)1521 4220 y FI(S)1592 4208 y FQ(=)23 b FJ(V)1728 4220 y FI(S)1777 4208 y FJ(=)p FA(g)p FQ(\()p FJ(C)1952 4220 y FI(S)2018 4208 y FL(\000)18 b FJ(p)p FQ(\()p FJ(S)5 b FQ(\)\))p FJ(V)2343 4220 y FI(S)2392 4208 y FJ(:)-1959 b FQ(\(7.4.2\))456 4359 y(It)34 b(is)f(ob)n(vious)g(that)h(lo)r(calizing)e FJ(\034)1542 4371 y FI(S)1624 4359 y FQ(at)i FJ(s)f FL(2)h FJ(S)5 b FQ(,)35 b(w)n(e)e(get)g(the)h(v)n(ector)f(space)g(of)g(coin)n(v)-5 b(arian)n(ts)456 4459 y FJ(\034)492 4471 y FI(s)527 4459 y FQ(.)68 b(The)38 b(coherency)e(of)i FJ(\034)1335 4471 y FI(S)1421 4459 y FQ(can)g(b)r(e)g(pro)n(v)n(ed)e(in)i(a)g(w)n(a)n(y)e (similar)h(to)h(the)g(pro)r(of)f(of)h(\014nite-)456 4559 y(dimensionalit)n(y)18 b(of)g(the)h(spaces)f FJ(\034)9 b FQ(\()p FJ(C)d FQ(\))21 b(in)d(the)i(previous)d(section,)j(using)f (the)g(follo)n(wing)e(lemma.)605 4717 y FP(Lemma)31 b FQ(7.4.3)p FP(.)40 b FO(L)l(et)28 b FJ(A)g FO(b)l(e)g(a)h (\014nite-dimensional)g(ve)l(ctor)f(bund)t(le)h(of)f(Lie)h(algebr)l(as) h(over)456 4817 y FJ(S)39 b FO(which)e(is)e(gener)l(ate)l(d)44 b FQ(\()p FO(as)36 b(a)g(Lie)f(algebr)l(a)6 b FQ(\))37 b FO(by)f(se)l(ctions)f FJ(x)2436 4829 y FM(1)2474 4817 y FJ(;)14 b(:)g(:)g(:)f(;)h(x)2705 4829 y FI(n)2751 4817 y FO(.)55 b(Denote)34 b(by)i FL(A)f FO(the)456 4917 y(she)l(af)j(of)f (se)l(ctions)g(of)h FJ(A)p FO(.)60 b(L)l(et)36 b FL(W)44 b FO(b)l(e)37 b(an)g FL(O)1923 4929 y FI(S)1971 4917 y FO(-mo)l(dule)g(with)g(an)g FL(O)2669 4929 y FI(S)2717 4917 y FO(-line)l(ar)g(action)h(of)g FL(A)p FO(.)456 5016 y(Assume)c(that)i FL(W)42 b FO(is)36 b(lo)l(c)l(al)t(ly)i(cyclic)j FQ(\()p FO(i.e.,)f(lo)l(c)l(al)t(ly)d(ther)l(e)f(exists)f(a)h(se)l (ction)g FJ(w)3001 5028 y FM(0)3072 5016 y FL(2)e(W)42 b FO(such)456 5116 y(that)31 b FL(W)j FQ(=)26 b FL(A)p FJ(w)959 5128 y FM(0)997 5116 y FQ(\))32 b FO(and)g(action)g(of)h FJ(x)1625 5128 y FI(i)1685 5116 y FO(is)f(lo)l(c)l(al)t(ly)h(nilp)l (otent:)42 b(for)33 b(every)g(se)l(ction)e FJ(w)r FO(,)j(one)e(has)456 5216 y FJ(x)503 5185 y FI(N)503 5237 y(i)566 5216 y FJ(w)26 b FQ(=)c(0)30 b FO(for)g FJ(N)i FL(\035)23 b FQ(0)p FO(.)38 b(Then)31 b FL(W)36 b FO(is)30 b FL(O)1742 5228 y FI(S)1791 5216 y FO(-c)l(oher)l(ent.)p eop %%Page: 176 10 176 179 bop 456 226 a FM(176)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 605 425 y FQ(T)-7 b(o)32 b(pro)n(v)n(e)e(this)i(lemma,)h(it)f (su\016ces)g(to)g(note)f(that)i(b)n(y)e(Gabb)r(er's)h(theorem,)g FJ(S)5 b(upp)p FQ(\()p FL(W)i FQ(\))456 525 y(is)33 b(the)h(zero)e (section)h(of)h(the)f(bundle)h FJ(A)1747 495 y FE(\003)1786 525 y FQ(,)h(and)e(that)h(ev)n(ery)e(mo)r(dule)i(o)n(v)n(er)d FL(O)2970 537 y FI(S)3019 525 y FQ([)p FJ(x)3089 537 y FM(1)3126 525 y FJ(;)14 b(:)g(:)g(:)g(;)g(x)3358 537 y FI(m)3421 525 y FQ(])456 624 y(whose)27 b(supp)r(ort)g(is)g(giv)n(en) g(b)n(y)h FJ(x)1471 636 y FI(i)1522 624 y FQ(=)22 b(0,)28 b(is)f FL(O)1851 636 y FI(S)1899 624 y FQ(-coheren)n(t.)p 3384 624 4 57 v 3388 572 50 4 v 3388 624 V 3437 624 4 57 v 605 792 a(W)-7 b(e)24 b(will)g(sho)n(w)f(that)g(the)h(sheaf)g FJ(\034)1656 804 y FI(S)1728 792 y FQ(has)f(a)g(natural)g(structure)g (of)g(a)h(t)n(wisted)f FL(D)3080 804 y FI(S)3128 792 y FQ(-mo)r(dule,)456 892 y(i.e.,)33 b(a)f(pro)5 b(jectiv)n(e)30 b(action)i(of)g(the)g(sheaf)g(\002)1861 904 y FI(S)1941 892 y FQ(of)g(v)n(ector)e(\014elds)i(on)g FJ(S)37 b FQ(whic)n(h)32 b(is)g(compatible)456 991 y(with)c(the)h FL(O)855 1003 y FI(S)903 991 y FQ(-mo)r(dule)f(structure:)38 b FJ(\030)t FQ(\()p FJ(\036\034)9 b FQ(\))26 b(=)d(\()p FJ(\030)t(\036)p FQ(\))p FJ(\034)30 b FQ(+)19 b FJ(\036)p FQ(\()p FJ(\030)t(\034)9 b FQ(\))p FJ(;)14 b(\030)30 b FL(2)24 b FQ(\002)2674 1003 y FI(S)2722 991 y FJ(;)14 b(\036)24 b FL(2)h(O)2978 1003 y FI(S)3026 991 y FQ(.)39 b(Since)28 b(it)h(is)456 1091 y(w)n(ell)24 b(kno)n(wn)g(that)h(ev)n(ery)e FL(O)r FQ(-coheren)n(t)g(t)n(wisted)i FL(D)r FQ(-mo)r(dule)g(is)f(in)h(fact)g (a)f(sheaf)g(of)h(sections)f(of)456 1191 y(a)h(v)n(ector)g(bundle)h (with)g(a)g(pro)5 b(jectiv)n(ely)24 b(\015at)i(connection,)g(this)g (will)g(establish)g(the)g(theorem.)605 1290 y(T)-7 b(o)28 b(construct)g(an)g(action)g(of)h(\002)1621 1302 y FI(S)1697 1290 y FQ(on)f(the)h(sheaf)g(of)f(coin)n(v)-5 b(arian)n(ts,)27 b(let)i(us)f(\014rst)h(consider)456 1390 y(the)23 b(case)f(when)h(w)n (e)g(ha)n(v)n(e)f(a)g(\014xed)h(curv)n(e)f FJ(C)30 b FQ(with)23 b(a)g(mark)n(ed)f(p)r(oin)n(t)h FJ(p)p FQ(,)h(and)f FJ(S)k FQ(is)c(the)h(set)f(of)g(all)456 1490 y(p)r(ossible)h(c)n (hoices)g(of)h(a)g(formal)f(lo)r(cal)g(parameter)g FJ(t)h FQ(at)g FJ(p)p FQ(.)36 b(This)25 b(set)g(has)f(a)h(natural)f(structure) 456 1589 y(of)31 b(a)h(pro)5 b(jectiv)n(e)30 b(limit)j(of)f(the)g(smo)r (oth)f(manifolds)h FJ(S)2200 1559 y FM(\()p FI(N)6 b FM(\))2344 1589 y FQ(=)30 b FL(f)p FJ(N)9 b FQ(-jets)31 b(of)h(lo)r(cal)f(parameters)456 1689 y(at)d FJ(p)p FL(g)p FQ(.)39 b(W)-7 b(e)29 b(ha)n(v)n(e)f(a)g(tautological)f(family)i(of)f (curv)n(es)g FJ(C)2229 1701 y FI(S)2302 1689 y FQ(=)c FJ(C)h FL(\002)19 b FJ(S)33 b FQ(o)n(v)n(er)27 b FJ(S)5 b FQ(,)29 b(with)g(the)g(same)456 1788 y(mark)n(ed)d(p)r(oin)n(t)i FJ(p)f FQ(and)h(with)g(the)g(formal)f(lo)r(cal)g(parameter)f (determined)i(b)n(y)f FJ(s)c FL(2)g FJ(S)5 b FQ(.)605 1888 y(This)26 b FJ(S)31 b FQ(is)c(a)e(torsor)g(o)n(v)n(er)f(the)j (pro-Lie)e(group)g(\(i.e.,)i(a)f(pro)5 b(jectiv)n(e)25 b(limit)i(of)f(Lie)h(groups\))456 1988 y FJ(K)527 2000 y FM(0)599 1988 y FQ(=)35 b(Aut)15 b FH(C)g FQ([[)p FJ(t)p FQ(]])41 b(of)35 b(c)n(hanges)f(of)h(lo)r(cal)g(parameter.)58 b(This)35 b(group)f(can)h(b)r(e)g(explicitly)h(de-)456 2087 y(scrib)r(ed)h(as)g(the)h(group)f(of)h(p)r(o)n(w)n(er)e(series)h (of)h(the)g(form)f FJ(a)2359 2099 y FM(1)2396 2087 y FJ(t)26 b FQ(+)e FJ(a)2585 2099 y FM(2)2622 2087 y FJ(t)2652 2057 y FM(2)2715 2087 y FQ(+)h FJ(:)14 b(:)g(:)f(;)h(a)2996 2099 y FM(1)3073 2087 y FL(6)p FQ(=)40 b(0,)g(with)456 2187 y(the)33 b(group)f(op)r(eration)g(b)r(eing)i(comp)r(osition;)h(it) f(acts)f(on)g(the)g(set)h(of)f(formal)f(lo)r(cal)h(param-)456 2287 y(eters)g(in)h(an)f(ob)n(vious)g(w)n(a)n(y)-7 b(.)54 b(The)34 b(corresp)r(onding)e(Lie)h(algebra)f FL(T)2608 2299 y FM(0)2679 2287 y FQ(=)h(Lie)14 b FJ(K)2974 2299 y FM(0)3044 2287 y FQ(is)34 b(giv)n(en)f(b)n(y)456 2386 y FL(T)501 2398 y FM(0)562 2386 y FQ(=)24 b FJ(t)p FH(C)15 b FQ([[)p FJ(t)p FQ(]])p FJ(@)901 2398 y FI(t)965 2386 y FQ(\(see)28 b([)p FK(TUY)q FQ(,)g(Section)h(1.4])e(for)h(precise)f (statemen)n(ts\).)39 b(Therefore,)27 b(the)i(tan-)456 2486 y(gen)n(t)d(space)h(to)g FJ(S)k FQ(at)c(ev)n(ery)f(p)r(oin)n(t)i (can)e(b)r(e)i(iden)n(ti\014ed)f(with)h FL(T)2438 2498 y FM(0)2475 2486 y FQ(.)37 b(or,)27 b(equiv)-5 b(alen)n(tly)e(,)26 b FL(T)3181 2498 y FM(0)3246 2486 y FQ(is)h(the)456 2585 y(space)e(of)g(all)h FJ(K)953 2597 y FM(0)1016 2585 y FQ(left-in)n(v)-5 b(arian)n(t)25 b(v)n(ector)f(\014elds)i(on)g FJ(S)5 b FQ(.)36 b(Th)n(us,)26 b(to)f(de\014ne)i(an)e(action)g(of)h (\002)3283 2597 y FI(S)3357 2585 y FQ(on)456 2685 y(the)i(bundle)g(of)f (coin)n(v)-5 b(arian)n(ts,)26 b(one)h(needs)h(to)f(de\014ne)h(an)f (action)g(of)h FL(T)2672 2697 y FM(0)2709 2685 y FQ(.)605 2785 y(Therefore,)d(w)n(e)h(see)f(that)h(the)h(k)n(ey)e(step)h(in)g (this)g(case)f(w)n(ould)g(b)r(e)i(to)e(de\014ne)i(an)e(action)h(of)456 2884 y FL(T)501 2896 y FM(0)564 2884 y FQ(=)g FJ(t)p FH(C)15 b FQ([[)p FJ(t)p FQ(]])p FJ(@)905 2896 y FI(t)970 2884 y FQ(on)29 b FJ(V)19 b FQ(.)42 b(In)30 b(the)g(general)e(case,)h (w)n(e)g(will)g(in)h(fact)f(need)h(an)f(action)g(of)g(a)g(larger)456 2984 y(Lie)c(algebra)g FL(T)44 b FQ(=)23 b FH(C)15 b FQ(\(\()p FJ(t)p FQ(\)\))q FJ(@)1317 2996 y FI(t)1352 2984 y FQ(,)26 b(whic)n(h)g(is)g(usually)g(called)f(the)h FO(Witt)i(algebr)l(a)p FQ(.)38 b(It)26 b(has)g(a)f(natural)456 3084 y(\(top)r(ological\))h(basis)h FJ(L)1203 3096 y FI(n)1271 3084 y FQ(=)22 b FL(\000)p FJ(t)1453 3053 y FI(n)p FM(+1)1582 3084 y FJ(@)1626 3096 y FI(t)1656 3084 y FQ(,)27 b FJ(n)c FL(2)h FH(Z)o FQ(,)e(with)28 b(the)g(comm)n(utation) f(relations)1481 3231 y([)p FJ(L)1561 3243 y FI(m)1623 3231 y FJ(;)14 b(L)1717 3243 y FI(n)1762 3231 y FQ(])23 b(=)g(\()p FJ(m)18 b FL(\000)g FJ(n)p FQ(\))p FJ(L)2241 3243 y FI(m)p FM(+)p FI(n)2396 3231 y FJ(:)-1963 b FQ(\(7.4.3\))456 3378 y(The)30 b(subalgebra)f FL(T)1093 3390 y FM(0)1162 3378 y FQ(is)h(generated)g(b)n(y)g FJ(L)1805 3390 y FI(n)1881 3378 y FQ(with)h FJ(n)d FL(\025)g 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FQ(])h(for)f(details)g(and)h(the)g(pro)r(of.)605 4033 y FP(Pr)n(oposition)j FQ(7.4.4)p FP(.)40 b FO(One)31 b(c)l(an)g(de\014ne)h(elements)g FJ(L)2357 4045 y FI(n)2402 4033 y FJ(;)14 b(n)26 b FL(2)h FH(Z)o FO(,)g(in)32 b(a)g(c)l(ertain)g (c)l(omple-)456 4133 y(tion)d(of)i FJ(U)9 b FQ(\()821 4129 y Fy(b)823 4133 y FA(g)p FQ(\))897 4145 y FI(k)968 4133 y FO(which)31 b(have)g(the)f(fol)t(lowing)i(pr)l(op)l(erties)p FQ(:)605 4233 y(\(i\))h FO(In)g(every)g(mo)l(dule)g FJ(V)52 b FO(fr)l(om)33 b(the)g(c)l(ate)l(gory)g FL(O)2176 4245 y FI(k)2217 4233 y FO(,)h(the)f(action)g(of)h FJ(L)2830 4245 y FI(n)2907 4233 y FO(is)f(wel)t(l-de\014ne)l(d,)456 4332 y(and)1131 4512 y FQ([)p FJ(L)1211 4524 y FI(m)1273 4512 y FJ(;)14 b(L)1367 4524 y FI(n)1412 4512 y FQ(])23 b(=)g(\()p FJ(m)18 b FL(\000)g FJ(n)p FQ(\))p FJ(L)1891 4524 y FI(m)p FM(+)p FI(n)2064 4512 y FQ(+)g FJ(\016)2184 4524 y FI(m)p FM(+)p FI(n;)p FM(0)2402 4456 y FJ(m)2475 4426 y FM(3)2531 4456 y FL(\000)g FJ(m)p 2402 4493 285 4 v 2503 4569 a FQ(12)2697 4512 y FJ(c;)-2300 b FQ(\(7.4.4\))456 4682 y FO(wher)l(e)1728 4852 y FJ(c)23 b FQ(=)1885 4796 y FJ(k)17 b FQ(dim)d FA(g)p 1885 4833 254 4 v 1890 4909 a FJ(k)21 b FQ(+)d FJ(h)2085 4885 y FE(_)2149 4852 y FJ(:)-1716 b FQ(\(7.4.5\))605 5034 y(\(ii\))27 b FO(The)h(op)l(er)l (ator)f FJ(L)1289 5046 y FI(n)1360 5034 y FO(has)g(de)l(gr)l(e)l(e)g FJ(n)g FO(with)g(r)l(esp)l(e)l(ct)f(to)g(the)h(homo)l(gene)l(ous)g(gr)l (ading,)i(and)1280 5182 y FQ([)p FJ(L)1360 5194 y FI(n)1404 5182 y FJ(;)14 b(a)p FQ([)p FJ(m)p FQ(]])23 b(=)g FL(\000)p FJ(m)14 b(a)p FQ([)p FJ(m)k FQ(+)g FJ(n)p FQ(])p FJ(;)183 b(a)23 b FL(2)g FA(g)p FJ(:)-2164 b FQ(\(7.4.6\))p eop %%Page: 177 11 177 180 bop 1563 226 a FM(7.4.)29 b(FLA)-5 b(T)29 b(CONNECTION)986 b(177)605 425 y FQ(\(iii\))31 b FO(In)e(the)h(Weyl)g(mo)l(dule)g FJ(V)1576 395 y FI(k)1557 449 y(\025)1647 425 y FQ(\()p FO(and)g(thus,)g(in)g FJ(L)2203 395 y FI(k)2203 449 y(\025)2246 425 y FQ(\))p FO(,)g(the)g(op)l(er)l(ator)h FJ(L)2855 437 y FM(0)2921 425 y FO(acts)f(by)1145 619 y FJ(L)1202 631 y FM(0)1239 619 y FJ(v)c FQ(=)d(\(\001)1494 631 y FI(\025)1556 619 y FL(\000)18 b FQ(deg)d FJ(v)s FQ(\))p FJ(v)s(;)184 b FQ(\001)2173 631 y FI(\025)2240 619 y FQ(=)2338 563 y FL(h)p FJ(\025;)14 b(\025)19 b FQ(+)f(2)p FJ(\032)p FL(i)p 2338 600 385 4 v 2355 676 a FQ(2\()p FJ(k)j FQ(+)d FJ(h)2624 652 y FE(_)2672 676 y FQ(\))2732 619 y FJ(:)-2299 b FQ(\(7.4.7\))605 822 y(P)n(art)26 b(\(i\))j(of)e(this)h(prop)r(osition)f(can)g(b)r(e)h(reform)n(ulated)e (as)h(follo)n(ws.)36 b(Let)1582 965 y FJ(V)19 b(ir)25 b FQ(=)e FH(C)15 b FQ(\(\()q FJ(t)p FQ(\)\))p FJ(@)2085 977 y FI(t)2139 965 y FL(\010)j FH(C)d FJ(c)-1856 b FQ(\(7.4.8\))456 1118 y(as)38 b(b)r(efore,)k(this)d(v)n(ector)f(space)g(has)h(top)r (ological)f(basis)g FJ(c;)14 b(L)2462 1130 y FI(n)2549 1118 y FQ(=)42 b FL(\000)p FJ(t)2751 1088 y FI(n)p FM(+1)2879 1118 y FJ(@)2923 1130 y FI(t)2953 1118 y FJ(;)14 b(n)42 b FL(2)g FH(Z)p FQ(.)65 b(W)-7 b(e)456 1218 y(de\014ne)33 b(the)h(structure)g(of)f(Lie)h(algebra)d(on)j FJ(V)19 b(ir)36 b FQ(b)n(y)e(\(7.4.4\))f(\(it)h(can)g(also)e(b)r(e)i(de\014ned) g(in)g(a)456 1317 y(co)r(ordinate-free)f(w)n(a)n(y)-7 b(,)36 b(with)g(the)f(cen)n(tral)g(term)g(giv)n(en)f(as)h(a)f(residue)h (of)g(the)h FJ(f)3064 1287 y FE(0)o(0)q(0)3125 1317 y FJ(g)s FQ(\).)60 b(This)456 1417 y(algebra)33 b(is)j(called)f(the)h FO(Vir)l(asor)l(o)i(algebr)l(a)f FQ(and)f(pla)n(ys)e(a)i(cen)n(tral)e (role)h(in)h(conformal)e(\014eld)456 1517 y(theory;)c(b)n(y)f (de\014nition,)i(it)f(is)f(a)h(cen)n(tral)e(extension)i(of)f(the)h (Witt)h(algebra)d FH(C)15 b FQ(\(\()q FJ(t)p FQ(\)\))p FJ(@)3137 1529 y FI(t)3172 1517 y FQ(.)44 b(Th)n(us,)456 1616 y(part)26 b(\(i\))i(claims)f(that)h(ev)n(ery)e(mo)r(dule)h FJ(V)42 b FL(2)24 b(O)1930 1628 y FI(k)1998 1616 y FQ(is)j(naturally)g (a)g(mo)r(dule)g(o)n(v)n(er)f FJ(V)19 b(ir)29 b FQ(with)f(the)456 1716 y(cen)n(tral)e(c)n(harge)g(equal)h(to)h FJ(k)16 b FQ(dim)f FA(g)p FJ(=)p FQ(\()p FJ(k)20 b FQ(+)e FJ(h)1834 1686 y FE(_)1883 1716 y FQ(\).)605 1815 y(Note)34 b(that)g(when)g (restricted)f(to)g FL(T)1749 1827 y FE(\000)p FM(1)1872 1815 y FQ(=)g FH(C)15 b FQ([[)p FJ(t)p FQ(]])p FJ(@)2190 1827 y FI(t)2225 1815 y FQ(,)35 b(the)g(cen)n(tral)d(term)i(in)g (\(7.4.4\))f(v)-5 b(an-)456 1915 y(ishes;)31 b(th)n(us,)h FL(T)937 1927 y FE(\000)p FM(1)1057 1915 y FQ(is)e(a)h(subalgebra)d(in) j FJ(V)19 b(ir)33 b FQ(and)e(therefore)e(acts)h(on)h FJ(V)19 b FQ(.)46 b(Hence,)31 b(the)g(same)456 2015 y(construction)e (also)h(de\014nes)g(an)h(action)f FL(T)1795 2027 y FM(0)1863 2015 y FQ(on)g FJ(V)19 b FQ(.)46 b(Considering)29 b FL(T)2622 2027 y FM(0)2690 2015 y FQ(as)h(the)h(Lie)g(algebra)e(of)456 2114 y(left-in)n(v)-5 b(arian)n(t)31 b(v)n(ector)f(\014elds)i(on)g(the) g(set)g FJ(S)37 b FQ(of)32 b(all)g(c)n(hoices)e(of)i(lo)r(cal)g (parameter)e(at)i FJ(p)p FQ(,)h(one)456 2214 y(easily)26 b(sees)g(that)i(this)f(action)g(can)g(b)r(e)g(uniquely)h(extended)f(to) g(the)h(action)e(of)h(the)h(sheaf)f(\002)3397 2226 y FI(S)456 2314 y FQ(of)g(all)g(v)n(ector)g(\014elds)g(on)h FJ(S)k FQ(on)27 b(the)h(sheaf)g FJ(V)1843 2326 y FI(S)1914 2314 y FQ(=)23 b FL(O)2068 2326 y FI(S)2135 2314 y FL(\012)18 b FJ(V)g FQ(.)605 2413 y(Let)32 b(us)f(no)n(w)g(consider)g(the)h (general)e(case,)h(when)h(not)g(only)f(the)h(lo)r(cal)f(parameter)f (but)456 2513 y(also)c(the)i(the)g(curv)n(e)f(itself)h(is)f(allo)n(w)n (ed)f(to)i(v)-5 b(ary)e(.)605 2612 y(First)28 b(of)g(all,)f(let)h FJ(C)35 b FQ(b)r(e)28 b(a)f(complex)h(curv)n(e,)f(and)g FJ(t)p FQ(|a)h(formal)f(parameter)f(at)i(the)g(p)r(oin)n(t)456 2712 y FJ(p)22 b FL(2)i FJ(C)6 b FQ(.)37 b(Denote)26 b(b)n(y)f(\002\()p FJ(C)d FL(\000)15 b FJ(p)p FQ(\))26 b(the)g(space)f(of)h(meromorphic)f(v)n(ector)g(\014elds)h(on)f FJ(C)33 b FQ(whic)n(h)25 b(are)456 2812 y(holomorphic)j(outside)h(of)h FJ(p)p FQ(.)43 b(Then)29 b(w)n(e)h(ha)n(v)n(e)e(a)h(Lie)h(algebra)d (homomorphism)i FJ(\015)3099 2824 y FI(p)3147 2812 y FQ(:)f(\002\()p FJ(C)e FL(\000)456 2911 y FJ(p)p FQ(\))d FL(!)g(T)45 b FQ(obtained)24 b(b)n(y)g(expanding)g(a)f(v)n(ector)g (\014eld)i(in)f(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)h FJ(p)g FQ(in)h(p)r(o)n(w)n(er)e(series)456 3011 y(in)k FJ(t)p FQ(.)37 b(Similarly)-7 b(,)28 b(if)g(w)n(e)f(ha)n(v)n(e)f(sev)n(eral)g (mark)n(ed)h(p)r(oin)n(ts)g FJ(p)2261 3023 y FM(1)2298 3011 y FJ(;)14 b(:)g(:)g(:)g(;)g(p)2525 3023 y FI(n)2570 3011 y FQ(,)27 b(w)n(e)h(can)f(de\014ne)h(a)f(map)1239 3166 y FJ(\015)1283 3179 y FI(~)-35 b(p)1344 3166 y FQ(=)1431 3087 y Fy(M)1571 3166 y FJ(\015)1614 3178 y FI(p)1648 3186 y FG(i)1688 3166 y FQ(:)27 b(\002\()p FJ(C)e FL(\000)19 b FJ(~)-43 b(p)p FQ(\))23 b FL(!)g(T)40 b(\010)18 b(\001)c(\001)g(\001) k(\010)g(T)k FJ(:)-2205 b FQ(\(7.4.9\))456 3321 y(On)32 b(the)g(other)g(hand,)i(Suga)n(w)n(ara)29 b(construction)j(giv)n(es)f (a)h(pro)5 b(jectiv)n(e)31 b(action)h(of)g(the)h(direct)456 3421 y(sum)25 b FL(T)35 b(\010)14 b(\001)g(\001)g(\001)f(\010)h(T)46 b FQ(on)25 b FJ(V)42 b FQ(=)22 b FJ(V)1406 3433 y FM(1)1458 3421 y FL(\012)14 b FJ(:)g(:)g(:)f FL(\012)h FJ(V)1774 3433 y FI(n)1819 3421 y FQ(;)26 b(th)n(us,)g(w)n(e)e(get)h(a)g(pro)5 b(jectiv)n(e)24 b(action)h(of)g(\002\()p FJ(C)20 b FL(\000)14 b FJ(~)-42 b(p)o FQ(\))456 3520 y(on)27 b FJ(V)19 b FQ(,)28 b(whic)n(h)f(w)n(e)g(will)h(also)f(denote)g(b)n(y)g FJ(\015)1798 3533 y FI(~)-35 b(p)1836 3520 y FQ(.)605 3674 y FP(Lemma)31 b FQ(7.4.5)p FP(.)40 b FQ(\(i\))30 b FO(The)h(action)g(of)f FQ(\002\()p FJ(C)25 b FL(\000)19 b FJ(~)-43 b(p)p FQ(\))30 b FO(on)g FJ(V)18 b FO(,)31 b(given)f(by)g FJ(\015)2767 3687 y FI(~)-35 b(p)2805 3674 y FO(,)30 b(is)g(a)g(true)f(action,)456 3774 y(not)g(a)h(pr)l(oje)l(ctive)h(one.)605 3874 y FQ(\(ii\))f FO(The)h(actions)g(of)f FQ(\002\()p FJ(C)25 b FL(\000)18 b FJ(p)p FQ(\))30 b FO(and)g FA(g)p FQ(\()p FJ(C)25 b FL(\000)18 b FJ(p)p FQ(\))29 b FO(on)h FJ(V)49 b FO(agr)l(e)l(e)30 b(as)g(fol)t(lows)p FQ(:)1044 4017 y([)p FJ(\015)1111 4030 y FI(~)-35 b(p)1148 4017 y FQ(\()p FJ(\030)t FQ(\))p FJ(;)14 b(a)19 b FL(\012)f FJ(f)9 b FQ(])23 b(=)f FJ(a)d FL(\012)f FJ(\030)t FQ(\()p FJ(f)9 b FQ(\))p FJ(;)99 b(\030)27 b FL(2)c FQ(\002\()p FJ(C)i FL(\000)18 b FJ(p)p FQ(\))p FJ(;)c(a)k FL(\012)g FJ(f)32 b FL(2)23 b FA(g)c FL(\012)f(O)3062 4029 y FI(S)3110 4017 y FJ(:)605 4165 y FQ(\(iii\))37 b FO(The)f(induc)l(e)l(d)h(action)f(of)h FQ(\002\()p FJ(C)29 b FL(\000)23 b FJ(~)-42 b(p)o FQ(\))37 b FO(on)e(the)h(sp)l(ac)l(e)h(of)g(c)l(oinvariants)g FJ(V)3120 4180 y Fk(g)p FM(\()p FI(C)t FE(\000)q FI(~)-35 b(p)p FM(\))3385 4165 y FO(is)456 4264 y(zer)l(o.)605 4419 y FP(Pr)n(oof.)41 b FQ(P)n(art)29 b(\(i\))i(follo)n(ws)f(from)g (the)h(fact)g(that)g(the)g(cen)n(tral)e(term)i(in)g(\(7.4.4\))f(can)g (b)r(e)456 4518 y(written)e(as)g(a)g(residue,)f(and)i(from)e(the)i (fact)g(that)f(the)h(sum)f(of)g(residues)g(of)g(a)g(meromorphic)456 4618 y(1-form)21 b(is)i(equal)f(to)g(zero.)35 b(The)22 b(pro)r(of)g(of)h(part)f(\(ii\))i(is)e(immediate)h(from)f(\(7.4.6\).)35 b(As)23 b(for)f(part)456 4717 y(\(iii\),)30 b(the)g(simplest)g(w)n(a)n (y)e(to)h(pro)n(v)n(e)f(it)i(is)f(to)g(note)h(that)f(\002\()p FJ(C)d FL(\000)20 b FJ(~)-43 b(p)p FQ(\))30 b(is)f(a)g(simple)h(Lie)f (algebra)456 4817 y(\(see)k([)p FK(BFM)p FQ(]\),)j(and)e(therefore)f (has)g(no)h(non-trivial)f(\014nite-dimensional)g(represen)n(tations.) 456 4917 y(Of)23 b(course,)g(this)g(is)h(a)f(v)n(ery)f(arti\014cial)g (pro)r(of.)35 b(A)24 b(more)e(natural)h(pro)r(of)f(can)h(b)r(e)h (obtained)f(from)456 5016 y(the)33 b(theory)e(of)i(c)n(hiral)f (algebras.)49 b(F)-7 b(or)32 b(readers)f(familiar)h(with)h(this)g (theory)-7 b(,)33 b(w)n(e)g(p)r(oin)n(t)f(out)456 5116 y(that)c(the)h(Suga)n(w)n(ara)d(construction)i(in)h(fact)f(sho)n(ws)g (that)h(the)g(generating)e(function)i FJ(L)p FQ(\()p FJ(z)t FQ(\))24 b(=)456 5153 y Fy(P)543 5240 y FI(n)p FE(2)p Fz(Z)686 5216 y FJ(L)743 5228 y FI(n)787 5216 y FJ(z)830 5185 y FE(\000)p FI(n)p FE(\000)p FM(2)1033 5216 y FQ(is)e(a)f(\014eld)h(in)g(the)g(v)n(ertex)e(op)r(erator)g (algebra)g(\(=c)n(hiral)g(algebra)g(on)h(a)g(formal)p eop %%Page: 178 12 178 181 bop 456 226 a FM(178)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(punctured)33 b(disk\))g(generated)e(b)n(y)i(the)g(Kac{Mo)r (o)r(dy)e(curren)n(ts)g FJ(a)p FQ(\()p FJ(z)t FQ(\))h(=)2797 363 y Fy(P)2885 450 y FI(n)p FE(2)p Fz(Z)3013 425 y FQ(\()p FJ(at)3119 395 y FI(n)3165 425 y FQ(\))p FJ(z)3240 395 y FE(\000)p FI(n)p FE(\000)p FM(1)3421 425 y FQ(,)456 525 y FJ(a)23 b FL(2)g FA(g)c FQ(\(see,)i(e.g.,)f([)p FK(K2)p FQ(]\);)i(similarly)-7 b(,)20 b(the)f(Lie)g(algebra)f(\002\()p FJ(C)7 b FL(\000)q FJ(p)p FQ(\))19 b(is)g(a)g(subalgebra)e(in)i(the)h (c)n(hiral)456 625 y(algebra)25 b(asso)r(ciated)h(with)i(the)g(curv)n (e)e FJ(C)f FL(\000)17 b FJ(p)p FQ(.)37 b(But)28 b(since)f(this)g(c)n (hiral)g(algebra)e(is)j(generated)456 724 y(\(in)33 b(an)f(appropriate) f(sense\))h(b)n(y)g(the)h(Kac{Mo)r(o)r(dy)e(curren)n(ts,)i(and)f(these) h(curren)n(ts)e(act)i(on)456 824 y(the)h(space)f(of)g(coin)n(v)-5 b(arian)n(ts)32 b(b)n(y)h(zero,)h(this)g(whole)g(c)n(hiral)e(algebra)g (acts)h(b)n(y)g(zero.)54 b(Details)456 924 y(can)27 b(b)r(e)h(found)g (in)g([)p FK(Gai)p FQ(].)p 3384 924 4 57 v 3388 871 50 4 v 3388 924 V 3437 924 4 57 v 605 1105 a(P)n(art)h(\(iii\))i(of)g(the) g(lemma)f(ma)n(y)g(seem)g(discouraging.)43 b(Note,)32 b(ho)n(w)n(ev)n(er,)d(that)h(what)h(w)n(e)456 1205 y(are)f(lo)r(oking)h (for)g(is)g(an)g(action)g(of)h(\002)1647 1217 y FI(S)1726 1205 y FQ(on)f(the)h(bundle)g(of)f(coin)n(v)-5 b(arian)n(ts,)31 b(not)g(an)h(action)f(of)456 1305 y(\002)521 1317 y FI(C)576 1305 y FQ(,)26 b(so)e(w)n(e)h(do)g(not)g(ha)n(v)n(e)e(a)i(problem)g (with)g(the)h(fact)f(that)g(\002\()p FJ(C)20 b FL(\000)13 b FJ(p)p FQ(\))25 b(acts)f(b)n(y)h(zero.)35 b(In)25 b(fact,)456 1404 y(it)j(will)f(b)r(e)h(useful)g(to)g(us.)605 1504 y(In)35 b(order)f(to)g(de\014ne)h(an)g(action)f(of)h(\002)1842 1516 y FI(S)1890 1504 y FQ(,)i(w)n(e)d(will)h(\014rst)g(lift)h(a)e(v)n (ector)g(\014eld)h(on)f FJ(S)40 b FQ(to)35 b(a)456 1604 y(v)n(ector)26 b(\014eld)i(on)f FJ(C)1059 1616 y FI(S)1108 1604 y FQ(,)g(and)h(then)g(restrict)f(to)g(a)g(formal)g(neigh)n(b)r (orho)r(o)r(d)g(of)g FJ(p)p FQ(.)605 1705 y(Let)d FJ(\022)j FQ(b)r(e)d(a)f(v)n(ector)g(\014eld)h(on)g FJ(S)5 b FQ(.)36 b(Let)24 b(us)g(lift)g(it)h(to)f(a)f(v)n(ector)g(\014eld)2686 1683 y(~)2679 1705 y FJ(\022)k FQ(on)c FJ(C)2915 1717 y FI(S)2975 1705 y FL(\000)11 b FJ(p)p FQ(\()p FJ(S)5 b FQ(\).)35 b(Suc)n(h)456 1805 y(a)29 b(lifting)h(is)g(alw)n(a)n(ys)d (p)r(ossible,)j(whic)n(h)g(follo)n(ws)e(from)i(the)g(fact)f(that)h FJ(\031)g FQ(:)d FJ(C)2853 1817 y FI(S)2921 1805 y FL(\000)19 b FJ(p)p FQ(\()p FJ(S)5 b FQ(\))27 b FL(!)f FJ(S)35 b FQ(is)456 1905 y(a\016ne,)28 b(and)g(therefore)g(de\014nes)g(an)g (exact)g(functor)g(on)g FL(O)r FQ(-coheren)n(t)f(shea)n(v)n(es)g (\(this)i(is)f(where)456 2004 y(w)n(e)f(need)h(to)f(allo)n(w)f(p)r (oles)i(at)f FJ(p)p FQ(\()p FJ(S)5 b FQ(\)!\).)605 2111 y(Let)24 b(us)g(consider)f(the)i(v)n(ector)e(\014eld)1742 2089 y(~)1735 2111 y FJ(\022)k FQ(in)d(a)g(neigh)n(b)r(orho)r(o)r(d)f (of)h(one)g(of)g(the)g(sections)g FJ(p)3297 2123 y FI(i)3324 2111 y FQ(\()p FJ(S)5 b FQ(\))456 2210 y(\(\\mark)n(ed)32 b(p)r(oin)n(t"\).)56 b(Then)34 b(the)h(c)n(hoice)e(of)h(lo)r(cal)f(co)r (ordinate)g FJ(t)2547 2222 y FI(i)2609 2210 y FQ(allo)n(ws)f(us)i(to)g (de\014ne)g(the)456 2310 y(notion)d(of)i(horizon)n(tal)d(v)n(ector)h (\014eld:)46 b(a)32 b(v)n(ector)f(\014eld)h FJ(v)k FQ(in)c(a)g (punctured)h(neigh)n(b)r(orho)r(o)r(d)e(of)456 2412 y FJ(p)498 2424 y FI(i)525 2412 y FQ(\()p FJ(S)5 b FQ(\))28 b(is)f(horizon)n(tal)f(if)j FJ(v)s FQ(\()p FJ(t)p FQ(\))23 b(=)g(0.)36 b(Then)28 b(w)n(e)f(can)h(de\014ne)g(\\v)n(ertical")d(comp) r(onen)n(t)i FJ(\015)3149 2424 y FI(p)3188 2412 y FQ(\()3227 2390 y(~)3220 2412 y FJ(\022)r FQ(\))h(b)n(y)1339 2557 y(~)1332 2579 y FJ(\022)e FQ(=)c FJ(\015)1527 2591 y FI(p)1561 2599 y FG(i)1592 2579 y FQ(\()1631 2557 y(~)1624 2579 y FJ(\022)r FQ(\))d(+)1806 2557 y(~)1799 2579 y FJ(\022)1840 2544 y FM(horiz)1989 2579 y FJ(;)2115 2557 y FQ(~)2109 2579 y FJ(\022)2150 2544 y FM(horiz)2298 2579 y FQ(\()p FJ(t)p FQ(\))24 b(=)e(0)p FJ(:)456 2741 y FQ(Note)27 b(that)h(while)g(one)g(can)f(easily)g(de\014ne)h(the)g (notion)g(of)g(a)f(v)n(ertical)g(v)n(ector)f(\014led)i(on)g FJ(C)3293 2753 y FI(S)3369 2741 y FQ(\()p FJ(v)456 2840 y FQ(is)i(v)n(ertical)g(if)i(its)f(pro)5 b(jection)30 b(to)h FJ(S)k FQ(is)c(zero\),)g(the)h(notion)e(of)h(horizon)n(tal)f(v)n (ector)f(\014eld,)k(nad)456 2947 y(th)n(us,)f(of)g(\\v)n(ertical)e (comp)r(onen)n(t")h FJ(\015)1619 2959 y FI(p)1653 2967 y FG(i)1684 2947 y FQ(\()1723 2925 y(~)1716 2947 y FJ(\022)r FQ(\))h(dep)r(ends)g(on)g(the)g(c)n(hoice)f(of)g(lo)r(cal)h(parameter)e FJ(t)3394 2959 y FI(i)3421 2947 y FQ(.)456 3046 y(If)k(w)n(e)g(c)n(ho)r (ose)f(lo)r(cal)h(co)r(ordinates)f FJ(x)1642 3058 y FI(i)1705 3046 y FQ(on)h FJ(S)5 b FQ(,)36 b(so)d(that)i FJ(\022)i FQ(=)2411 2984 y Fy(P)2513 3046 y FJ(f)2554 3058 y FI(i)2581 3046 y FQ(\()p FJ(x)p FQ(\))p FJ(@)2736 3058 y FI(x)2774 3066 y FG(i)2805 3046 y FQ(,)g(then)d(\()p FJ(x)3139 3058 y FI(i)3168 3046 y FJ(;)14 b(t)p FQ(\))34 b(giv)n(e)456 3153 y(a)29 b(co)r(ordinate)g(system)g(in)h(a)f(neigh)n(b)r(orho)r(o)r (d)g(of)h FJ(p)2048 3165 y FI(i)2075 3153 y FQ(\()p FJ(S)5 b FQ(\),)31 b(and)f(w)n(e)f(can)g(write)2913 3131 y(~)2906 3153 y FJ(\022)g FQ(=)d FJ(g)s FQ(\()p FJ(x;)14 b(t)p FQ(\))p FJ(@)3330 3165 y FI(t)3380 3153 y FQ(+)456 3197 y Fy(P)557 3259 y FJ(f)598 3271 y FI(i)625 3259 y FQ(\()p FJ(x)p FQ(\))p FJ(@)780 3271 y FI(x)818 3279 y FG(i)850 3259 y FQ(.)37 b(Then)28 b FJ(\015)1170 3271 y FI(p)1204 3279 y FG(i)1235 3259 y FQ(\()1274 3237 y(~)1267 3259 y FJ(\022)s FQ(\))23 b(=)h FJ(g)s FQ(\()p FJ(x;)14 b(t)p FQ(\))p FJ(@)1718 3271 y FI(t)1747 3259 y FQ(.)38 b(The)28 b(function)h FJ(g)s FQ(\()p FJ(x;)14 b(t)p FQ(\))28 b(can)g(ha)n(v)n(e) e(p)r(oles)i(at)g FJ(t)3241 3271 y FI(i)3292 3259 y FQ(=)23 b(0,)456 3369 y(so)k(it)h(can)f(b)r(e)h(view)n(ed)f(as)g(a)g(lo)r(cal)g (section)g(of)h FL(O)1983 3381 y FI(S)2031 3369 y FQ(\(\()p FJ(t)2125 3381 y FI(i)2153 3369 y FQ(\)\),)g(and)g(th)n(us)g FJ(\015)2656 3381 y FI(p)2690 3389 y FG(i)2720 3369 y FQ(\()2759 3347 y(~)2752 3369 y FJ(\022)s FQ(\))23 b FL(2)h(O)2994 3381 y FI(S)3060 3369 y FL(\012)18 b(T)k FQ(.)605 3468 y(Rep)r(eating)28 b(this)f(for)h(all)f(p)r(oin)n(ts)g FJ(p)1696 3480 y FI(i)1724 3468 y FQ(,)g(w)n(e)h(de\014ne)1220 3638 y FJ(\015)1264 3651 y FI(~)-35 b(p)1302 3638 y FQ(\()1341 3616 y(~)1334 3638 y FJ(\022)r FQ(\))24 b(=)1518 3559 y Fy(X)1652 3638 y FJ(\015)1695 3650 y FI(p)1729 3658 y FG(i)1760 3638 y FQ(\()1799 3616 y(~)1792 3638 y FJ(\022)r FQ(\))g FL(2)f(O)2033 3650 y FI(S)2100 3638 y FL(\012)18 b FQ(\()p FL(T)40 b(\010)18 b(\001)c(\001)g(\001)k(\010)g(T)j FQ(\))-2223 b(\(7.4.10\))456 3811 y(\(for)27 b FJ(S)h FQ(=)22 b FL(f)p FJ(pt)p FL(g)p FQ(,)27 b(this)h(coincides)f(with)h (the)g(de\014nition)g(\(7.4.9\))o(\).)605 3918 y(No)n(w,)f(let)h(us)g (de\014ne)g(the)g(action)f(of)1777 3896 y(~)1770 3918 y FJ(\022)j FQ(on)d FJ(V)2002 3930 y FI(S)2074 3918 y FQ(=)c FJ(V)37 b FL(\012)18 b(O)2396 3930 y FI(S)2472 3918 y FQ(b)n(y)1374 4065 y(~)1368 4087 y FJ(\022)r FQ(\()p FJ(f)9 b(v)s FQ(\))23 b(=)g(\()p FJ(\022)r FQ(\()p FJ(f)9 b FQ(\)\))p FJ(v)23 b FQ(+)18 b FJ(f)2105 4008 y Fy(X)2153 4185 y FI(i)2239 4087 y FJ(\015)2282 4099 y FI(p)2316 4107 y FG(i)2346 4087 y FQ(\()2385 4065 y(~)2378 4087 y FJ(\022)s FQ(\))p FJ(v)s(;)456 4336 y FQ(where)27 b FJ(\015)739 4348 y FI(p)773 4356 y FG(i)803 4336 y FQ(\()842 4314 y(~)835 4336 y FJ(\022)s FQ(\))h(acts)f(on)g FJ(V)1271 4348 y FI(i)1327 4336 y FQ(b)n(y)g(the)h(Suga)n(w)n(ara)d (construction.)605 4509 y FP(Lemma)31 b FQ(7.4.6)p FP(.)40 b FO(The)25 b(ab)l(ove)g(de\014ne)l(d)f(action)g(of)2155 4487 y FQ(~)2148 4509 y FJ(\022)i FO(on)e FJ(V)2374 4521 y FI(S)2446 4509 y FO(has)h(teh)f(fol)t(lowing)i(pr)l(op)l(erties:)605 4637 y FQ(1.)41 b FO(It)33 b(is)h(c)l(omp)l(atible)g(with)g(the)g (structur)l(e)d(of)j FL(O)2151 4649 y FI(S)2200 4637 y FO(-mo)l(dule:)45 b(for)35 b FJ(f)j FL(2)29 b(O)2918 4649 y FI(S)2967 4637 y FJ(;)14 b(v)32 b FL(2)e FJ(V)3209 4649 y FI(S)3258 4637 y FO(,)k(one)711 4739 y(has)866 4717 y FQ(~)860 4739 y FJ(\022)r FQ(\()p FJ(f)9 b(v)s FQ(\))23 b(=)g(\()p FJ(\022)r FQ(\()p FJ(f)9 b FQ(\)\))p FJ(v)23 b FQ(+)18 b FJ(f)1590 4717 y FQ(~)1584 4739 y FJ(\022)r FQ(\()p FJ(v)s FQ(\))p FO(.)605 4838 y FQ(2.)41 b FO(It)25 b(is)i(c)l(omp)l(atible)g(with)f(the)g(action)h(of)f FA(g)p FQ(\()p FJ(C)2068 4850 y FI(S)2127 4838 y FL(\000)10 b FJ(~)-42 b(p)2244 4850 y FI(S)2291 4838 y FQ(\))26 b FO(on)g FJ(V)2512 4850 y FI(S)2560 4838 y FO(:)37 b(if)27 b FJ(f)k FL(2)24 b(O)2916 4853 y FI(C)2964 4861 y FG(S)3005 4853 y FE(\000)q FI(~)-35 b(p)p FM(\()p FI(S)s FM(\))3191 4838 y FJ(;)14 b(x)24 b FL(2)f FA(g)p FO(,)711 4954 y(then)30 b FQ([)926 4932 y(~)919 4954 y FJ(\022)r(;)14 b(f)9 b(x)p FQ(])23 b(=)g(\()1267 4932 y(~)1260 4954 y FJ(\022)r FQ(\()p FJ(f)9 b FQ(\)\))p FJ(x)p FO(.)605 5116 y FP(Pr)n(oof.)41 b FQ(The)25 b(\014rst)g(part)g(immediately)g(follo)n(ws)g(from)f(the)i (de\014nition;)h(the)e(second)g(one)456 5216 y(follo)n(ws)h(from)h (Theorem)g(7.4.5\(ii\).)p 3384 5216 V 3388 5163 50 4 v 3388 5216 V 3437 5216 4 57 v eop %%Page: 179 13 179 182 bop 1563 226 a FM(7.4.)29 b(FLA)-5 b(T)29 b(CONNECTION)986 b(179)605 425 y FQ(It)26 b(immediately)g(follo)n(ws)f(from)g(part)g (\(ii\))i(of)f(this)g(lemma)f(that)h(w)n(e)g(ha)n(v)n(e)e(a)i(w)n (ell-de\014ned)456 532 y(action)h(of)806 510 y(~)799 532 y FJ(\022)j FQ(on)d(the)h(bundle)h(of)e(coin)n(v)-5 b(arian)n(ts)26 b FJ(\034)1989 544 y FI(S)2060 532 y FQ(=)d FJ(V)2196 544 y FI(S)2244 532 y FJ(=)p FA(g)p FQ(\()p FJ(C)2419 544 y FI(S)2486 532 y FL(\000)c FJ(~)-43 b(p)2611 544 y FI(S)2659 532 y FQ(\))p FJ(V)2739 544 y FI(S)2788 532 y FQ(.)605 711 y FP(Pr)n(oposition)31 b FQ(7.4.7)p FP(.)40 b FO(The)28 b(induc)l(e)l(d)g(action)g(of)2194 689 y FQ(~)2187 711 y FJ(\022)i FO(on)e(the)f(bund)t(le)h(of)g(c)l (oinvariants)h(de-)456 813 y(p)l(ends)h(only)g(on)g FJ(\022)i FO(and)f(not)e(on)h(the)g(choic)l(e)i(of)f(lifting)2215 791 y FQ(~)2208 813 y FJ(\022)s FO(.)39 b(It)29 b(de\014nes)h(a)g(pr)l (oje)l(ctive)i(action)f(of)456 912 y(the)i(Lie)h(algebr)l(a)h FQ(\002)1096 924 y FI(S)1177 912 y FO(on)e(the)h(bund)t(le)g(of)g(c)l (oinvariants,)i(which)f(agr)l(e)l(es)f(with)g(the)f(structur)l(e)456 1012 y(of)d FL(O)619 1024 y FI(S)667 1012 y FO(-mo)l(dule.)605 1185 y FP(Pr)n(oof.)41 b FQ(The)35 b(only)f(non-trivial)g(statemen)n(t) h(is)g(the)g(indep)r(endence)g(of)g(the)g(c)n(hoice)g(of)456 1284 y(lifting.)h(It)25 b(follo)n(ws)f(from)h(the)g(fact)g(that)g(an)n (y)f(t)n(w)n(o)h(liftings)g(di\013er)g(b)n(y)f(a)h(v)n(ertical)f(v)n (ector)f(\014eld.)456 1384 y(On)35 b(the)i(other)e(hand,)j(it)f(follo)n (ws)e(from)g(Theorem)g(7.4.5\(iii\))h(that)g(v)n(ertical)f(\014elds)h (act)g(b)n(y)456 1484 y(zero.)p 3384 1484 4 57 v 3388 1431 50 4 v 3388 1484 V 3437 1484 4 57 v 605 1691 a(This)28 b(completes)f(the)h(pro)r(of)f(of)g(Theorem)g(7.4.1.)p 3384 1691 V 3388 1638 50 4 v 3388 1691 V 3437 1691 4 57 v 605 1791 a(More)32 b(careful)g(analysis)f(also)h(allo)n(ws)f(one)h (to)h(calculate)f(explicitly)g(the)h(failure)g(of)f(the)456 1890 y(connection)c(to)i(b)r(e)f(\015at.)43 b(Using)29 b(the)g(language)f(of)h(t)n(wisted)h FL(D)r FQ(-mo)r(dules)f(dev)n (elop)r(ed)g(in)h(Sec-)456 1990 y(tion)g(6.6)g(and)g(the)h(notion)g(of) f(determinan)n(t)h(line)g(bundle)g FJ(Q)2408 2002 y FI(S)2486 1990 y FQ(de\014ned)g(in)g(Section)g(6.7,)f(the)456 2089 y(result)d(can)g(b)r(e)h(form)n(ulated)f(as)g(follo)n(ws:)605 2262 y FP(Theorem)32 b FQ(7.4.8)p FP(.)40 b FO(Under)25 b(the)g(assumptions)h(of)g(The)l(or)l(em)g FQ(7.4.1)p FO(,)g(the)f(she)l(af)i FJ(\034)3127 2274 y FI(S)3200 2262 y FO(c)l(arries)456 2362 y(a)h(natur)l(al)f(structur)l(e)f(of)j(a) f FL(D)1387 2374 y FI(Q)1439 2358 y FG(c)1475 2362 y FO(-mo)l(dule,)h(wher)l(e)f FJ(c)g FO(is)g(the)g(Vir)l(asor)l(o)h(c)l (entr)l(al)e(char)l(ge)i(de\014ne)l(d)456 2461 y(by)37 b FQ(\(7.4.5\))o FO(.)605 2634 y FQ(W)-7 b(e)39 b(do)e(not)h(giv)n(e)g (a)f(pro)r(of)h(of)g(this)g(theorem,)i(referring)d(the)h(reader)f(to)h ([)p FK(BS)p FQ(].)68 b(The)456 2734 y(pro)r(of)28 b(is)g(based)h(on)f (the)h(fact)g(that)g(the)h(cen)n(tral)d(extension)i(de\014ning)g(the)g (Virasoro)d(algebra)456 2833 y(can)37 b(b)r(e)h(de\014ned)g(using)f (the)i(action)e(of)g(the)h(Lie)g(algebra)e(of)i(v)n(ector)e(\014elds)i (on)f(the)h(space)456 2933 y FH(C)15 b FQ(\(\()569 2918 y FJ(~)574 2933 y(t)6 b FQ(\)\))24 b(=)e FL(\010)850 2945 y FI(i)878 2933 y FH(C)15 b FQ(\(\()p FJ(t)1026 2945 y FI(i)1060 2933 y FQ(\)\))24 b(and)g(the)h(\\univ)n(ersal")c(co)r (cycle)j(de\014ned)g(b)n(y)g(the)g(the)g(subspace)g FH(C)15 b FQ([[)3270 2918 y FJ(~)3275 2933 y(t)6 b FQ(]])23 b(=)456 3037 y FL(\010)521 3049 y FI(i)548 3037 y FH(C)15 b FQ([[)p FJ(t)678 3049 y FI(i)712 3037 y FQ(]])23 b FL(\032)g FH(C)15 b FQ(\(\()982 3021 y FJ(~)987 3037 y(t)6 b FQ(\)\).)38 b(This)28 b(co)r(cycle)f(w)n(as)g(\014rst)h(disco)n(v)n(ered)d(b)n(y)j (T)-7 b(ate)28 b([)p FK(T)-8 b(a)p FQ(])28 b(and)g(redisco)n(v)n(ered) 456 3136 y(under)21 b(di\013eren)n(t)h(names)g(b)n(y)f(man)n(y)h (authors)e(\(see)i([)p FK(BS)p FQ(],)h([)p FK(A)m(CK)q FQ(]\).)35 b(On)22 b(the)g(other)f(hand,)i(it)g(is)456 3240 y(w)n(ell)c(kno)n(wn)g(that)h(for)f(a)g(connected)g(smo)r(oth)h (curv)n(e)e FJ(C)26 b FQ(one)19 b(has)h FH(C)14 b FQ(\()q(\()2584 3225 y FJ(~)2589 3240 y(t)6 b FQ(\)\))p FJ(=)p FQ(\()p FH(C)15 b FQ([[)2858 3225 y FJ(~)2863 3240 y(t)6 b FQ(]])r(+)r FL(O)r FQ(\()p FJ(C)i FL(\000)s FJ(~)-43 b(p)r FQ(\)\))24 b(=)456 3339 y FJ(H)532 3309 y FM(1)569 3339 y FQ(\()p FJ(C)q(;)14 b FL(O)r FQ(\).)72 b(This)39 b(giv)n(es)f(a)h(relation)f(b) r(et)n(w)n(een)h(this)g(co)r(cycle)f(and)h(the)g(determinan)n(t)g(line) 456 3439 y(bundle)24 b(\(recall)f(that)h FJ(Q)1216 3451 y FI(s)1274 3439 y FQ(=)f(det\()p FJ(H)1585 3409 y FM(1)1623 3439 y FQ(\()p FJ(C)1714 3451 y FI(s)1750 3439 y FJ(;)14 b FL(O)r FQ(\)\)\).)37 b(Details)24 b(can)f(b)r(e)h(found)h(in)f([)p FK(BS)p FQ(])g(or)f([)p FK(BFM)o FQ(].)605 3612 y FP(Example)31 b FQ(7.4.9)p FP(.)40 b FQ(Let)19 b(us)h(calculate)e(this)i(\015at)g (connection)f(explicitly)g(in)h(the)g(case)e(when)456 3711 y(the)25 b(curv)n(e)f FJ(C)32 b FQ(is)24 b(\014xed)i(but)f(the)h (p)r(oin)n(t)f FJ(p)g FQ(is)g(allo)n(w)n(ed)e(to)i(mo)n(v)n(e.)35 b(Let)25 b FJ(u)g FQ(b)r(e)h(a)e(lo)r(cal)h(co)r(ordinate)456 3811 y(on)30 b FJ(C)6 b FQ(,)32 b(i.e.)46 b(a)30 b(biholomorphic)f(map) i FJ(u)9 b FQ(:)28 b FJ(C)1826 3781 y FM(0)1892 3811 y FL(!)g FJ(U)9 b FQ(,)31 b(where)f FJ(C)2431 3781 y FM(0)2500 3811 y FQ(is)g(some)g(op)r(en)h(subset)f(of)h FJ(C)6 b FQ(,)456 3911 y(and)23 b FJ(U)33 b FQ(an)23 b(op)r(en)h(subset)f(of)h FH(C)15 b FQ(.)42 b(W)-7 b(e)24 b(will)g(denote)f(b)n(y)g FJ(z)k FQ(a)d(global)e(co)r(ordinate)h(on)g FH(C)45 b FQ(and)23 b(th)n(us,)456 4010 y(on)i FJ(U)9 b FQ(.)37 b(Let)26 b(us)g(de\014ne)g(the)g(follo)n(wing)f(family)h(of)g (punctured)h(curv)n(es)d(o)n(v)n(er)h FJ(U)9 b FQ(:)36 b FJ(C)3029 4022 y FI(U)3108 4010 y FQ(=)22 b FJ(C)g FL(\002)15 b FJ(U)9 b FQ(,)456 4110 y FJ(p)p FQ(\()p FJ(z)t FQ(\))30 b(=)g FJ(u)778 4080 y FE(\000)p FM(1)867 4110 y FQ(\()p FJ(z)t FQ(\),)j(and)f(the)h(lo)r(cal)e(parameter)g(at)h FJ(p)g FQ(giv)n(en)f(b)n(y)h FJ(t)f FQ(=)f FJ(u)21 b FL(\000)g FJ(z)36 b FQ(\(considered)31 b(as)h(a)456 4209 y(function)j(on)g FJ(C)30 b FL(\002)23 b FJ(U)9 b FQ(\).)60 b(Note)35 b(that)g(b)r(oth)h(\()p FJ(z)t(;)14 b(u)p FQ(\))34 b(and)h(\()p FJ(z)t(;)14 b(t)p FQ(\))35 b(can)g(b)r(e)g(considered)g (as)f(lo)r(cal)456 4309 y(co)r(ordinates)26 b(on)h FJ(C)e FL(\002)18 b FJ(U)9 b FQ(.)605 4409 y(In)23 b(this)g(case,)h(ev)n(ery)d (v)n(ector)h(\014eld)h FJ(f)9 b FQ(\()p FJ(z)t FQ(\))p FJ(@)1892 4421 y FI(z)1953 4409 y FQ(on)23 b FJ(U)31 b FQ(admits)23 b(a)g(canonical)f(horizon)n(tal)f(lifting)456 4508 y(to)k FJ(C)d FL(\002)15 b FJ(U)9 b FQ(;)26 b(in)g(terms)g(of)g (the)g(co)r(ordinate)f(system)h(\()p FJ(z)t(;)14 b(u)p FQ(\))25 b(this)h(lifting)h(is)f(giv)n(en)f(b)n(y)h FJ(f)9 b FQ(\()p FJ(z)t FQ(\))p FJ(@)3301 4520 y FI(z)3361 4508 y FL(7!)456 4608 y FJ(f)g FQ(\()p FJ(z)t FQ(\))p FJ(@)657 4620 y FI(z)720 4608 y FQ(+)25 b(0)g FL(\001)h FJ(@)970 4620 y FI(u)1013 4608 y FQ(.)69 b(When)38 b(w)n(e)g(rewrite)g(this)g (in)h(terms)f(of)g(\()p FJ(z)t(;)14 b(t)p FQ(\),)40 b(w)n(e)e(get)g FJ(f)9 b FQ(\()p FJ(z)t FQ(\)\()p FJ(@)3162 4620 y FI(z)3226 4608 y FL(\000)25 b FJ(@)3360 4620 y FI(t)3389 4608 y FQ(\).)456 4708 y(Therefore,)31 b(the)h(action)g(of)g(suc)n(h)f(a)g(v)n (ector)g(\014eld)h(on)f(the)i(bundle)f(of)g(coin)n(v)-5 b(arian)n(ts)30 b(is)h(giv)n(en)456 4807 y(b)n(y)c(\()p FJ(f)9 b(@)697 4819 y FI(z)735 4807 y FQ(\)\()p FJ(\036v)s FQ(\))24 b(=)f FJ(f)9 b FQ(\()p FJ(@)1161 4819 y FI(z)1199 4807 y FJ(\036)p FQ(\))p FJ(v)22 b FQ(+)17 b FJ(f)9 b(\036L)1580 4819 y FE(\000)p FM(1)1669 4807 y FJ(v)30 b FQ(\(recall)d(that)g FJ(L)2229 4819 y FE(\000)p FM(1)2341 4807 y FL(2)c FJ(V)c(ir)30 b FQ(corresp)r(onds)25 b(to)i FL(\000)p FJ(@)3247 4819 y FI(t)3276 4807 y FQ(\).)37 b(In)456 4907 y(other)22 b(w)n(ords,)g(the)h(corresp)r(onding)e(\015at)i(connection)f(on)h FJ(U)31 b FQ(is)23 b(induced)g(from)g(the)g(connection)456 5006 y(on)k FJ(V)37 b FL(\012)18 b(O)805 5018 y FI(S)881 5006 y FQ(giv)n(en)27 b(b)n(y)1661 5182 y FL(r)c FQ(=)g FJ(d)18 b FQ(+)g FJ(L)2042 5194 y FE(\000)p FM(1)2131 5182 y FJ(dz)t(:)p eop %%Page: 180 14 180 183 bop 456 226 a FM(180)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(It)g(is)f(easy)g(to)g(see)g(that)h(for)f(sev)n(eral)f(p)r (oin)n(ts,)i(w)n(e)f(get)1536 580 y FL(r)c FQ(=)g FJ(d)18 b FQ(+)1860 501 y Fy(X)1908 678 y FI(i)1980 580 y FQ(\()p FJ(L)2069 592 y FE(\000)p FM(1)2158 580 y FQ(\))2190 592 y FI(i)2218 580 y FJ(dz)2300 592 y FI(i)2327 580 y FJ(;)-1894 b FQ(\(7.4.11\))456 800 y(where)27 b(\()p FJ(L)785 812 y FE(\000)p FM(1)874 800 y FQ(\))906 812 y FI(i)961 800 y FQ(stands)g(for)g FJ(L)1404 812 y FE(\000)p FM(1)1521 800 y FQ(acting)g(in)h FJ(V)1915 812 y FI(i)1943 800 y FQ(.)605 900 y(Note)h(that)g(in)h(this)f(case)f(ev)n(ery)g(v)n (ector)g(\014eld)h(on)g FJ(S)34 b FQ(can)29 b(b)r(e)g(lifted)h(to)f(a)f (regular)g(v)n(ector)456 1000 y(\014eld)23 b(on)f FJ(C)800 1012 y FI(S)849 1000 y FQ(.)35 b(Therefore,)22 b(w)n(e)h(only)f(need)h (to)g(use)f(the)h(Suga)n(w)n(ara)d(construction)i(for)g(the)h(\014elds) 456 1099 y(from)30 b FH(C)15 b FQ([[)p FJ(t)p FQ(]])p FJ(@)875 1111 y FI(t)939 1099 y FQ(=)28 b FL(T)1077 1111 y FE(\000)p FM(1)1166 1099 y FQ(.)47 b(Since)31 b(the)g(cen)n(tral)f (term)h(in)g(\(7.4.4\))f(v)-5 b(anishes)30 b(when)h(restricted)g(to)456 1199 y FL(T)501 1211 y FE(\000)p FM(1)590 1199 y FQ(,)d(w)n(e)f(get)g (a)g(true)h(action,)f(not)h(a)f(pro)5 b(jectiv)n(e)26 b(one.)605 1353 y(Let)31 b(us)h(consider)e(ev)n(en)h(more)f(sp)r(ecial) h(case)f(than)i(in)f(the)h(previous)e(example,)i(namely)456 1453 y(when)f FJ(C)36 b FQ(=)29 b FH(P)917 1423 y FM(1)953 1453 y FQ(,)k(with)f(mark)n(ed)e(p)r(oin)n(ts)i FJ(z)1794 1465 y FM(1)1830 1453 y FJ(;)14 b(:)g(:)g(:)g(;)g(z)2054 1465 y FI(n)2128 1453 y FL(6)p FQ(=)29 b FL(1)j FQ(and)f(lo)r(cal)g (parameters)f(giv)n(en)h(b)n(y)456 1552 y FJ(t)486 1564 y FI(i)536 1552 y FQ(=)23 b FJ(z)9 b FL(\000)c FJ(z)781 1564 y FI(i)807 1552 y FQ(.)35 b(This)20 b(de\014nes)h(a)g(family)g(of) g(curv)n(es)e(o)n(v)n(er)g FJ(X)2198 1564 y FI(n)2266 1552 y FQ(=)k FH(C)2408 1522 y FI(n)2464 1552 y FL(n)5 b FQ(diagonals)n(.)34 b(Assign)21 b(to)f(these)456 1652 y(p)r(oin)n(ts)i(W)-7 b(eyl)22 b(mo)r(dules)h FJ(V)1292 1622 y FI(k)1273 1675 y(\025)1312 1683 y FF(1)1349 1652 y FJ(;)14 b(:)g(:)g(:)g(;)g(V)1601 1622 y FI(k)1582 1675 y(\025)1621 1683 y FG(n)1666 1652 y FQ(.)35 b(Then,)24 b(b)n(y)e(Prop)r(osition)f(7.3.8,)h(the)h(v)n(ector)e(bundle)i(of)456 1763 y(coin)n(v)-5 b(arian)n(ts)24 b FJ(\034)9 b FQ(\()p FH(P)1043 1732 y FM(1)1080 1763 y FJ(;)14 b(z)1156 1775 y FM(1)1192 1763 y FJ(;)g(:)g(:)g(:)g(;)g(z)1416 1775 y FI(n)1461 1763 y FJ(;)g(V)1564 1732 y FI(k)1546 1786 y(\025)1585 1794 y FF(1)1622 1763 y FJ(;)g(:)g(:)g(:)f(;)h(V)1873 1732 y FI(k)1854 1786 y(\025)1893 1794 y FG(n)1939 1763 y FQ(\))26 b(is)g(a)g(quotien)n(t)g(of)g(the)g(trivial)f(v)n(ector)g (bundle)456 1865 y(with)e(the)h(\014b)r(er)g(\()p FJ(V)1047 1877 y FI(\025)1086 1885 y FF(1)1133 1865 y FL(\012)10 b FJ(:)k(:)g(:)9 b FL(\012)h FJ(V)1437 1877 y FI(\025)1476 1885 y FG(n)1521 1865 y FQ(\))1553 1877 y Fk(g)1616 1865 y FQ(o)n(v)n(er)22 b FJ(X)1859 1877 y FI(n)1904 1865 y FQ(.)35 b(Therefore,)23 b(the)h(construction)f(ab)r(o)n(v)n(e)e (de\014nes)456 1965 y(a)j(\015at)h(connection)f(in)i(this)f(quotien)n (t)f(bundle.)37 b(P)n(assing)23 b(to)i(the)g(dual)g(v)n(ector)e (bundle,)j(w)n(e)f(see)456 2064 y(get)i(a)g(\015at)h(connection)f(in)h (the)g(v)n(ector)e(subbundle)587 2241 y Fy(\000)625 2309 y FJ(\034)9 b FQ(\()p FH(P)754 2274 y FM(1)791 2309 y FJ(;)14 b(z)867 2321 y FM(1)904 2309 y FJ(;)g(:)g(:)g(:)f(;)h(z)1127 2321 y FI(n)1172 2309 y FQ(;)g FJ(V)1276 2274 y FI(k)1257 2329 y(\025)1296 2337 y FF(1)1333 2309 y FJ(;)g(:)g(:)g(:)g(;)g(V)1585 2274 y FI(k)1566 2329 y(\025)1605 2337 y FG(n)1650 2309 y FQ(\))1682 2241 y Fy(\001)1721 2259 y FE(\003)1782 2309 y FL(\032)1870 2241 y Fy(\000)1908 2309 y FJ(V)1956 2321 y FI(\025)1995 2329 y FF(1)2050 2309 y FL(\012)k FJ(:)c(:)g(:)19 b FL(\012)f FJ(V)2380 2321 y FI(\025)2419 2329 y FG(n)2465 2241 y Fy(\001)2503 2259 y FE(\003)2503 2342 y Fk(g)2565 2309 y FQ(=)k(\()p FJ(V)2752 2274 y FE(\003)2732 2329 y FI(\025)2771 2337 y FF(1)2828 2309 y FL(\012)c FJ(:)c(:)g(:)k FL(\012)g FJ(V)3176 2274 y FE(\003)3157 2329 y FI(\025)3196 2337 y FG(n)3242 2309 y FQ(\))3274 2274 y Fk(g)605 2478 y FP(Theorem)32 b FQ(7.4.10)d(\([)p FK(KZ)q FQ(]\))p FP(.)41 b FO(The)g(\015at)f(c)l(onne)l(ction)g (describ)l(e)l(d)i(ab)l(ove)f(c)l(oincides)h(with)456 2578 y(the)30 b(r)l(estriction)g(of)g(the)g(KZ)f(c)l(onne)l(ction)h(in) g FJ(V)1957 2548 y FE(\003)1938 2601 y FI(\025)1977 2609 y FF(1)2033 2578 y FL(\012)18 b FJ(:)c(:)g(:)k FL(\012)g FJ(V)2381 2548 y FE(\003)2362 2601 y FI(\025)2401 2609 y FG(n)2447 2578 y FO(,)30 b(de\014ne)l(d)g(by)38 b FQ(\(KZ)3045 2590 y FI(n)3090 2578 y FQ(\))q FO(.)605 2735 y FQ(A)29 b(pro)r(of)f(of)h(this)g(theorem)f(can)g(b)r(e)i(found)f(in)g(the)g (original)e(pap)r(er)h([)p FK(KZ)p FQ(])h(\(only)g(recom-)456 2834 y(mended)f(for)g(those)g(familiar)f(with)i(the)f(basics)g(of)g (conformal)f(\014eld)h(theory\).)38 b(This)28 b(pro)r(of)g(is)456 2934 y(also)i(rep)r(eated)g(in)i(a)e(n)n(um)n(b)r(er)h(of)g(sources,)g (for)f(example,)i(in)f([)p FK(EFK)p FQ(],)h(in)f(a)g(language)e(more) 456 3034 y(familiar)k(to)h(mathematicians.)57 b(This)34 b(theorem)g(and)g(comparison)f(of)h(the)g(gluing)g(isomor-)456 3133 y(phisms,)29 b(whic)n(h)g(w)n(e)g(will)g(do)g(later,)f(will)i(b)r (e)f(used)g(to)g(sho)n(w)f(that)h(for)g FJ(k)37 b(=)-51 b FL(2)26 b FH(Q)40 b FQ(the)30 b(functor)f(of)456 3233 y(coin)n(v)-5 b(arian)n(ts)28 b(de\014ned)i(ab)r(o)n(v)n(e)f(for)g(gen) n(us)h(zero)e(curv)n(es)h(coincides)h(with)g(the)h(mo)r(dular)e(func-) 456 3333 y(tor)h(de\014ning)h(Drinfeld's)h(category|see)d(Theorem)h (7.9.12.)46 b(In)31 b(particular,)g(this)g(mo)r(dular)456 3432 y(functor)24 b(can)h(b)r(e)g(de\014ned)g(in)g(a)g(w)n(a)n(y)e (whic)n(h)i(do)r(esn't)g(refer)f(to)h(the)g(a\016ne)f(Lie)h(algebras)e (at)i(all.)456 3532 y(Note,)f(ho)n(w)n(ev)n(er,)e(that)i(for)g FJ(k)35 b(=)-52 b FL(2)24 b FH(Q)35 b FQ(this)24 b(mo)r(dular)f (functor)h(can)f(not)g(b)r(e)i(extended)e(to)h(p)r(ositiv)n(e)456 3631 y(gen)n(us.)605 3786 y FP(Example)31 b FQ(7.4.11)p FP(.)39 b FQ(Let)e FJ(C)q(;)16 b(~)-44 b(p)q(;)1607 3770 y(~)1612 3786 y(t)37 b FQ(b)r(e)g(as)f(b)r(efore.)65 b(Cho)r(ose)36 b(one)h(of)g(the)g(p)r(oin)n(ts)g FJ(p)3239 3798 y FI(j)3311 3786 y FQ(and)456 3885 y(consider)20 b(the)h(family)g(of)g(curv)n(es)f FJ(C)11 b FL(\002)5 b FH(C)1686 3855 y FE(\002)1769 3885 y FQ(o)n(v)n(er)19 b FH(C)1994 3855 y FE(\002)2056 3885 y FQ(,)j(with)g(the)f(the)g(mark)n (ed)f(p)r(oin)n(ts)h FJ(p)3130 3897 y FI(i)3157 3885 y FQ(\()p FJ(z)t FQ(\))i(=)g FJ(p)3417 3897 y FI(i)456 3985 y FQ(and)31 b(lo)r(cal)h(parameters)e FJ(t)1286 3997 y FI(i)1314 3985 y FQ(\()p FJ(x;)14 b(z)t FQ(\))30 b(=)g FJ(t)1660 3997 y FI(i)1688 3985 y FQ(\()p FJ(x)p FQ(\))p FJ(;)14 b(x)32 b FL(2)f FJ(C)q(;)14 b(z)34 b FL(2)d FH(C)15 b FQ(,)39 b(except)32 b(for)f FJ(i)g FQ(=)f FJ(j)37 b FQ(when)32 b(w)n(e)f(set)456 4085 y FJ(t)486 4097 y FI(i)513 4085 y FQ(\()p FJ(x;)14 b(z)t FQ(\))36 b(=)g FJ(t)871 4097 y FI(i)898 4085 y FQ(\()p FJ(x)p FQ(\))p FJ(=z)t FQ(.)60 b(By)35 b(the)h(construction)e(of)h(this)h (section,)h(the)e(corresp)r(onding)f(v)n(ector)456 4184 y(bundle)41 b(of)h(coin)n(v)-5 b(arian)n(ts)39 b FJ(\034)51 b FQ(has)41 b(a)g(canonical)f(\015at)h(connection.)78 b(An)42 b(easy)e(calculation,)456 4284 y(similar)23 b(to)i(the)g(one)f (in)g(Example)g(7.4.9,)g(sho)n(ws)f(that)i(this)g(connection)f(is)g (induced)h(from)f(the)456 4383 y(connection)1638 4549 y FL(r)g FQ(=)e FJ(d)d FQ(+)f(\()p FJ(L)2052 4561 y FM(0)2089 4549 y FQ(\))2121 4561 y FI(j)2166 4493 y FJ(dz)p 2166 4530 86 4 v 2188 4606 a(z)456 4715 y FQ(in)26 b(the)h(trivial)e(v)n (ector)g(bundle)i(with)g(\014b)r(er)f FJ(V)1883 4727 y FM(1)1937 4715 y FL(\012)15 b FJ(:)f(:)g(:)i FL(\012)f FJ(V)2258 4727 y FI(n)2304 4715 y FQ(.)36 b(In)27 b(particular,)e(the)i (mono)r(drom)n(y)456 4815 y(of)e(this)h(connection)f(around)g FJ(z)h FQ(=)c(0)k(is)f(giv)n(en)g(b)n(y)g FJ(e)2069 4785 y FM(2)p FI(\031)r FM(i)p FI(L)2208 4793 y FF(0)2244 4815 y FQ(,)h(so)f(if)h FJ(V)2515 4827 y FI(j)2576 4815 y FQ(is)f(an)g(irreducible)g(mo)r(dule)456 4917 y(with)j(highest)f(w)n (eigh)n(t)g FJ(\025)p FQ(,)h(the)g(mono)r(drom)n(y)e(op)r(erator)g(is)i (constan)n(t)f(and)g(equals)g FJ(e)3101 4887 y FM(2)p FI(\031)r FM(i\001)3249 4896 y FG(\025)3291 4917 y FQ(.)605 5016 y(Note)35 b(that)g(if)g(w)n(e)f(pass)g(from)g(1-jet)g(of)h(lo)r (cal)f(parameter)f(to)h(tangen)n(t)h(v)n(ector,)g(w)n(e)f(see)456 5116 y(that)j(the)h(tangen)n(t)e(v)n(ector)g(is)h(giv)n(en)f(b)n(y)h FJ(z)t(@)1899 5128 y FI(t)1924 5136 y FG(j)1959 5116 y FQ(,)i(and)e(th)n(us,)j(as)c FJ(z)41 b FQ(go)r(es)36 b(around)g(the)i(origin)456 5216 y(coun)n(terclo)r(cwise,)19 b(so)g(do)r(es)h(the)g(tangen)n(t)f(v)n(ector.)33 b(Recalling)19 b(the)h(relation)e(b)r(et)n(w)n(een)i(mo)r(dular)p eop %%Page: 181 15 181 184 bop 996 226 a FM(7.5.)30 b(FR)n(OM)f(LOCAL)g(P)-5 b(ARAMETERS)28 b(TO)h(T)-5 b(ANGENT)30 b(VECTORS)419 b(181)456 425 y FQ(functor)28 b(and)h(tensor)f(categories,)f(w)n(e)h (see)h(that)g(in)g(the)g(tensor)f(category)f(corresp)r(onding)g(to)456 525 y(the)h(WZW)g(mo)r(dular)f(functor,)g(the)h(univ)n(ersal)e(t)n (wist)i(is)g(giv)n(en)e(b)n(y)1623 697 y FJ(\022)1662 717 y FI(L)1708 697 y FG(k)1708 737 y(\025)1774 697 y FQ(=)d FJ(e)1901 662 y FM(2)p FI(\031)r FM(i\001)2049 671 y FG(\025)2105 697 y FQ(id)2174 717 y FI(L)2220 697 y FG(k)2220 737 y(\025)456 697 y FQ(\(7.4.12\))456 880 y(\(compare)k(with)j(Remark)e(3.1.20\),)g(whic)n(h)g(agrees)f(with)j (the)f(form)n(ulas)f(for)g(univ)n(ersal)f(t)n(wist)456 979 y(in)i(Drinfeld's)h(category)e(\(Theorem)g(2.2.7\))h(and)g(in)h (the)f(category)f(of)h(represen)n(tations)f(of)h(a)456 1079 y(quan)n(tum)f(group)f(Exercise)f(2.2.6|whic)n(h)h(is)h(another)f (argumen)n(t)g(con\014rming)g(equiv)-5 b(alence)456 1178 y(of)27 b(these)h(categories.)605 1278 y(In)38 b(fact,)k(this)c(v)n (ector)f(bundle)i(on)f FH(C)1815 1248 y FE(\002)1915 1278 y FQ(admits)g(a)g(canonical)f(extension)h(to)g(a)g(v)n(ector)456 1378 y(bundle)26 b(on)g FH(P)892 1348 y FM(1)928 1378 y FQ(,)h(and)f(the)g(connection)g(has)f(logarithmic)g(singularities)g (at)h(0)p FJ(;)14 b FL(1)p FQ(.)36 b(Indeed,)27 b(w)n(e)456 1477 y(can)i(assume)g(that)g FJ(V)1128 1489 y FI(j)1190 1477 y FQ(=)d FJ(L)1338 1447 y FI(k)1338 1501 y(\025)1381 1477 y FQ(.)43 b(Denote)29 b FJ(V)45 b FQ(=)26 b FL(\012)1982 1489 y FI(i)p FE(6)p FM(=)p FI(j)2091 1477 y FJ(V)2139 1489 y FI(i)2167 1477 y FQ(.)43 b(The)30 b(\014b)r(er)f(of)h FJ(\034)39 b FQ(at)29 b(p)r(oin)n(t)h FJ(z)f FL(2)e FH(C)3297 1447 y FE(\002)3389 1477 y FQ(is)456 1580 y(giv)n(en)i(b)n(y)h FJ(\034)829 1592 y FI(z)894 1580 y FQ(=)d FJ(W)1064 1592 y FI(z)1103 1580 y FJ(=)p FA(g)p FQ(\()p FJ(C)f FL(\000)20 b FJ(~)-42 b(p)o FQ(\))p FJ(W)1540 1592 y FI(z)1579 1580 y FQ(,)31 b(where)f FJ(W)1954 1592 y FI(z)2020 1580 y FQ(=)c FJ(V)39 b FL(\012)20 b FJ(L)2340 1550 y FI(k)2340 1603 y(\025)2413 1580 y FQ(do)r(es)30 b(not)g(dep)r(end)h(on)f FJ(z)t FQ(.)43 b(Note)456 1680 y(that)32 b(the)h(subspace)f FA(g)p FQ(\()p FJ(C)c FL(\000)22 b FJ(~)-43 b(p)p FQ(\))p FJ(W)1539 1692 y FI(z)1610 1680 y FQ(dep)r(ends)33 b(on)g FJ(z)t FQ(,)g(since)f(the)h(c)n(hoice)e(of)i(lo)r(cal)f(co)r(ordinate) 456 1779 y(at)i FJ(p)606 1791 y FI(j)675 1779 y FQ(dep)r(ends)g(on)g FJ(z)t FQ(.)56 b(Let)35 b(us)f(c)n(ho)r(ose)f(a)h(di\013eren)n(t)g (trivialization)f(of)h(the)h(v)n(ector)e(bundle)456 1882 y FJ(V)k FL(\012)18 b FJ(L)681 1852 y FI(k)681 1905 y(\025)724 1882 y FQ(,)28 b(namely)-7 b(,)27 b(let)h(us)g(iden)n(tify)1519 2060 y FJ(V)37 b FL(\012)18 b FJ(L)1744 2026 y FI(k)1744 2081 y(\025)1810 2060 y FL(!)23 b FQ(\()p FJ(V)38 b FL(\012)18 b FJ(L)2174 2026 y FI(k)2174 2081 y(\025)2217 2060 y FQ(\))2249 2072 y FI(z)2287 2060 y FJ(;)1567 2202 y(v)k FL(\012)c FJ(v)1752 2214 y FI(j)1810 2202 y FL(7!)23 b FJ(z)1959 2168 y FM(deg)11 b FI(v)2103 2176 y FG(j)2138 2202 y FJ(v)22 b FL(\012)c FJ(v)2323 2214 y FI(j)2358 2202 y FJ(:)456 2387 y FQ(In)33 b(other)g(w)n(ords,)h(in)g(this)g (trivialization)f(constan)n(t)f(sections)h(are)g(giv)n(en)g(b)n(y)g FJ(z)3014 2357 y FM(deg)11 b FI(v)3158 2365 y FG(j)3193 2387 y FJ(v)26 b FL(\012)c FJ(v)3386 2399 y FI(j)3421 2387 y FQ(.)456 2487 y(Then)28 b(one)h(easily)f(sees)g(that)h(in)g (this)g(trivialization,)e(the)i(subspace)f FA(g)p FQ(\()p FJ(C)e FL(\000)19 b FJ(~)-42 b(p)o FQ(\))p FJ(W)3068 2499 y FI(z)3136 2487 y FQ(do)r(es)28 b(not)456 2586 y(dep)r(end)f(on)f FJ(z)t FQ(;)h(th)n(us,)g(it)g(also)e(giv)n(es)h(a)g (trivialization)g(of)g(the)h(v)n(ector)f(bundle)h(of)f(coin)n(v)-5 b(arian)n(ts)456 2686 y(on)30 b FH(C)627 2656 y FE(\002)690 2686 y FQ(,)h(and)f(in)g(this)h(trivialization)e(the)h(\015at)g (connection)g(is)g(giv)n(en)g(b)n(y)g FL(r)d FQ(=)g FJ(d)21 b FQ(+)e(\001)3208 2698 y FI(\025)3252 2686 y FJ(dz)t(=z)t FQ(.)456 2786 y(Therefore,)k(this)h(giv)n(es)e(an)h(extension)g(of)h (our)f(v)n(ector)f(bundle)i(with)g(a)f(\015at)h(connection)f(to)g FH(P)3385 2755 y FM(1)3421 2786 y FQ(,)456 2885 y(and)k(the)h (connection)f(has)g(logarithmic)g(singularities)f(at)h(0)p FJ(;)14 b FL(1)p FQ(.)605 2985 y(Note)39 b(that)h(for)e(this)i (de\014nition)f(of)h(extension)e(to)h FJ(z)46 b FQ(=)c(0,)g(a)c (function)i(of)f(the)h(form)456 3084 y FJ(f)9 b FQ(\()p FJ(z)t FQ(\)\()p FJ(v)685 3096 y FM(1)741 3084 y FL(\012)18 b FJ(:)c(:)g(:)19 b FL(\012)g FJ(v)1064 3096 y FI(j)1118 3084 y FL(\012)g FJ(:)14 b(:)g(:)19 b FL(\012)f FJ(v)1441 3096 y FI(n)1487 3084 y FQ(\))29 b(de\014nes)f(a)g(section)g (holomorphic)g(at)g(0)g(i\013)h FJ(z)2961 3054 y FE(\000)11 b FM(deg)g FI(v)3168 3062 y FG(j)3204 3084 y FJ(f)e FQ(\()p FJ(z)t FQ(\))28 b(is)456 3184 y(regular)d(at)j FJ(z)e FQ(=)d(0)k(\(w)n(e)h(assume)e(that)i FJ(v)1722 3196 y FI(j)1785 3184 y FQ(is)g(homogeneous\).)985 3460 y FK(7.5.)47 b(F)-8 b(rom)30 b(lo)s(cal)h(parameters)h(to)f(tangen)m(t)i(v)m(ectors) 605 3609 y FQ(In)41 b(the)g(previous)e(section,)k(w)n(e)e(ha)n(v)n(e)e (studied)i(prop)r(erties)e(of)i(the)g(v)n(ector)e(spaces)h(of)456 3709 y(coin)n(v)-5 b(arian)n(ts)21 b(for)i(a)g(curv)n(e)f FJ(C)30 b FQ(with)24 b(mark)n(ed)e(p)r(oin)n(ts)h(and)g(c)n(hosen)g(lo) r(cal)g(parameters)e(at)i(these)456 3809 y(p)r(oin)n(ts,)g(or)f(a)h (family)g(of)g(suc)n(h)f(curv)n(es.)35 b(In)23 b(this)g(section)f(w)n (e)h(will)g(sho)n(w)f(that)h(the)h(v)n(ector)d(space)456 3908 y(of)32 b(coin)n(v)-5 b(arian)n(ts)30 b(only)i(dep)r(ends)h(on)f (the)h(1-jet)f(of)h(lo)r(cal)e(parameter:)45 b(if)33 b FJ(t)2846 3920 y FI(i)2874 3908 y FJ(;)14 b(t)2941 3878 y FE(0)2941 3930 y FI(i)3001 3908 y FQ(are)31 b(di\013eren)n(t)456 4008 y(c)n(hoices)36 b(of)i(lo)r(cal)f(parameter)f(at)h FJ(p)1616 4020 y FI(i)1681 4008 y FQ(suc)n(h)g(that)h FJ(d)2111 4020 y FI(p)2145 4028 y FG(i)2176 4008 y FJ(t)2206 4020 y FI(i)2273 4008 y FQ(=)i FJ(d)2421 4020 y FI(p)2455 4028 y FG(i)2485 4008 y FJ(t)2515 3978 y FE(0)2515 4029 y FI(i)2543 4008 y FQ(,)g(then)e(the)g(v)n(ector)e(spaces)456 4120 y FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)705 4104 y(~)710 4120 y(t;)14 b FL(L)p FQ(\))29 b(and)g FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)1311 4099 y(~)1312 4120 y(t)1342 4096 y FE(0)1365 4120 y FJ(;)14 b FL(L)p FQ(\))30 b(are)d(canonically)h(isomorphic,)g(and)g(similarly)g(for)h (families)456 4219 y(of)e(curv)n(es.)605 4319 y(Let)41 b(us)f(start)g(with)h(the)f(case)g(when)g(w)n(e)g(only)g(ha)n(v)n(e)f (one)h(curv)n(e)g FJ(C)6 b FQ(;)47 b(as)40 b(b)r(efore,)j(for)456 4419 y(simplicit)n(y)33 b(w)n(e)f(assume)g(that)i(it)f(has)f(only)h (one)f(mark)n(ed)g(p)r(oin)n(t)h FJ(p)p FQ(.)53 b(Let)33 b(us)g(\014x)g(a)f(non-zero)456 4518 y(tangen)n(t)26 b(v)n(ector)g FJ(v)g FL(2)d FJ(T)1199 4530 y FI(p)1237 4518 y FJ(C)33 b FQ(and)27 b(consider)f(only)h(suc)n(h)f(formal)g(lo)r (cal)h(parameters)e FJ(t)i FQ(at)f FJ(p)h FQ(that)456 4618 y FJ(@)500 4630 y FI(v)539 4618 y FJ(t)c FQ(=)g(1;)j(the)i(set)e (of)h(formal)f(lo)r(cal)g(parameter)f(form)i(a)f(pro-v)-5 b(ariet)n(y)24 b FJ(M)9 b FQ(.)37 b(W)-7 b(e)27 b(w)n(an)n(t)f(to)g (sho)n(w)456 4717 y(that)k(for)g(suc)n(h)g(lo)r(cal)g(parameters)f FJ(t)p FQ(,)i(the)g(v)n(ector)e(spaces)g FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(p;)g(t;)g FL(L)p FQ(\))32 b(can)e(b)r(e)h (canonically)456 4817 y(iden)n(ti\014ed.)54 b(In)33 b(order)f(to)h(do)g (that,)i(consider)d(the)i(family)f(of)h(curv)n(es)e FJ(C)2794 4829 y FI(M)2900 4817 y FQ(=)g FJ(C)d FL(\002)21 b FJ(M)42 b FQ(o)n(v)n(er)456 4917 y FJ(M)9 b FQ(,)27 b(with)h(a)f(mark)n(ed)f(p) r(oin)n(t)i FJ(p)f FQ(\(whic)n(h)h(do)r(es)f(not)g(dep)r(end)h(on)f FJ(m)p FQ(\))h(and)f(the)h(lo)r(cal)f(parameter)456 5016 y(at)i FJ(p)d FL(2)g FJ(C)767 5028 y FI(m)860 5016 y FQ(de\014ned)j(b)n(y)h FJ(m)25 b FL(2)i FJ(M)9 b FQ(.)42 b(As)29 b(discussed)g(in)g(the)h(previous)e(section,)i(this)g (de\014nes)f(a)456 5116 y(canonical)f(\015at)i(connection)f(on)h(the)g (bundle)g(of)g(coin)n(v)-5 b(arian)n(ts)28 b FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(p;)g(t;)g FL(L)p FQ(\).)45 b(W)-7 b(e)30 b(will)g(sho)n(w)456 5216 y(that)j(this)g(v)n(ector)f(bundle)h (with)h(a)e(\015at)h(connection)f(is)h(trivial.)52 b(Indeed,)35 b(it)e(is)g(easy)f(to)h(see)p eop %%Page: 182 16 182 185 bop 456 226 a FM(182)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(that)f FJ(M)37 b FQ(is)27 b(a)g(torsor)f(o)n(v)n(er)g(the) i(group)1268 562 y FJ(K)1339 574 y FM(1)1399 562 y FQ(=)22 b FL(f)p FJ(k)k FL(2)d FQ(Aut)15 b FH(C)g FQ([[)p FJ(t)p FQ(]])29 b FL(j)23 b FQ(\()p FJ(k)s FQ(\()p FJ(t)p FQ(\)\))2285 527 y FE(0)2309 562 y FQ(\(0\))g(=)g(1)p FL(g)p FJ(:)456 699 y FQ(This)h(group)e(can)i(b)r(e)g(explicitly)g(describ)r(ed)g(as)g (the)g(group)f(of)h(all)f(formal)g(p)r(o)n(w)n(er)g(series)g(of)h(the) 456 798 y(form)32 b(1)21 b(+)807 736 y Fy(P)894 757 y FE(1)894 823 y FM(2)978 798 y FJ(a)1022 810 y FI(i)1050 798 y FJ(t)1080 768 y FI(i)1107 798 y FQ(,)34 b(with)g(the)f(group)e (op)r(eration)h(b)r(eing)g(substitution)h(of)g(one)f(series)g(in)n(to) 456 899 y(another.)k(The)27 b(corresp)r(onding)f(Lie)h(algebra)f(is)i (Lie)13 b FJ(K)29 b FQ(=)22 b FL(T)2368 911 y FM(1)2429 899 y FQ(=)h FJ(t)2547 869 y FM(2)2584 899 y FH(C)15 b FQ([[)p FJ(t)p FQ(]])p FJ(@)2804 911 y FI(t)2839 899 y FQ(.)605 999 y(No)n(w)35 b(the)g(trivialit)n(y)f(of)h(the)h(\015at)f (connection)f(follo)n(ws)g(from)h(the)g(follo)n(wing)f(t)n(w)n(o)g (easy)456 1098 y(lemmas)27 b(whose)g(pro)r(ofs)g(are)f(omitted.)605 1249 y FP(Lemma)31 b FQ(7.5.1)p FP(.)40 b FO(L)l(et)31 b(a)i(manifold)g FJ(M)41 b FO(b)l(e)31 b(a)i(torsor)f(over)g(a)g(Lie)h (gr)l(oup)f FJ(K)6 b FO(,)32 b(and)g FJ(E)37 b FO(b)l(e)32 b(a)456 1349 y(ve)l(ctor)c(bund)t(le)h(with)f(a)h(\015at)f(c)l(onne)l (ction)g(over)h FJ(M)9 b FO(.)38 b(Then)28 b(this)h(\015at)f(c)l(onne)l (ction)g(is)h(trivial)g(i\013)456 1448 y(the)h(action)g(of)h FQ(Lie)13 b FJ(K)35 b FO(by)c(ve)l(ctor)f(\014elds)g(on)g FJ(E)k FO(c)l(an)c(b)l(e)g(lifte)l(d)h(to)f(an)f(action)i(of)f FJ(K)36 b FO(on)29 b FJ(E)5 b FO(.)605 1599 y FP(Lemma)31 b FQ(7.5.2)p FP(.)40 b FO(The)35 b(action)h(of)f FQ(Lie)14 b FJ(K)1884 1611 y FM(1)1952 1599 y FQ(=)31 b FL(T)2093 1611 y FM(1)2165 1599 y FO(on)j(an)h(inte)l(gr)l(able)g(mo)l(dule)g FL(L)p FO(,)h(de\014ne)l(d)456 1698 y(by)30 b(the)g(Sugawar)l(a)g(c)l (onstruction,)g(c)l(an)f(b)l(e)h(inte)l(gr)l(ate)l(d)g(to)g(an)g (action)g(of)h FJ(K)2847 1710 y FM(1)2913 1698 y FO(on)f FL(L)p FO(.)605 1849 y FQ(Com)n(bining)40 b(these)h(t)n(w)n(o)f (lemmas,)j(w)n(e)d(get)h(that)g(in)f(our)g(case,)j(the)e(\015at)g (connection)456 1949 y(on)35 b(the)h(bundle)g(of)f(conformal)g(blo)r(c) n(ks)g(is)g(trivial,)i(and)e(th)n(us)h(all)f(the)h(spaces)f FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(p;)g(t;)g FL(L)p FQ(\))456 2048 y(are)39 b(canonically)h(isomorphic.)75 b(Therefore,)43 b(w)n(e)d(can)h(de\014ne)g(the)g(space)f(of)g(coin)n(v)-5 b(arian)n(ts)456 2148 y FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(p;)g(v)s(;)g FL(L)p FQ(\))29 b(as)e(the)h(space)f(of)g(global)g (\015at)h(sections)e(of)i(the)g(bundle)g FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(p;)g(t;)g FL(L)p FQ(\))29 b(on)f FJ(M)9 b FQ(.)605 2299 y FP(Remark)32 b FQ(7.5.3)p FP(.)39 b FQ(Note)24 b(that)h(the)f(action)f(of)h FL(T)2097 2311 y FM(0)2158 2299 y FQ(usually)g(can)f(not)h(b)r(e)g(in)n(tegrated)f(to) h(the)456 2398 y(action)d(of)i(Aut)14 b FH(C)h FQ([[)p FJ(t)p FQ(]].)42 b(Indeed,)23 b(in)g(Aut)15 b FH(C)g FQ([[)p FJ(t)p FQ(]])28 b(one)22 b(has)g FJ(e)2252 2368 y FM(2)p FI(\031)r FM(i)p FI(L)2391 2376 y FF(0)2450 2398 y FQ(=)g(1,)h(but)g(in)g(a)f(highest)g(w)n(eigh)n(t)453 2494 y Fy(b)456 2498 y FA(g)27 b FQ(mo)r(dule)h(with)g(highest)g(w)n (eigh)n(t)e FJ(\025)p FQ(,)j(one)e(has)1565 2634 y FJ(e)1604 2600 y FM(2)p FI(\031)r FM(i)p FI(L)1743 2608 y FF(0)1802 2634 y FQ(=:)22 b FJ(\022)1951 2646 y FI(\025)2018 2634 y FQ(=)h FJ(e)2145 2600 y FM(2)p FI(\031)r FM(i\001)2293 2609 y FG(\025)456 2771 y FQ(whic)n(h)28 b(is)h(not)f(equal)g(to)h(1)f (unless)g(\001)1635 2783 y FI(\025)1704 2771 y FL(2)d FH(Z)o FQ(.)34 b(Therefore,)27 b(w)n(e)i(do)f(need)h(to)f(sp)r(ecify)h (a)f(1-jet)h(of)456 2871 y(lo)r(cal)e(parameter.)605 3021 y(No)n(w)f(let)h(us)g(consider)e(families)i(of)g(curv)n(es.)35 b(Let)27 b FJ(C)2229 3033 y FI(S)2277 3021 y FJ(;)14 b(p)p FQ(\()p FJ(S)5 b FQ(\))27 b(b)r(e)g(a)f(family)h(of)g(curv)n(es)e (with)456 3121 y(a)31 b(\014xed)h(1-jet)g(of)f(lo)r(cal)h(parameter)e FJ(t)i FQ(at)g FJ(p)p FQ(\()p FJ(S)5 b FQ(\).)50 b(If)32 b(w)n(e)g(\014x)f(a)h(formal)f(lo)r(cal)g(parameter)g FJ(t)h FQ(at)456 3221 y FJ(p)p FQ(\()p FJ(S)5 b FQ(\))32 b(with)g(giv)n(en)f(1-jet,)i(then,)g(b)n(y)f(the)g(construction)f(of)g (the)i(previous)d(section,)j(w)n(e)e(get)h(a)456 3320 y(v)n(ector)24 b(bundle)i(of)f(coin)n(v)-5 b(arian)n(ts)24 b(with)i(a)f(\015at)h(connection)f(o)n(v)n(er)f FJ(S)5 b FQ(.)36 b(Let)25 b(us)h(sho)n(w)f(that)g(these)456 3420 y(v)n(ector)h(bundles)i(for)f(di\013eren)n(t)h(c)n(hoices)e(of)i FJ(t)f FQ(can)g(b)r(e)h(canonically)f(iden)n(ti\014ed.)605 3520 y(Using)h(the)g(same)f(idea)h(as)f(in)h(the)h(case)e FJ(S)h FQ(=)23 b FL(f)p FJ(point)p FL(g)p FQ(,)k(consider)g(the)h (pro-v)-5 b(ariet)n(y)26 b FJ(M)32 b FQ(=)456 3619 y FL(f)p FQ(\()p FJ(s;)14 b(t)p FQ(\))23 b FL(j)h FJ(s)g FL(2)g FJ(S)5 b FL(g)p FQ(;)28 b(ob)n(viously)-7 b(,)26 b FJ(M)37 b FQ(is)28 b(a)g(principal)f FJ(K)2101 3631 y FM(1)2138 3619 y FQ(-bundle)h(o)n(v)n(er)f FJ(S)5 b FQ(.)38 b(The)28 b(family)g FJ(C)3218 3631 y FI(S)3294 3619 y FQ(o)n(v)n(er)456 3719 y FJ(S)37 b FQ(de\014nes)32 b(a)g(family)g FJ(C)1212 3731 y FI(M)1319 3719 y FQ(o)n(v)n(er)e FJ(M)41 b FQ(and)32 b(therefore)g(de\014nes)g(a)g(bundle)h(of)f(coin)n (v)-5 b(arian)n(ts)30 b FJ(\034)3370 3731 y FI(M)456 3818 y FQ(with)36 b(a)g(\015at)g(connection)f(o)n(v)n(er)g FJ(M)9 b FQ(.)61 b(Our)36 b(goal)e(is)i(to)g(sho)n(w)f(that)i(this)f (\015at)g(connection)g(is)456 3918 y(trivial)26 b(along)f(the)j(\014b)r (ers)e(of)h(the)g(pro)5 b(jection)26 b FJ(M)31 b FL(!)24 b FJ(S)5 b FQ(.)36 b(A)27 b(con)n(v)n(enien)n(t)f(framew)n(ork)f(for)h (suc)n(h)456 4018 y(pro)r(ofs)g(is)i(pro)n(vided)e(b)n(y)i(the)g (formalism)f(of)g(Harish{Chandra)e(pairs.)605 4168 y FP(Definition)32 b FQ(7.5.4)p FP(.)40 b FQ(A)34 b FO(Harish{Chandr)l(a) 39 b(p)l(air)45 b FQ(is)34 b(a)g(pair)f(\()p FA(g)p FJ(;)14 b(K)6 b FQ(\),)36 b(where)e FA(g)g FQ(is)h(a)f(Lie)456 4268 y(algebra,)25 b(and)i FJ(K)33 b FQ(is)26 b(a)h(Lie)g(group)f(with) i(the)f(Lie)g(algebra)e(Lie)14 b FJ(K)29 b FQ(=)22 b FA(k)i FL(\032)e FA(g)p FQ(.)37 b(W)-7 b(e)27 b(also)f(assume)456 4368 y(that)g(w)n(e)h(are)e(giv)n(en)h(an)g(action)g FJ(Ad)i FQ(of)e FJ(K)32 b FQ(on)26 b FA(g)h FQ(whic)n(h)f(agrees)f (with)i(b)r(oth)g(the)g(standard)f FJ(Ad)456 4467 y FQ(action)h(of)g FJ(K)33 b FQ(on)28 b FA(k)g FQ(and)f FJ(ad)h FQ(action)f(of)g FA(k)h FQ(on)g FA(g)p FQ(.)605 4618 y(As)36 b(usual,)h(w)n(e)f (de\014ne)f(a)h(mo)r(dule)g FJ(V)54 b FQ(o)n(v)n(er)34 b(a)h(Harish{Chandra)f(pair)h(\()p FA(g)p FJ(;)14 b(K)6 b FQ(\))35 b(to)h(b)r(e)g(a)456 4717 y(v)n(ector)26 b(space)h(whic)n(h) g(has)g(an)h(action)f(of)g(b)r(oth)h FA(g)g FQ(and)f FJ(K)6 b FQ(,)27 b(and)h(these)f(actions)g(agree)f(on)i FA(k)p FQ(.)605 4817 y(These)41 b(de\014nitions)g(can)g(b)r(e)g (suitably)g(reform)n(ulated)f(if)h(w)n(e)g(w)n(an)n(t)f(to)h(replace)f (a)h(Lie)456 4917 y(algebra)21 b FA(g)i FQ(b)n(y)g(the)h(sheaf)f(of)g (v)n(ector)f(\014elds)h(on)g(a)g(manifold)g FJ(M)32 b FQ(\(or,)24 b(more)e(generally)-7 b(,)23 b(b)n(y)g(a)g(Lie)456 5016 y(algebroid)i(o)n(v)n(er)g FJ(M)9 b FQ(|see)26 b([)p FK(BFM)p FQ(]\).)37 b(Let)27 b(us)g(assume)f(that)h(w)n(e)g(ha)n(v)n(e) e(a)i(manifold)g FJ(M)35 b FQ(with)28 b(a)456 5116 y(free)g(action)g (of)g(a)h(Lie)f(group)f FJ(K)34 b FQ(suc)n(h)29 b(that)g FJ(M)37 b FQ(is)28 b(a)g(principal)g FJ(K)6 b FQ(-bundle)29 b(o)n(v)n(er)d(a)i(manifold)456 5216 y FJ(S)5 b FQ(.)53 b(W)-7 b(e)33 b(denote)g(b)n(y)g FJ(p)f FQ(:)g FJ(M)41 b FL(!)32 b FJ(S)38 b FQ(the)33 b(pro)5 b(jection.)52 b(Denote)33 b(b)n(y)g(\002)2652 5228 y FI(M)2759 5216 y FQ(the)g(sheaf)g(of)g(v)n(ector)p eop %%Page: 183 17 183 186 bop 1155 226 a FM(7.6.)29 b(F)-7 b(AMILIES)28 b(OF)h(CUR)-7 b(VES)29 b(O)n(VER)g(F)n(ORMAL)g(BASE)577 b(183)456 425 y FQ(\014elds)32 b(on)g FJ(M)9 b FQ(.)52 b(Then)33 b(for)f(ev)n(ery)f FJ(U)40 b FL(\032)31 b FJ(M)9 b FQ(,)33 b(w)n(e)f(ha)n(v)n(e)g(a)g(natural)g(em)n(b)r(edding)g FA(k)g FL(\032)f FQ(\002)3218 437 y FI(M)3291 425 y FQ(\()p FJ(U)9 b FQ(\),)456 525 y(whic)n(h)23 b(is)g(a)g(Lie)g(algebra)f (homomorphism.)34 b(W)-7 b(e)24 b(also)e(ha)n(v)n(e)g(an)h(adjoin)n(t)g (action)g(of)g FJ(K)29 b FQ(on)23 b(\002)3359 537 y FI(X)3421 525 y FQ(.)456 624 y(Therefore,)j(the)h(pair)g(\(\002)1265 636 y FI(M)1338 624 y FJ(;)14 b(K)6 b FQ(\))27 b(is)g(a)g(natural)f (sheaf)h(analogue)e(of)i(a)g(Harish-Chandra)e(pair.)605 765 y FP(Definition)32 b FQ(7.5.5)p FP(.)40 b FQ(Let)27 b FJ(M)t(;)14 b(K)q(;)g FQ(\002)1758 777 y FI(M)1858 765 y FQ(b)r(e)28 b(as)f(ab)r(o)n(v)n(e.)35 b(A)28 b (\014nite-dimensional)f(\(\002)3196 777 y FI(M)3269 765 y FJ(;)14 b(K)6 b FQ(\))p FO(-)456 865 y(mo)l(dule)27 b FQ(is)g(a)g(\014nite-dimensional)g(v)n(ector)e(bundle)j FJ(V)46 b FQ(with)28 b(a)e(\015at)h(connection)g(o)n(v)n(er)e FJ(M)36 b FQ(with)456 964 y(an)28 b(action)g(of)g FJ(K)34 b FQ(on)28 b FJ(V)19 b FQ(,)28 b(whic)n(h)h(agrees)d(an)i(ob)n(vious)f (sense)h(with)h(b)r(oth)g(the)g(action)f(of)g FJ(K)34 b FQ(on)456 1064 y FJ(M)i FQ(and)27 b(with)h(the)g(action)f(of)h FA(k)23 b FL(\032)g FQ(\002)1618 1076 y FI(M)1719 1064 y FQ(b)n(y)k(v)n(ector)g(\014elds)g(on)h FJ(V)18 b FQ(.)605 1205 y(\(A)26 b(not)g(necessarily)e(\014nite-dimensional)h(\(\002)2047 1217 y FI(M)2121 1205 y FJ(;)14 b(K)6 b FQ(\)-mo)r(dule)25 b(can)g(b)r(e)h(de\014ned)g(in)g(a)f(sim-)456 1304 y(ilar)h(w)n(a)n(y) -7 b(,)27 b(replacing)f(\\v)n(ector)g(bundle)i(with)g(a)g(\015at)f (connection")g(b)n(y)g(\\)p FL(D)r FQ(-mo)r(dule.\))605 1404 y(Our)g(main)g(reason)f(in)i(dev)n(eloping)f(this)h(tec)n(hnique)f (is)h(the)g(follo)n(wing)e(lemma.)605 1545 y FP(Lemma)31 b FQ(7.5.6)p FP(.)40 b FO(A)n(ny)24 b(\014nite-dimensional)h FQ(\(\002)2092 1557 y FI(M)2166 1545 y FJ(;)14 b(K)6 b FQ(\))p FO(-mo)l(dule)24 b FJ(V)43 b FO(de\014nes)24 b(a)h(ve)l(ctor)f(bun-)456 1644 y(d)t(le)30 b(with)h(a)f(\015at)f(c)l (onne)l(ction)h FJ(V)1473 1614 y FI(K)1567 1644 y FO(on)g FJ(S)d FQ(=)c FJ(M)t(=K)6 b FO(.)605 1785 y FP(Pr)n(oof.)41 b FQ(F)-7 b(or)32 b(ev)n(ery)f FJ(s)g FL(2)g FJ(S)5 b FQ(,)34 b(de\014ne)e(the)h(v)n(ector)e(space)h FJ(V)2520 1755 y FI(K)2501 1805 y(s)2615 1785 y FQ(=)e(\(\000\()p FJ(M)2907 1797 y FI(s)2943 1785 y FJ(;)14 b(V)19 b FQ(\)\))3111 1755 y FI(K)3175 1785 y FQ(,)34 b(where)456 1884 y FJ(M)537 1896 y FI(s)599 1884 y FQ(=)27 b FJ(p)733 1854 y FE(\000)p FM(1)822 1884 y FQ(\()p FJ(s)p FQ(\))j(is)g(the)h(\014b)r(er)f(of)g (the)g(pro)5 b(jection)29 b FJ(p)9 b FQ(:)29 b FJ(M)36 b FL(!)27 b FJ(S)5 b FQ(.)44 b(It)30 b(is)g(easy)f(to)h(see)g(that)g (these)456 1984 y(v)n(ector)i(spaces)g(form)h(a)g(v)n(ector)f(bundle)i (o)n(v)n(er)e FJ(S)38 b FQ(of)33 b(the)h(same)f(dimension)g(as)g(the)h (original)456 2084 y(bundle)i FJ(V)56 b FQ(\(it)37 b(su\016ces)f(to)g (c)n(ho)r(ose)f(lo)r(cally)g(a)h(section)g(of)g(the)h(pro)5 b(jection)35 b(to)h(sho)n(w)g(this\).)456 2183 y(Note)30 b(that)h(an)n(y)f(section)g FJ(\036)h FQ(of)f(this)h(bundle)g(is)f (killed)h(b)n(y)f(the)h(v)n(ertical)e(v)n(ector)g(\014elds;)j(th)n(us,) 456 2283 y(the)c(quotien)n(t)f(\002)991 2295 y FI(M)1064 2283 y FJ(=)p FQ(\002)1171 2253 y FI(v)1171 2306 y(M)1267 2283 y FQ(=)c(\002)1420 2295 y FI(S)1495 2283 y FQ(acts)k(on)h FJ(V)1848 2253 y FI(K)1912 2283 y FQ(.)p 3384 2283 4 57 v 3388 2230 50 4 v 3388 2283 V 3437 2283 4 57 v 605 2432 a(No)n(w)f(w)n(e)g(ha)n(v)n(e)g(all)g(the)h(prerequisites)e(to)i (pro)n(v)n(e)e(the)i(follo)n(wing)e(theorem.)605 2573 y FP(Theorem)32 b FQ(7.5.7)p FP(.)40 b FO(L)l(et)33 b FJ(C)1455 2585 y FI(S)1537 2573 y FO(b)l(e)h(a)h(family)g(of)g(p)l (ointe)l(d)g(curves)f(over)g(a)h(smo)l(oth)f(b)l(ase)g FJ(S)5 b FO(,)456 2673 y(and)35 b(let)g FJ(L)801 2643 y FI(k)801 2693 y FM(1)841 2673 y FJ(;)14 b(:)g(:)g(:)g(;)g(L)1083 2643 y FI(k)1083 2693 y(n)1162 2673 y FO(b)l(e)35 b(some)g(inte)l(gr)l (able)h(mo)l(dules)f(assigne)l(d)h(to)e(these)h(p)l(oints.)55 b(Then)35 b(we)456 2772 y(have)j(a)f(bund)t(le)g(of)h(c)l(oinvariants)g FJ(\034)1623 2784 y FI(S)1708 2772 y FO(over)g FJ(S)j FO(which)e(c)l(arries)f(a)f(natur)l(al)g(pr)l(oje)l(ctively)i(\015at) 456 2872 y(c)l(onne)l(ction,)24 b(and)e(this)h(bund)t(le)g(is)f (functorial)h(in)f FJ(S)27 b FO(in)22 b(the)h(same)f(sense)g(as)h(in)f (The)l(or)l(em)h FQ(7.4.1)p FO(.)605 3013 y FP(Pr)n(oof.)41 b FQ(T)-7 b(ak)n(e)33 b FJ(M)43 b FQ(=)33 b FL(f)p FQ(\()p FJ(s;)14 b(t)p FQ(\))p FL(g)p FJ(;)g(s)34 b FL(2)g FJ(S;)14 b(t)p FQ({a)34 b(lo)r(cal)f(parameter)g(at)h FJ(p)g FL(2)g FJ(C)2989 3025 y FI(s)3060 3013 y FQ(with)g(giv)n(en)456 3112 y(di\013eren)n(tial.)53 b(Ob)n(viously)-7 b(,)33 b FJ(M)42 b FQ(is)32 b(a)h FJ(K)1696 3082 y FI(n)1741 3112 y FQ({torsor)d(o)n(v)n(er)i FJ(S)5 b FQ(,)34 b(where)f FJ(K)k FQ(=)32 b(Aut)2917 3124 y FM(1)2968 3112 y FH(C)15 b FQ([[)p FJ(t)p FQ(]])39 b(and)33 b(w)n(e)456 3212 y(ha)n(v)n(e)f(a)h (tautological)f(family)i FJ(C)1505 3224 y FI(M)1613 3212 y FQ(of)f(curv)n(es)g(o)n(v)n(er)f FJ(M)42 b FQ(with)34 b(mark)n(ed)f(p)r(oin)n(ts)g(and)h(a)f(lo)r(cal)456 3312 y(parameters)20 b(at)i(these)g(p)r(oin)n(ts.)35 b(By)22 b(the)g(construction)g(of)g(the)g(previous)f(section,)i(this)f (de\014nes)456 3411 y(a)28 b(v)n(ector)f(bundle)i(with)g(a)g(pro)5 b(jectiv)n(ely)27 b(\015at)h(connection)g(o)n(v)n(er)f FJ(M)9 b FQ(.)40 b(By)28 b(Lemma)h(7.5.2,)e(this)456 3511 y(connection)h(is)h(in)n(tegrates)f(to)h(an)g(action)f(of)h FJ(K)6 b FQ(.)41 b(Therefore,)28 b(b)n(y)h(Lemma)g(7.5.6,)f(w)n(e)h(ha) n(v)n(e)f(a)456 3610 y(\015at)f(connection)g(on)h FJ(S)f FQ(=)c FJ(M)t(=K)6 b FQ(.)p 3384 3610 V 3388 3558 50 4 v 3388 3610 V 3437 3610 4 57 v 605 3760 a FP(Cor)n(ollar)-6 b(y)33 b FQ(7.5.8)p FP(.)39 b FO(F)-6 b(or)36 b(a)g(\014xe)l(d)f (\014nite)g(set)g FJ(A)h FO(and)g(a)f(c)l(ol)t(le)l(ction)i(of)f(mo)l (dules)g FJ(L)3315 3730 y FI(k)3315 3780 y(a)3389 3760 y FL(2)456 3859 y(O)524 3829 y FM(in)n(t)522 3883 y FI(k)608 3859 y FO(,)29 b(we)g(have)g(a)g(ve)l(ctor)g(bund)t(le)f(of)i(c)l (oinvariants)f FJ(\034)9 b FQ(\()p FL(f)p FJ(L)2292 3829 y FI(k)2292 3880 y(a)2333 3859 y FL(g)p FQ(\))28 b FO(over)h(the)g(mo)l (duli)g(stack)g FJ(M)3312 3871 y FE(\003)p FI(;A)3419 3859 y FO(,)456 3959 y(which)i(c)l(arries)g(a)f(natur)l(al)f(pr)l(oje)l 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FQ(is)g(a)g(\014xed)g(non-negativ)n(e)e(in)n(teger.)35 b(As)23 b(b)r(efore,)g(w)n(e)g(will)g(not)456 624 y(really)30 b(de\014ne)j FJ(D)1004 594 y FM(\()p FI(n)p FM(\))1101 624 y FQ(;)h(instead,)f(w)n(e)e(will)i(de\014ne)f(the)g(structure)g (sheaf)f(of)h FJ(D)2897 594 y FM(\()p FI(n)p FM(\))2994 624 y FQ(,)h FL(O)r FQ(-mo)r(dules)456 724 y(on)27 b FJ(D)642 694 y FM(\()p FI(n)p FM(\))739 724 y FQ(,)h(family)f(of)h (curv)n(es)e(o)n(v)n(er)g FJ(D)1641 694 y FM(\()p FI(n)p FM(\))1738 724 y FQ(,)i(etc.)605 888 y FP(Definition)k FQ(7.6.1)p FP(.)40 b FQ(The)24 b(structure)h(sheaf)f(of)g FJ(D)2204 858 y FM(\()p FI(n)p FM(\))2326 888 y FQ(is)h(the)g(sheaf)f (of)h(algebras)e FL(O)3235 845 y FM(\()p FI(n)p FM(\))3233 912 y FI(D)3357 888 y FQ(on)456 1007 y FJ(D)29 b FQ(de\014ned)f(b)n(y)g FL(O)1024 964 y FM(\()p FI(n)p FM(\))1022 1032 y FI(D)1144 1007 y FQ(=)23 b FL(O)1298 1019 y FI(S)1346 1007 y FJ(=q)1428 977 y FI(n)p FM(+1)1557 1007 y FL(O)1623 1019 y FI(S)1671 1007 y FQ(.)605 1179 y(One)32 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FL(D)1790 3808 y FM(0)1788 3861 y FI(S)1836 3838 y FQ(,)h(it)f(is)g(natural)f(to)g(giv)n(e)g(the)h (follo)n(wing)f(de\014nition.)605 3996 y FP(Definition)32 b FQ(7.6.4)p FP(.)40 b FQ(A)28 b(\015at)g(connection)g(on)f FJ(D)2154 3966 y FM(\()p FI(n)p FM(\))2279 3996 y FQ(with)i (logarithmic)e(singularities)g(at)456 4096 y FJ(D)j FQ(\()p FJ(l)r(og)16 b(D)r FQ(-connnection)27 b(for)h(short\))f(is)h(a)g(lisse) g(sheaf)f(on)h FJ(D)2371 4066 y FM(\()p FI(n)p FM(\))2496 4096 y FQ(with)h(a)e(structure)h(of)g FL(D)3273 4066 y FM(0)3271 4127 y FI(D)3327 4110 y FF(\()p FG(n)p FF(\))3417 4096 y FQ(-)456 4195 y(mo)r(dule.)605 4348 y(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(follo)n(wing)f(ob)n(vious)f(lemma.)605 4500 y FP(Lemma)31 b FQ(7.6.5)p FP(.)40 b FQ(\(i\))26 b FO(Every)h FJ(l)r(og)16 b(D)27 b FO(\015at)f(c)l(onne)l(ction)f(on)h FJ(S)k FO(de\014nes)c(a)g FJ(l)r(og)16 b(D)27 b FO(\015at)f(c)l(onne)l (c-)456 4603 y(tion)j FJ(D)698 4573 y FM(\()p FI(n)p FM(\))825 4603 y FO(by)h FL(F)h(7!)24 b(F)1199 4573 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FQ({)k(a)g(sheaf)g(of)g(algebras)f FL(O)1454 493 y FM(\()p FI(n)p FM(\))1452 561 y FI(C)1500 569 y FG(D)1586 536 y FQ(on)h FJ(C)1763 548 y FI(D)1854 536 y FQ(with)h(a)f(structure)f(of)i(a)f(\015at)g FL(O)2864 493 y FM(\()p FI(n)p FM(\))2862 561 y FI(D)2961 536 y FQ(-mo)r(dule)h(suc)n(h)456 664 y(that)c FL(O)703 621 y FM(\()p FI(n)p FM(\))701 689 y FI(C)749 697 y FG(D)805 664 y FJ(=q)s FL(O)955 621 y FM(\()p FI(n)p FM(\))953 689 y FI(C)1001 697 y FG(D)1079 664 y FQ(=)c FL(O)1233 676 y FI(C)1281 684 y FG(D)1336 664 y FQ(.)605 764 y(The)28 b(family)f(is)h(called)f FO(non-singular)h FQ(if)g(the)g(family)g FJ(C)2361 776 y FI(D)2449 764 y FQ(is)f(non-singular.)605 937 y(In)33 b(a)f(similar)g(w)n(a)n(y)-7 b(,)33 b(one)f(can)g(de\014ne) h(a)g(notion)f(of)h(families)f(with)i(mark)n(ed)d(p)r(oin)n(ts)i(and) 456 1037 y(lo)r(cal)26 b(parameters)f(at)h(these)h(p)r(oin)n(ts)g(b)n (y)f(adding)h(to)f(the)i(data)e(ab)r(o)n(v)n(e)f(a)i(collection)f(of)h (p)r(oin)n(ts)456 1148 y FJ(p)498 1160 y FI(i)560 1148 y FL(2)35 b FJ(C)709 1160 y FM(0)781 1148 y FQ(and)f(lo)r(cal)g (parameters)f FJ(t)1619 1160 y FI(i)1681 1148 y FL(2)j(O)1840 1105 y FM(\()p FI(n)p FM(\))1838 1158 y FI(p)1872 1166 y FG(i)1972 1148 y FQ(suc)n(h)e(that)h FJ(t)2383 1160 y FI(i)2410 1148 y FQ(\()p FJ(p)2484 1160 y FI(i)2512 1148 y FQ(\))g(=)f(0)55 b(mo)r(d)28 b FJ(q)s(;)14 b FQ(\()p FJ(dt)3144 1160 y FI(i)3172 1148 y FQ(\))3204 1160 y FI(p)3238 1168 y FG(i)3304 1148 y FL(6)p FQ(=)34 b(0)456 1248 y(mo)r(d)27 b FJ(q)s FQ(.)54 b(W)-7 b(e)34 b(can)f(also)f (de\014ne)i(an)f(analogue)e(of)j(the)f FL(O)2271 1260 y FI(S)2319 1248 y FQ(-sheaf)g FL(O)r FQ(\()p FJ(C)2722 1260 y FI(S)2793 1248 y FL(\000)23 b FJ(~)-43 b(p)p FQ(\()p FJ(S)5 b FQ(\)\).)55 b(Namely)-7 b(,)456 1364 y(w)n(e)25 b(de\014ne)i FL(O)883 1320 y FM(\()p FI(n)p FM(\))881 1388 y FI(D)980 1364 y FQ(-mo)r(dule)f FL(O)1367 1333 y FM(\()p FI(n)p FM(\))1465 1364 y FQ(\()p FJ(C)c FL(\000)16 b FJ(~)-43 b(p)p FQ(\))26 b(to)g(b)r(e)h(the)f(space)g(of)g(global)f (sections)g(on)h FJ(C)3149 1376 y FI(D)3236 1364 y FQ(of)g(the)456 1483 y(sheaf)h FL(O)734 1440 y FM(\()p FI(n)p FM(\))732 1507 y FI(C)831 1483 y FQ([)p FJ(t)884 1447 y FE(\000)p FM(1)884 1506 y FI(i)974 1483 y FQ(].)605 1589 y(Ob)n(viously)-7 b(,)30 b(ev)n(ery)f(family)h(of)h(curv)n(es)e(o)n(v)n(er)f FJ(S)35 b FQ(de\014nes)c(a)f(family)g(of)h(curv)n(es)e(o)n(v)n(er)f FJ(D)3324 1559 y FM(\()p FI(n)p FM(\))3421 1589 y FQ(:)456 1701 y(it)h(su\016ces)h(to)f(tak)n(e)g FL(O)1178 1657 y FM(\()p FI(n)p FM(\))1176 1725 y FI(C)1301 1701 y FQ(=)d FL(O)1458 1713 y FI(C)1506 1721 y FG(S)1552 1701 y FJ(=q)1634 1670 y FI(n)p FM(+1)1763 1701 y FL(O)1829 1713 y FI(C)1877 1721 y FG(S)1922 1701 y FQ(;)31 b(w)n(e)e(will)g(call)h(this)f FO(r)l(estriction)h FQ(of)g(the)f(family)456 1807 y FJ(C)515 1819 y FI(S)595 1807 y FQ(to)j FJ(D)772 1777 y FM(\()p FI(n)p FM(\))869 1807 y FQ(\).)51 b(It)32 b(turns)g(out)g(that)h(if)f FJ(C)1767 1819 y FI(D)1860 1807 y FQ(is)g(non-singular,)f(then)i(this)f (statemen)n(t)g(can)g(b)r(e)456 1907 y(rev)n(ersed.)605 2080 y FP(Lemma)f FQ(7.6.7)p FP(.)40 b FO(L)l(o)l(c)l(al)t(ly)30 b(in)e FJ(S)5 b FO(,)29 b(every)g(non-singular)f(family)i(of)f(curves)f (over)h FJ(D)3196 2050 y FM(\()p FI(n)p FM(\))3321 2080 y FO(c)l(an)456 2180 y(b)l(e)i(obtaine)l(d)i(as)e(a)h(r)l(estriction)g (of)g(an)g(analytic)h(family)g(of)f(curves)f(over)i(a)f(neighb)l(orho)l (o)l(d)i(of)456 2279 y FJ(D)d FO(in)f FJ(S)5 b FO(.)3536 2353 y Fq(?!)605 2453 y FQ(Let)28 b(us)f(giv)n(e)g(an)g(example)g(of)h (a)f(singular)g(family)g(o)n(v)n(er)f FJ(D)2452 2423 y FM(\()p FI(n)p FM(\))2549 2453 y FQ(.)605 2627 y FP(Example)31 b FQ(7.6.8)p FP(.)40 b FQ(Let)29 b(dim)14 b FJ(S)31 b FQ(=)26 b(1,)j(and)g(let)h FJ(C)2149 2639 y FI(S)2227 2627 y FQ(b)r(e)f(a)g(family)h(of)f(curv)n(es)f(o)n(v)n(er)g FJ(S)34 b FQ(suc)n(h)456 2726 y(that)26 b FJ(C)693 2738 y FI(s)756 2726 y FQ(is)g(smo)r(oth)g(for)g FJ(s)d FL(6)p FQ(=)f FJ(D)r FQ(,)27 b(and)g FJ(C)1746 2738 y FI(D)1833 2726 y FQ(is)f(the)h(curv)n(e)e(with)i(one)f(double)g(p)r(oin)n(t)h FJ(a)p FQ(,)f(so)g(that)456 2826 y(in)31 b(a)g(neigh)n(b)r(oho)r(o)r(d) g(of)g FJ(a)p FQ(,)i FJ(C)1379 2838 y FI(S)1458 2826 y FQ(has)e(lo)r(cal)g(co)r(ordinates)f FJ(t)2285 2838 y FM(1)2322 2826 y FJ(;)14 b(t)2389 2838 y FM(2)2458 2826 y FQ(and)31 b(the)h(pro)5 b(jection)30 b(is)h(giv)n(en)456 2926 y(b)n(y)c FJ(q)f FQ(=)d FJ(t)752 2938 y FM(1)789 2926 y FJ(t)819 2938 y FM(2)856 2926 y FQ(;)28 b(th)n(us,)g FJ(C)1172 2938 y FM(0)1237 2926 y FQ(is)f(giv)n(en)g(b)n(y)g(equation)g FJ(t)2021 2938 y FM(1)2059 2926 y FJ(t)2089 2938 y FM(2)2149 2926 y FQ(=)c(0.)605 3025 y(Let)28 b(us)f(describ)r(e)g(the)h(corresp)r (onding)d(family)j(of)f(curv)n(es)f(o)n(v)n(er)g FJ(D)2708 2995 y FM(\()p FI(n)p FM(\))2805 3025 y FQ(.)37 b(In)28 b(this)f(case,)g(the)456 3136 y(curv)n(e)f FJ(C)736 3148 y FI(D)824 3136 y FQ(is)h(singular|it)g(has)g(double)g(p)r(oin)n(t)h FJ(a)p FQ(.)37 b(T)-7 b(o)27 b(describ)r(e)g(the)h(sheaf)f FL(O)2960 3093 y FM(\()p 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FQ(\))2281 3616 y FI(n)p FM(+1)2411 3651 y FJ(;)-1978 b FQ(\(7.6.2\))456 3834 y(where)33 b FL(O)r FQ(\()p FJ(t)832 3846 y FM(1)870 3834 y FJ(;)14 b(t)937 3846 y FM(2)974 3834 y FQ(\))34 b(is)f(the)h(ring)f(of)h(germs)e(of)i(analytic)f(functions)h(in)g FJ(t)2722 3846 y FM(1)2759 3834 y FJ(;)14 b(t)2826 3846 y FM(2)2897 3834 y FQ(near)33 b(the)h(origin)456 3934 y FJ(t)486 3946 y FM(1)546 3934 y FQ(=)23 b FJ(t)664 3946 y FM(2)724 3934 y FQ(=)f(0.)605 4033 y(T)-7 b(o)35 b(relate)f(the)i(stalk)f(at)g(the)g(double)h(p)r(oin)n(t)f(with)h(the)f (stalks)g(at)g(nearb)n(y)f(p)r(oin)n(ts,)j(let)456 4133 y(us)32 b(describ)r(e)g FL(O)961 4103 y FM(\()p FI(n)p FM(\))1059 4133 y FQ(\()p FJ(U)9 b FQ(\),)34 b(where)e FJ(U)42 b FQ(is)32 b(a)g(punctured)h(neigh)n(b)r(orho)r(o)r(d)f(of)g FJ(a)h FQ(in)g FJ(C)3018 4145 y FI(D)3078 4133 y FQ(.)52 b(Since)33 b(in)456 4233 y(a)27 b(neigh)n(b)r(orho)r(o)r(d)g(of)h FJ(a)p FQ(,)h(the)f(curv)n(e)f FJ(C)1662 4245 y FI(D)1751 4233 y 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5149 y FM(1)1485 5137 y FQ(\))c FL(\012)f FQ(\()p FH(C)d FQ([)p FJ(t)1758 5149 y FM(2)1801 5137 y FQ(])p FJ(=)p FQ(\()p FJ(t)1928 5149 y FM(2)1965 5137 y FQ(\))1997 5102 y FI(n)p FM(+1)2127 5137 y FQ(\))2159 5020 y Fy(\023)2239 5137 y FL(\010)2322 5020 y Fy(\022)2383 5137 y FL(O)r FQ(\()p FJ(U)2540 5149 y FM(2)2577 5137 y FQ(\))k FL(\012)f FQ(\()p FH(C)d FQ([)p FJ(t)2850 5149 y FM(1)2894 5137 y FQ(])p FJ(=)p FQ(\()p FJ(t)3021 5149 y FM(1)3058 5137 y FQ(\))3090 5102 y FI(n)p FM(+1)3219 5137 y FQ(\))3251 5020 y Fy(\023)p eop %%Page: 186 20 186 189 bop 456 226 a FM(186)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 605 429 y FQ(No)n(w)g(it)i(is)e(easy)g(to)h(see)f(that)h(for)f FJ(f)9 b FQ(\()p FJ(t)1807 441 y FM(1)1845 429 y FJ(;)14 b(t)1912 441 y FM(2)1949 429 y FQ(\))25 b FL(2)g(O)2154 386 y FM(\()p FI(n)p FM(\))2152 453 y FI(C;a)2260 429 y FQ(,)30 b(its)f(restriction)e(to)i(the)g(punctured)456 531 y(neigh)n(b)r(orho)r(o)r(d)d(of)i FJ(a)f FQ(is)h(giv)n(en)e(b)n(y) 1138 669 y FJ(t)1168 634 y FI(k)1168 689 y FM(1)1209 669 y FJ(t)1239 634 y FI(l)1239 689 y FM(2)1299 669 y FL(7!)d FQ(\()p FJ(t)1467 634 y FI(k)1467 689 y FM(1)1508 669 y FJ(t)1538 634 y FI(l)1538 689 y FM(2)1575 669 y FQ(\))c FL(\010)f FQ(\()p FJ(t)1771 634 y FI(k)1771 689 y FM(1)1812 669 y FJ(t)1842 634 y FI(l)1842 689 y FM(2)1880 669 y FQ(\))23 b(=)f(\()p FJ(t)2084 681 y FM(1)2122 669 y FQ(\))2154 634 y FI(k)q FE(\000)p FI(l)2268 669 y FJ(q)2308 634 y FI(l)2352 669 y FL(\010)c FQ(\()p FJ(t)2497 681 y FM(2)2535 669 y FQ(\))2567 634 y FI(l)p FE(\000)p FI(k)2681 669 y FJ(q)2721 634 y FI(k)456 669 y FQ(\(7.6.3\))605 820 y(In)28 b(particular,)f(if)h FJ(l)d(>)e(n)p FQ(,)28 b(then)h(restriction)e(of)h FJ(t)2144 790 y FI(k)2144 840 y FM(1)2185 820 y FJ(t)2215 790 y FI(l)2215 840 y FM(2)2280 820 y FQ(to)g FJ(U)2439 832 y FM(1)2503 820 y FQ(is)g(zero,)f(and)h(if)g FJ(k)f(>)c(n)p FQ(,)28 b(then)456 920 y(restriction)e(of)i FJ(t)978 889 y FI(k)978 940 y FM(1)1019 920 y FJ(t)1049 889 y FI(l)1049 940 y FM(2)1113 920 y FQ(to)g FJ(U)1272 932 y FM(2)1337 920 y FQ(is)f(zero.)605 1071 y(F)-7 b(or)22 b(ev)n(ery)g(family)g FJ(C)1271 1083 y FI(S)1343 1071 y FQ(o)n(v)n(er)f FJ(S)27 b FQ(with)c(mark)n(ed)f(p)r (oin)n(ts)h(and)f(mo)r(dules)h FJ(V)2839 1083 y FI(i)2890 1071 y FL(2)h(O)3037 1041 y FM(in)n(t)3035 1095 y FI(k)3144 1071 y FQ(assigned)456 1171 y(to)h(these)g(p)r(oin)n(ts)g(w)n(e)g(ha)n (v)n(e)f(a)g(sheaf)h(of)g(coin)n(v)-5 b(arian)n(ts)23 b FJ(\034)9 b FQ(\()p FJ(C)2280 1183 y FI(S)2330 1171 y FQ(\))25 b(o)n(v)n(er)f FJ(S)30 b FQ(whic)n(h)25 b(giv)n(es)f(rise)g (to)h(the)456 1273 y(sheaf)32 b FJ(\034)716 1243 y FM(\()p FI(n)p FM(\))847 1273 y FQ(o)n(v)n(er)f FJ(D)1101 1243 y FM(\()p FI(n)p FM(\))1199 1273 y FQ(;)k(if)f FJ(C)1398 1285 y FI(S)1479 1273 y FQ(is)f(a)g(smo)r(oth)g(family)-7 b(,)34 b(then)g FJ(\034)2458 1243 y FM(\()p FI(n)p FM(\))2589 1273 y FQ(is)f(lisse.)52 b(It)34 b(follo)n(ws)e(from)456 1373 y(Lemma)27 b(7.6.2\(ii\))h(that)g(this)g(mo)r(dule)h(can)e(b)r(e)i (de\014ned)f(in)g(terms)g(of)g(the)g FJ(n)p FQ(-th)g(in\014nitesimal) 456 1473 y(neigh)n(b)r(orho)r(o)r(d)e(of)i FJ(D)r FQ(,)f(namely)1414 1710 y FJ(\034)1459 1676 y FM(\()p FI(n)p FM(\))1580 1710 y FQ(=)c FJ(V)1734 1676 y FM(\()p FI(n)p FM(\))1832 1710 y FJ(=)p FA(g)1916 1676 y FM(\()p FI(n)p FM(\))2012 1710 y FQ(\()p FJ(C)i FL(\000)19 b FJ(~)-43 b(p)p FQ(\))p FJ(V)2352 1676 y FM(\()p FI(n)p FM(\))2449 1710 y FJ(;)-2016 b FQ(\(7.6.4\))456 1878 y(where)27 b FA(g)738 1848 y FM(\()p FI(n)p FM(\))835 1878 y FQ(\()p FJ(C)e FL(\000)18 b FJ(~)-42 b(p)o FQ(\))24 b(=)e FA(g)d FL(\012)f(O)1430 1848 y FM(\()p FI(n)p FM(\))1527 1878 y FQ(\()p FJ(C)25 b FL(\000)19 b FJ(~)-43 b(p)p FQ(\),)28 b(and)f FJ(V)2079 1848 y FM(\()p FI(n)p FM(\))2199 1878 y FQ(=)c FJ(V)37 b FL(\012)18 b(O)2523 1835 y FM(\()p FI(n)p FM(\))2521 1902 y FI(D)2621 1878 y FQ(.)605 1977 y(Therefore,)36 b(it)f(is)g(natural)f(to)h(tak)n(e)g(this)g(form)n(ula)f(as)g(the)i (de\014nition)f(of)g(the)h(sheaf)f(of)456 2077 y(coin)n(v)-5 b(arian)n(ts)25 b(for)i(families)h(o)n(v)n(er)e FJ(D)1594 2047 y FM(\()p FI(n)p FM(\))1691 2077 y FQ(.)605 2229 y FP(Pr)n(oposition)31 b FQ(7.6.9)p FP(.)40 b FO(L)l(et)33 b FJ(C)1584 2248 y FI(D)1640 2231 y FF(\()p FG(n)p FF(\))1764 2229 y FO(b)l(e)g(a)h(family)i(of)e(curves)g(with)g(marke)l(d)h(p)l (oints)f(over)456 2330 y FJ(D)527 2300 y FM(\()p FI(n)p FM(\))624 2330 y FO(,)k(with)f(lo)l(c)l(al)h(p)l(ar)l(ameters)f(at)f (these)h(p)l(oints,)i(and)e(inte)l(gr)l(able)2644 2326 y Fy(b)2647 2330 y FA(g)o FO(-mo)l(dules)g(assigne)l(d)g(to)456 2441 y(these)e(p)l(oints.)54 b(L)l(et)34 b FJ(\034)1157 2411 y FM(\()p FI(n)p FM(\))1289 2441 y FO(b)l(e)h(the)g FL(O)1607 2398 y FM(\()p FI(n)p FM(\))1605 2465 y FI(D)1705 2441 y FO(-mo)l(dule)g(de\014ne)l(d)g(by)42 b FQ(\(7.6.4\))o FO(.)54 b(Assume)34 b(that)h FJ(C)3290 2453 y FI(D)3385 2441 y FO(is)456 2547 y(nonsingular.)47 b(Then)33 b FJ(\034)1215 2517 y FM(\()p FI(n)p FM(\))1346 2547 y FO(is)f(lisse)i(and)f(has)g(a)g (natur)l(al)f(structur)l(e)f(of)j(a)f(pr)l(oje)l(ctive)h FL(D)3271 2517 y FM(0)3269 2578 y FI(D)3325 2562 y FF(\()p FG(n)p FF(\))3415 2547 y FO(-)456 2660 y(mo)l(dule)c(such)g(that)f(the) h(action)h(of)f FJ(q)s(@)1668 2672 y FI(q)1735 2660 y FO(on)g FJ(\034)1899 2630 y FM(\(0\))2012 2660 y FQ(=)22 b FJ(\034)2144 2630 y FM(\()p FI(n)p FM(\))2242 2660 y FJ(=q)s(\034)2369 2630 y FM(\()p FI(n)p FM(\))2496 2660 y FO(is)30 b(zer)l(o.)605 2812 y FP(Pr)n(oof.)41 b FQ(By)36 b(Lemma)h(7.6.7,)g(suc)n(h)g(a)f(family)h(can)f(b)r(e)h (obtained)g(as)f(a)g(restriction)g(of)456 2911 y(some)24 b(analytic)h(family)-7 b(.)36 b(No)n(w)25 b(existence)g(of)g(the)h (\015at)f(connection)g(and)g(the)h(fact)f(that)g FJ(\034)3265 2881 y FM(\()p FI(n)p FM(\))3389 2911 y FQ(is)456 3011 y(lisse)j(immediately)h(follo)n(w)g(from)f(Theorem)g(7.4.1)g(and)h (Lemma)g(7.6.7.)39 b(T)-7 b(o)29 b(pro)n(v)n(e)e(that)j FJ(q)s(@)3408 3023 y FI(q)456 3117 y FQ(acts)f(b)n(y)g(zero)g(on)g FJ(\034)1086 3087 y FM(\(0\))1176 3117 y FQ(,)h(just)h(note)e(that)h (for)f(the)h(analytic)f(family)-7 b(,)31 b(w)n(e)e(ha)n(v)n(e)f(a)i(w)n (ell-de\014ned)456 3216 y(action)d(of)g FJ(@)843 3228 y FI(q)880 3216 y FQ(,)h(and)f(th)n(us)h FJ(q)s(@)1359 3228 y FI(q)1419 3216 y FQ(=)22 b(0)55 b(mo)r(d)28 b FJ(q)s FQ(.)605 3316 y(It)f(is)g(also)g(imp)r(ortan)n(t)f(to)h(note)g (that)h(the)f(structure)g(of)g FL(D)2458 3286 y FM(0)2456 3347 y FI(D)2512 3330 y FF(\()p FG(n)p FF(\))2602 3316 y FQ(-mo)r(dule)g(can)g(b)r(e)g(de\014ned)456 3429 y(completely)c(in)g (terms)g(of)g FJ(D)1349 3399 y FM(\()p FI(n)p FM(\))1446 3429 y FQ(,)h(without)g(extending)f(this)g(to)g(a)g(family)g(on)g FJ(S)5 b FQ(.)35 b(Let)23 b(\002)3176 3399 y FM(\()p FI(n)p FM(\))3273 3429 y FQ(\()p FJ(C)16 b FL(\000)456 3532 y FJ(~)-42 b(p)o FQ(\))21 b(b)r(e)f(the)g(space)f(of)h(global)e (sections)i(\(on)f FJ(C)1828 3544 y FI(D)1889 3532 y FQ(\))h(of)g(the)g(sheaf)g(of)f(deriv)-5 b(ations)19 b(of)h FL(O)3022 3502 y FM(\()p FI(n)p FM(\))3119 3532 y FQ(\()p FJ(C)9 b FL(\000)t FJ(~)-43 b(p)p FQ(\)|)456 3631 y(this)22 b(is)f(the)h(in\014nitesimal)g(analogue)e(of)i(the)g (algebra)e(of)h(v)n(ector)f(\014elds.)35 b(Then)22 b(w)n(e)f(can)h (lift)g(an)n(y)456 3731 y(v)n(ector)j(\014eld)j FJ(\022)h FQ(on)e FJ(S)k FQ(whic)n(h)c(is)g(tangen)n(t)g(to)g FJ(D)r FQ(|in)g(particular,)f(the)h(v)n(ector)f(\014eld)h FJ(q)s(@)3182 3743 y FI(q)3219 3731 y FQ(|to)g(a)456 3840 y(\\v)n(ector)21 b(\014eld")966 3819 y(~)959 3840 y FJ(\022)26 b FL(2)d FQ(\002)1167 3810 y FM(\()p FI(n)p FM(\))1264 3840 y FQ(\()p FJ(C)16 b FL(\000)11 b FJ(~)-43 b(p)p FQ(\).)35 b(The)24 b(easiest)e(w)n(a)n(y)g(to)i(pro)n(v)n(e)d(this)j(is)f(to)g (use)h(Lemma)f(7.6.7.)605 3940 y(As)29 b(in)g(the)g(analytic)g(case)f (\(see)g(pro)r(of)h(of)g(Theorem)f(7.4.1\),)g(de\014ne)h(the)g(action)g (of)g FJ(\022)i FQ(on)456 4040 y(the)d(bundle)g(of)f(coin)n(v)-5 b(arian)n(ts)26 b(b)n(y)1342 4277 y FJ(\022)r FQ(\()p FJ(f)9 b(v)s FQ(\))24 b(=)f(\()p FJ(\022)r FQ(\()p FJ(f)9 b FQ(\)\))p FJ(v)22 b FQ(+)c FJ(f)2080 4199 y Fy(X)2128 4375 y FI(i)2213 4277 y FJ(\015)2256 4289 y FI(p)2290 4297 y FG(i)2321 4277 y FQ(\()2360 4255 y(~)2353 4277 y FJ(\022)s FQ(\)\()p FJ(v)s FQ(\))p FJ(:)605 4488 y FQ(The)30 b(same)f(argumen)n(ts)f(as)h(in)h(Theorem)f(7.4.1)f(sho)n(w)h (that)h(this)g(is)f(indeed)h(de\014nes)g(the)456 4588 y(structure)d(of)g(a)g(pro)5 b(jectiv)n(e)27 b FL(D)1427 4558 y FM(0)1425 4619 y FI(D)1481 4602 y FF(\()p FG(n)p FF(\))1571 4588 y FQ(-mo)r(dule)g(on)h(the)g(sheaf)f(of)g(coin)n(v)-5 b(arian)n(ts.)p 3384 4588 4 57 v 3388 4535 50 4 v 3388 4588 V 3437 4588 4 57 v 1188 4767 a FK(7.7.)46 b(Coin)m(v)-5 b(arian)m(ts)32 b(for)g(singular)f(curv)m(es)605 4917 y FQ(In)22 b(this)f(section,)h(w)n(e)f(giv)n(e)g(a)g(description)g(of)g (the)h(v)n(ector)e(space)g FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)r(;)14 b(V)k FQ(\))k(for)f(a)g(singular)456 5016 y(curv)n(e)d FJ(C)6 b FQ(.)35 b(This)19 b(description)g(will)h(b)r (e)g(used)g(in)f(the)h(next)g(section)f(to)h(pro)n(v)n(e)e(that)h(the)h (bundle)h(of)456 5116 y(conformal)c(blo)r(c)n(ks)h(satis\014es)g(the)h (gluing)f(axiom)f(and)i(in)g(particular)e(has)h(regular)f (singularities)456 5216 y(on)27 b(the)h(b)r(oundary)f(of)g(the)h(mo)r (duli)g(space.)p eop %%Page: 187 21 187 190 bop 1215 226 a FM(7.7.)29 b(COINV)-7 b(ARIANTS)29 b(F)n(OR)h(SINGULAR)f(CUR)-7 b(VES)637 b(187)605 425 y FQ(Let)25 b FJ(C)q(;)16 b(~)-44 b(p)q(;)923 410 y(~)928 425 y(t)25 b FQ(b)r(e)h(stable)f(singular)f(curv)n(e)g(with)i(mark)n (ed)e(p)r(oin)n(ts)h(and)g(lo)r(cal)g(parameters)f(at)456 525 y(these)e(p)r(oin)n(ts.)35 b(Cho)r(ose)22 b(mo)r(dules)g FJ(V)1595 537 y FM(1)1633 525 y FJ(;)14 b(:)g(:)g(:)g(;)g(V)1866 537 y FI(n)1934 525 y FL(2)24 b(O)2081 495 y FM(in)n(t)2079 548 y FI(k)2187 525 y FQ(assigned)d(to)i(these)f(p)r(oin)n(ts.)35 b(W)-7 b(e)23 b(de\014ne)456 624 y(the)30 b(space)f(of)g(coin)n(v)-5 b(arian)n(ts)28 b FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)q(;)14 b(V)19 b FQ(\))30 b(\(or,)g(for)f(brevit)n(y)-7 b(,)30 b FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(V)20 b FQ(\)\))31 b(b)n(y)e(the)h(same)f(form)n(ula)456 724 y(as)23 b(for)h(non-singular) f(curv)n(es)g(\(see)i(De\014nition)g(7.3.1\).)35 b(F)-7 b(or)24 b(simplicit)n(y)-7 b(,)25 b(let)g(us)f(only)h(consider)456 824 y(the)j(case)e(when)i FJ(C)34 b FQ(has)27 b(only)g(one)g(double)h (p)r(oin)n(t;)g(general)e(case)h(is)g(completely)h(parallel.)605 923 y(Denote)g(b)n(y)f FJ(C)1070 893 y FE(_)1147 923 y FQ(the)h FO(normalization)i FQ(of)d FJ(C)6 b FQ(,)28 b(i.e.)37 b(the)28 b(non-singular)e(curv)n(e)h(suc)n(h)g(that)h FJ(C)456 1023 y FQ(is)33 b(obtained)g(b)n(y)h(iden)n(tifying)g(p)r(oin) n(ts)f FJ(a)1732 993 y FE(0)1755 1023 y FJ(;)14 b(a)1836 993 y FE(00)1912 1023 y FL(2)33 b FJ(C)6 b FQ(.)55 b(Let)34 b(us)g(c)n(ho)r(ose)e(the)i(lo)r(cal)f(co)r(ordinates)456 1123 y FJ(t)486 1092 y FE(0)509 1123 y FJ(;)14 b(t)576 1092 y FE(00)646 1123 y FQ(near)27 b FJ(a)875 1092 y FE(0)898 1123 y FJ(;)14 b(a)979 1092 y FE(00)1021 1123 y FQ(.)605 1281 y FP(Theorem)32 b FQ(7.7.1)p FP(.)40 b FO(The)30 b(map)1644 1444 y FJ(V)42 b FL(!)1840 1365 y Fy(M)1883 1544 y FI(\025)1979 1444 y FJ(V)c FL(\012)18 b FQ(\()p FJ(L)2237 1410 y FI(k)2237 1465 y(\025)2299 1444 y FL(\012)g FJ(D)r(L)2510 1410 y FI(k)2510 1465 y(\025)2553 1444 y FQ(\))1248 1673 y FJ(v)1288 1685 y FM(1)1344 1673 y FL(\012)g FJ(:)c(:)g(:)19 b FL(\012)f FJ(v)1666 1685 y FI(n)1734 1673 y FL(7!)1840 1594 y Fy(M)1979 1673 y FJ(v)2019 1685 y FM(1)2075 1673 y FL(\012)g FJ(:)c(:)g(:)19 b FL(\012)f FJ(v)2397 1685 y FI(n)2461 1673 y FL(\012)g FQ(1)2586 1685 y FI(\025)2629 1673 y FJ(;)456 1848 y FO(wher)l(e)38 b FJ(D)r(L)826 1818 y FI(k)826 1872 y(\025)907 1848 y FO(is)f(de\014ne)l(d)h(as)g(in)g(Se)l(ction)g FQ(7.1)p FO(,)h(and)f FQ(1)2196 1860 y FI(\025)2276 1848 y FL(2)g FJ(V)2417 1860 y FI(\025)2485 1848 y FL(\012)24 b FJ(V)2640 1818 y FE(\003)2622 1872 y FI(\025)2716 1848 y FL(\032)37 b FJ(L)2875 1818 y FI(k)2875 1872 y(\025)2942 1848 y FL(\012)24 b FJ(D)r(L)3159 1818 y FI(k)3159 1872 y(\025)3239 1848 y FO(is)38 b(the)456 1948 y(c)l(anonic)l(al)30 b FA(g)p FO(-invariant)g(ve)l(ctor,)g(induc)l(es)f(an)h(isomorphism)h (of)g(the)e(sp)l(ac)l(es)h(of)g(c)l(oinvariants)1263 2111 y FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(V)20 b FQ(\))j FL(')1672 2032 y Fy(M)1648 2219 y FI(\025)p FE(2)p FI(P)1783 2199 y FG(k)1774 2238 y FF(+)1835 2111 y FJ(\034)9 b FQ(\()p FJ(C)1977 2076 y FE(_)2027 2111 y FJ(;)14 b(V)37 b FL(\012)18 b FJ(L)2289 2076 y FI(k)2289 2131 y(\025)2351 2111 y FL(\012)g FJ(D)r(L)2562 2076 y FI(k)2562 2131 y(\025)2605 2111 y FQ(\))456 2383 y FO(with)30 b(the)g(mo)l(dules)g FJ(L)1147 2353 y FI(k)1147 2407 y(\025)1190 2383 y FJ(;)14 b(D)r(L)1355 2353 y FI(k)1355 2407 y(\025)1428 2383 y FO(assigne)l(d)30 b(to)g(the)g(p)l(oints)g FJ(a)2280 2353 y FE(0)2303 2383 y FJ(;)14 b(a)2384 2353 y FE(00)2456 2383 y FO(r)l(esp)l(e)l(ctively.)605 2542 y FP(Pr)n(oof.)41 b FQ(The)30 b(basic)g(observ)-5 b(ation)29 b(is)i(that)f FL(O)r FQ(\()p FJ(C)e FL(\000)20 b FJ(~)-43 b(p)p FQ(\))28 b(=)g FL(f)p FJ(f)36 b FL(2)28 b(O)r FQ(\()p FJ(C)2863 2512 y FE(_)2933 2542 y FL(\000)21 b FJ(~)-43 b(p)p FQ(\))28 b FL(j)g FJ(f)9 b FQ(\()p FJ(a)3297 2512 y FE(0)3320 2542 y FQ(\))28 b(=)456 2641 y FJ(f)9 b FQ(\()p FJ(a)582 2611 y FE(0)o(0)624 2641 y FQ(\))p FL(g)p FQ(.)37 b(Therefore,)932 2798 y FA(g)p FQ(\()p FJ(C)25 b FL(\000)18 b FJ(~)-42 b(p)o FQ(\))24 b(=)e FL(f)p FJ(f)32 b FL(2)1555 2777 y FQ(~)1550 2798 y(\000)23 b FL(j)g FQ(\()p FJ(\015)1746 2810 y FI(a)1782 2794 y Fx(0)1827 2798 y FL(\010)18 b FJ(\015)1953 2810 y FI(a)1989 2794 y Fx(0)q(0)2034 2798 y FQ(\))p FJ(f)32 b FL(2)2218 2731 y Fy(\000)2254 2794 y(b)2256 2798 y FA(g)2298 2764 y FM(+)2371 2798 y FL(\010)2452 2794 y Fy(b)2454 2798 y FA(g)2496 2764 y FM(+)2569 2798 y FL(\010)19 b FQ(\001\()p FA(g)p FQ(\))2828 2731 y Fy(\001)2889 2798 y FL(\032)2975 2794 y Fy(b)2977 2798 y FA(g)f FL(\010)3118 2794 y Fy(b)3120 2798 y FA(g)p FL(g)-2748 b FQ(\(7.7.1\))456 2968 y(where)27 b(\001\()p FA(g)p FQ(\))c(=)g FL(f)p FJ(x)18 b FL(\010)g FJ(x)p FL(g)p FJ(;)c(x)24 b FL(2)f FA(g)p FQ(,)1545 2947 y(~)1540 2968 y(\000)g(=)f FA(g)p FQ(\()p FJ(C)1841 2938 y FE(_)1909 2968 y FL(\000)d FJ(~)-43 b(p)18 b FL(\000)g FJ(a)2179 2938 y FE(0)2221 2968 y FL(\000)g FJ(a)2348 2938 y FE(00)2390 2968 y FQ(\).)605 3068 y(Next,)28 b(let)g(us)g(de\014ne)f(the)h FJ(U)1504 3064 y Fy(b)1506 3068 y FA(g)1548 3080 y FI(k)1607 3068 y FL(\012)18 b FJ(U)1754 3064 y Fy(b)1756 3068 y FA(g)1798 3080 y FI(k)1839 3068 y FQ(-mo)r(dule)27 b FJ(U)37 b FQ(as)27 b(follo)n(ws:)1589 3227 y FJ(U)32 b FQ(=)23 b(Ind)1888 3188 y FI(U)1938 3185 y Fu(b)1939 3188 y Fk(g)1974 3197 y FG(k)2011 3188 y FE(\012)p FI(U)2113 3185 y Fu(b)2114 3188 y Fk(g)2149 3197 y FG(k)1899 3251 y FM(~)1888 3266 y FI(U)2203 3227 y FH(C)15 b FK(1)456 3402 y FQ(where)702 3381 y(~)687 3402 y FJ(U)32 b FL(\032)23 b FJ(U)921 3414 y FI(k)959 3398 y Fy(b)961 3402 y FA(g)r FL(\012)r FJ(U)1129 3414 y FI(k)1167 3398 y Fy(b)1169 3402 y FA(g)c FQ(is)g(the)h (subalgebra)d(generated)h(b)n(y)2322 3398 y Fy(b)2324 3402 y FA(g)2366 3372 y FM(+)2423 3402 y FL(\012)r FQ(1)p FJ(;)c FQ(1)r FL(\012)2678 3398 y Fy(b)2680 3402 y FA(g)2722 3372 y FM(+)2774 3402 y FJ(;)g(x)r FL(\012)r FQ(1)r(+)r(1)r FL(\012)r FJ(x;)g(x)21 b FL(2)j FA(g)p FQ(,)456 3501 y(whic)n(h)j(acts)g(trivially)g(on)g FH(C)15 b FQ(:)1106 3652 y(\()1136 3648 y Fy(b)1138 3652 y FA(g)1180 3618 y FM(+)1253 3652 y FL(\012)j FQ(1\))p FK(1)23 b FQ(=)g(\(1)18 b FL(\012)1742 3648 y Fy(b)1744 3652 y FA(g)1786 3618 y FM(+)1841 3652 y FQ(\))p FK(1)23 b FQ(=)g(\()p FJ(a)18 b FL(\012)g FQ(1)g(+)g(1)g FL(\012)g FJ(a)p FQ(\))p FK(1)23 b FQ(=)g(0)p FJ(:)-2339 b FQ(\(7.7.2\))605 3813 y(\(By)34 b(P)n(oincare-Birkho\013-Witt)e(theorem,)k FJ(U)43 b FQ(is)34 b(isomorphic)f(to)h FJ(U)9 b FA(g)23 b FL(\012)f FQ(\()p FJ(U)3038 3809 y Fy(b)3040 3813 y FA(g)3082 3783 y FE(\000)3138 3813 y FQ(\))3170 3783 y FE(\012)p FM(2)3294 3813 y FQ(as)34 b(a)456 3913 y(graded)26 b(v)n(ector)g(space.\))605 4012 y(Since)20 b FJ(U)28 b FQ(is)19 b(a)h(\()p FJ(U)9 b FQ(\()1164 4008 y Fy(b)1166 4012 y FA(g)p FQ(\))1240 4024 y FI(k)1281 4012 y FQ(\))1313 3982 y FE(\012)p FM(2)1402 4012 y FQ(-mo)r(dule,)21 b(w)n(e)e(can)h(de\014ne)f(the)h(space)f(of)g (coin)n(v)-5 b(arian)n(ts)18 b FJ(\034)9 b FQ(\()p FJ(C)3258 3982 y FE(_)3308 4012 y FJ(;)15 b(~)-43 b(p)r FL([)456 4112 y FJ(a)500 4082 y FE(0)541 4112 y FL([)19 b FJ(a)659 4082 y FE(00)701 4112 y FJ(;)14 b(V)38 b FL(\012)18 b FJ(U)9 b FQ(\).)605 4278 y FP(Lemma)31 b FQ(7.7.2)p FP(.)40 b FO(The)24 b(map)f FJ(v)k FL(7!)c FJ(v)6 b FL(\012)s FK(1)22 b FO(is)i(an)e(isomorphism)j FJ(\034)9 b FQ(\()p FJ(C)q(;)14 b(V)21 b FQ(\))2851 4231 y FE(\030)2822 4278 y FL(\000)-39 b(!)23 b FJ(\034)9 b FQ(\()p FJ(C)3096 4248 y FE(_)3146 4278 y FJ(;)14 b(V)22 b FL(\012)s FJ(U)9 b FQ(\))p FO(.)605 4437 y FQ(The)32 b(pro)r(of)f(of)g(this)h(lemma)f (is)h(more)f(or)f(less)h(standard:)44 b(one)31 b(has)g(to)h(c)n(hec)n (k)e(that)i(this)456 4536 y(map)21 b(is)h(w)n(ell-de\014ned,)g(whic)n (h)g(follo)n(ws)e(from)i(\(7.7.2\))o(;)i(injectivit)n(y)e(follo)n(ws)e (from)h(the)h(fact)g(that)456 4636 y FJ(U)41 b FQ(is)33 b(free)f(o)n(v)n(er)f FJ(U)1054 4632 y Fy(b)1056 4636 y FA(g)1098 4606 y FE(\000)1175 4636 y FL(\012)22 b FJ(U)1326 4632 y Fy(b)1328 4636 y FA(g)1370 4606 y FE(\000)1425 4636 y FQ(.)52 b(Pro)r(of)32 b(of)g(surjectivit)n(y)g(is)h(is)f(only)h (sligh)n(tly)f(more)g(di\016cult:)456 4735 y(it)38 b(su\016ces)g(to)g (pro)n(v)n(e)e(that)i(for)g(ev)n(ery)f FJ(v)43 b FL(2)e FJ(V)5 b(;)14 b(u)40 b FL(2)h FJ(u)d FQ(one)f(can)h(\014nd)h FJ(v)2830 4705 y FE(0)2894 4735 y FL(2)h FJ(V)57 b FQ(suc)n(h)38 b(that)456 4836 y FJ(v)16 b FL(\012)e FJ(u)22 b FL(\021)h FJ(v)792 4806 y FE(0)829 4836 y FL(\012)14 b FK(1)54 b FQ(mo)r(d)28 b(Im)1355 4815 y(~)1349 4836 y(\000.)36 b(It)26 b(follo)n(ws)e(from)h(the)h(fact)f(that)h(for)e(ev)n(ery)g FJ(a)14 b FL(\010)g FJ(b)22 b FL(2)3104 4832 y Fy(b)3106 4836 y FA(g)14 b FL(\010)3239 4832 y Fy(b)3241 4836 y FA(g)o FJ(;)g(u)22 b FL(2)456 4936 y FJ(u)37 b FQ(there)h(exists)g(a)g (function)h FJ(f)49 b FL(2)42 b FQ(\000)c(suc)n(h)g(that)g(\()p FJ(\015)2160 4948 y FI(q)2192 4932 y Fx(0)2245 4936 y FL(\010)25 b FJ(\015)2378 4948 y FI(q)2410 4932 y Fx(0)q(0)2456 4936 y FQ(\)\()p FJ(f)9 b FQ(\))p FJ(u)40 b FQ(=)h FJ(au)p FQ(,)f(and)e(therefore)456 5036 y FJ(v)21 b FL(\012)d FQ(\()p FJ(a)h FL(\010)f FJ(b)p FQ(\))p FJ(u)k FL(\021)h(\000)p FQ(\()p FJ(\015)1145 5049 y FI(~)-35 b(p)1182 5036 y FJ(f)9 b FQ(\))p FJ(v)22 b FL(\012)c FJ(u)p FQ(.)605 5198 y FP(Lemma)31 b FQ(7.7.3)p FP(.)40 b FO(Maximal)30 b(inte)l(gr)l(able)f(quotient)f(of)h FJ(U)37 b FO(is)29 b(e)l(qual)f(to)2769 5136 y Fy(L)2861 5223 y FI(\025)p FE(2)p FI(P)2996 5203 y FG(k)2987 5242 y FF(+)3052 5198 y FJ(L)3109 5168 y FI(k)3109 5222 y(\025)3168 5198 y FL(\012)15 b FJ(D)r(L)3376 5168 y FI(k)3376 5222 y(\025)3419 5198 y FO(.)p eop %%Page: 188 22 188 191 bop 456 226 a FM(188)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 605 425 y FQ(Indeed,)23 b(let)f(us)f(de\014ne)h(the)f(homomorphism)g (of)g(\()p FJ(U)9 b FQ(\()2274 421 y Fy(b)2276 425 y FA(g)p FQ(\))2350 437 y FI(k)2391 425 y FQ(\))2423 395 y FE(\012)p FM(2)2534 425 y FQ(mo)r(dules)21 b FJ(\031)27 b FQ(:)c FJ(U)32 b FL(!)3168 363 y Fy(L)3274 425 y FJ(L)3331 395 y FI(k)3331 449 y(\025)3380 425 y FL(\012)456 527 y FJ(D)r(L)584 497 y FI(k)584 551 y(\025)649 527 y FQ(b)n(y)23 b(1)f FL(7!)930 465 y Fy(L)1036 527 y FQ(1)1078 539 y FI(\025)1144 527 y FQ(\(since)h FJ(U)31 b FQ(is)23 b(the)g(induced)g (mo)r(dule,)h(this)f(uniquely)g(de\014nes)f FJ(\031)s FQ(\).)36 b(It)23 b(is)g(an)456 630 y(easy)i(exercise)g(to)i(sho)n(w)e (that)1427 568 y Fy(L)1533 630 y FQ(1)1575 642 y FI(\025)1645 630 y FQ(is)h(a)g(cyclic)g(v)n(ector)g(in)2366 568 y Fy(L)2472 630 y FJ(L)2529 600 y FI(k)2529 653 y(\025)2589 630 y FL(\012)15 b FJ(D)r(L)2797 600 y FI(k)2797 653 y(\025)2867 630 y FQ(\(with)27 b(resp)r(ect)g(to)456 732 y(the)22 b(action)g(of)924 728 y Fy(b)926 732 y FA(g)8 b FL(\010)1047 728 y Fy(b)1049 732 y FA(g)o FQ(\),)23 b(and)f(therefore,)h(the)f(ab)r(o)n(v)n(e)f(map)h(is)g(surjectiv)n(e;)i (th)n(us,)2909 670 y Fy(L)3015 732 y FJ(L)3072 702 y FI(k)3072 756 y(\025)3123 732 y FL(\012)8 b FJ(D)r(L)3324 702 y FI(k)3324 756 y(\025)3389 732 y FQ(is)456 832 y(indeed)23 b(an)g(in)n(tegrable)f(quotien)n(t)h(of)g FJ(U)9 b FQ(.)35 b(On)23 b(the)g(other)f(hand,)i(ev)n(ery)e(in)n(tegrable)g(\()p FJ(U)9 b FQ(\()3178 828 y Fy(b)3180 832 y FA(g)p FQ(\))3254 844 y FI(k)3295 832 y FQ(\))3327 802 y FE(\012)p FM(2)3417 832 y FQ(-)456 931 y(mo)r(dule)32 b(is)g(of)g(the)h(form)1289 869 y Fy(L)1381 956 y FI(\025;\026)p FE(2)p FI(P)1576 936 y FG(k)1567 975 y FF(+)1632 931 y FJ(N)1699 943 y FI(\025\026)1783 931 y FJ(L)1840 901 y FI(k)1840 955 y(\025)1904 931 y Fw(\002)21 b FJ(L)2047 901 y FI(k)2047 952 y(\026)2091 931 y FQ(.)51 b(Since)32 b FJ(U)41 b FQ(is)32 b(generated)f(b)n(y)h(a)g(v)n(ector)f(1)456 1053 y(whic)n(h)j(is)h(\001\()p FA(g)p FQ(\))g(in)n(v)-5 b(arian)n(t,)35 b(it)g(easily)f(follo)n(ws)g(that)h(an)n(y)e(in)n (tegrable)h(quotien)n(t)g(of)h FJ(U)43 b FQ(m)n(ust)456 1152 y(ha)n(v)n(e)26 b FJ(N)714 1164 y FI(\025;\026)840 1152 y FL(\024)d FJ(\016)965 1164 y FI(\025;\026)1064 1148 y Fx(\003)1103 1152 y FQ(.)37 b(Details)28 b(are)e(left)j(to)e (the)h(reader.)605 1252 y(These)23 b(t)n(w)n(o)g(lemmas,)h(com)n(bined) g(with)g(Lemma)f(7.3.3,)h(giv)n(e)e(the)i(pro)r(of)f(of)h(the)g (theorem.)p 3384 1352 4 57 v 3388 1299 50 4 v 3388 1352 V 3437 1352 4 57 v 946 1521 a FK(7.8.)46 b(Bundle)31 b(of)h(coin)m(v)-5 b(arian)m(ts)33 b(for)f(a)g(singular)f(family)605 1670 y FQ(In)d(this)g(section,)g(w)n(e)f(con)n(tin)n(ue)g(the)i(study)f (of)f(coin)n(v)-5 b(arian)n(ts)26 b(for)i(singular)e(curv)n(es.)37 b(This)456 1770 y(time,)f(w)n(e)e(will)g(consider)f(a)h(family)g(of)g (p)r(oin)n(ted)g(curv)n(es)f FJ(C)2361 1782 y FI(S)2444 1770 y FQ(o)n(v)n(er)f(a)i(smo)r(oth)f(base)h FJ(S)39 b FQ(suc)n(h)456 1870 y(that)26 b FJ(C)693 1882 y FI(s)755 1870 y FQ(is)f(stable)h(and)g(non-singular)e(for)h FJ(S)20 b FL(n)15 b FJ(D)r FQ(,)26 b(and)g FJ(C)2301 1882 y FI(s)2362 1870 y FQ(is)g(a)g(stable)f(singular)g(curv)n(e)g(with)456 1969 y(one)36 b(double)g(p)r(oin)n(t)g(for)g FJ(s)h FL(2)h FJ(D)r FQ(,)h(where)d FJ(D)i FQ(is)e(a)g(smo)r(oth)g(divisor)f(in)i FJ(S)k FQ(\(without)c(loss)f(of)456 2069 y(generalit)n(y)e(w)n(e)h(ma)n (y)g(assume)g(that)h FJ(D)i FQ(is)e(connected\).)61 b(As)36 b(b)r(efore,)h(w)n(e)f(assume)f(that)h(w)n(e)456 2169 y(ha)n(v)n(e)26 b(some)g(in)n(tegrable)g(mo)r(dules)h FJ(V)1612 2181 y FM(1)1650 2169 y FJ(;)14 b(:)g(:)g(:)g(;)g(V)1883 2181 y FI(n)1955 2169 y FQ(assigned)26 b(to)h(the)h(mark)n(ed)e(p)r (oin)n(ts)h FJ(p)3113 2181 y FM(1)3150 2169 y FJ(;)14 b(:)g(:)g(:)f(;)h(p)3376 2181 y FI(n)3421 2169 y FQ(.)456 2268 y(Then,)23 b(b)n(y)e(the)i(construction)e(of)g(the)i(previous)d (sections,)j(this)f(data)f(de\014nes)h(a)f(v)n(ector)g(bundle)456 2368 y(of)27 b(coin)n(v)-5 b(arian)n(ts)26 b FJ(\034)33 b FQ(=)22 b FJ(\034)9 b FQ(\()p FJ(C)1302 2380 y FI(S)1352 2368 y FJ(;)14 b(~)-42 b(p)o(;)14 b(V)19 b FQ(\))28 b(o)n(v)n(er)e FJ(S)d FL(n)18 b FJ(D)r FQ(.)605 2467 y(Let)27 b(us)h(extend)f FJ(\034)37 b FQ(to)27 b(the)h(whole)e(of)i FJ(S)k FQ(as)26 b(an)h FL(O)r FQ(-mo)r(dule.)37 b(De\014ne)28 b(the)f(sheaf)g FJ(\034)37 b FQ(on)27 b FJ(S)32 b FQ(in)456 2567 y(the)c(ob)n(vious)f (w)n(a)n(y)-7 b(,)27 b(as)h(in)g(Lemma)g(7.4.2.)38 b(The)28 b(restriction)f(of)h(this)h(sheaf)e(to)i FJ(S)23 b FL(n)18 b FJ(D)31 b FQ(is)d(lisse,)456 2667 y(and)c(its)h(\014b)r(er)f(at)g(a)g (p)r(oin)n(t)h FJ(s)e FL(2)h FJ(D)i FQ(is)e(the)h(v)n(ector)f(space)f FJ(\034)9 b FQ(\()p FJ(C)2350 2679 y FI(s)2387 2667 y FJ(;)15 b(~)-43 b(p;)14 b(V)k FQ(\))25 b(whic)n(h)g(w)n(as)e(discussed) h(in)456 2766 y(the)31 b(previous)e(section.)45 b(The)31 b(same)f(argumen)n(ts)f(as)h(b)r(efore)h(sho)n(w)e(that)i FJ(\034)2830 2778 y FI(S)2909 2766 y FQ(is)g FL(O)3062 2778 y FI(S)3110 2766 y FQ(-coheren)n(t)456 2866 y(sheaf.)42 b(The)29 b(goal)f(of)i(this)f(section)h(is)f(to)g(pro)n(v)n(e)f(the)i (follo)n(wing)e(theorem,)i(whic)n(h)f(is)g(the)h(k)n(ey)456 2966 y(step)d(in)h(pro)n(ving)e(the)i(gluing)f(axiom.)605 3099 y FP(Theorem)32 b FQ(7.8.1)p FP(.)40 b FO(Under)29 b(the)h(assumptions)g(ab)l(ove,)i(the)e(she)l(af)g FJ(\034)2743 3111 y FI(S)2822 3099 y FO(is)g(lisse.)605 3232 y FQ(The)22 b(remaining)g(part)f(of)i(this)f(section)g(is)g(dev)n(oted)g(to)g(the)g (pro)r(of)g(of)g(this)h(theorem.)34 b(Note)456 3331 y(that)27 b(b)n(y)g(Theorem)f(7.4.1,)g(the)h(restriction)g(of)g FJ(\034)36 b FQ(to)27 b FJ(S)c FL(n)16 b FJ(D)30 b FQ(is)d(lisse,)f(so) h(the)g(only)g(problem)g(is)456 3431 y(analyzing)f(the)i(b)r(eha)n (vior)e(of)i FJ(\034)37 b FQ(at)28 b FJ(D)r FQ(.)605 3580 y FP(Pr)n(oof.)41 b FQ(The)23 b(pro)r(of)g(consists)g(of)g(sev)n (eral)f(steps.)35 b(The)24 b(main)f(idea)g(is)h(to)f(use)g(the)h (results)456 3680 y(of)k(the)h(previous)f(section,)g(relating)g(coin)n (v)-5 b(arian)n(ts)27 b(for)h(the)h(singular)e(\014b)r(ers)i FJ(C)2967 3692 y FI(s)3002 3680 y FJ(;)14 b(s)25 b FL(2)g FJ(D)31 b FQ(with)456 3780 y(the)36 b(coin)n(v)-5 b(arian)n(ts)33 b(for)j(nonsingular)e(curv)n(e)g FJ(C)1960 3749 y FE(_)1954 3800 y FI(s)2045 3780 y FQ(obtained)i(b)n(y)f(normalization)f(of)i FJ(C)3214 3792 y FI(s)3250 3780 y FQ(,)i(and)456 3879 y(extend)e(it)g(to)f(an)h(isomorphism)f(of)g(shea)n(v)n(es)f(of)i(coin) n(v)-5 b(arian)n(ts)34 b(in)i(some)f(neigh)n(b)r(orho)r(o)r(d)g(of)456 3979 y FJ(D)r FQ(.)44 b(Unfortunately)-7 b(,)30 b(it)h(is)f(imp)r (ossible)f(to)h(do)g(this)g(directly:)42 b(w)n(e)29 b(can)h(not)g (extend)g FJ(C)3220 3949 y FE(_)3299 3979 y FQ(to)g(a)456 4078 y(family)h(of)g(nonsingular)e(curv)n(es)h FJ(C)1585 4048 y FE(_)1579 4101 y FI(S)1666 4078 y FQ(o)n(v)n(er)f FJ(S)36 b FQ(with)31 b(a)g(natural)f(map)h FJ(C)2746 4048 y FE(_)2740 4101 y FI(S)2825 4078 y FL(!)d FJ(C)2995 4090 y FI(S)3044 4078 y FQ(.)47 b(Ho)n(w)n(ev)n(er,)456 4178 y(this)22 b(b)r(ecomes)h(p)r(ossible)f(if)h(instead)f(of)h (constructing)f(a)g(family)g(o)n(v)n(er)f FJ(S)27 b FQ(w)n(e)c (restrict)e(ourselv)n(es)456 4278 y(to)26 b(an)h(in\014nitesimal)g (neigh)n(b)r(orho)r(o)r(d)e(of)i FJ(D)r FQ(,)g(as)f(de\014ned)h(in)g (Section)g(7.6,)f(whic)n(h)h(is)f(su\016cien)n(t)456 4377 y(for)32 b(our)h(purp)r(oses.)54 b(F)-7 b(or)32 b(simplicit)n(y)-7 b(,)36 b(w)n(e)d(will)g(assume)g(that)h FJ(S)k FQ(is)33 b(a)g(disk)g(in)h(the)g(complex)456 4477 y(plane)39 b(with)h(co)r(ordinate)f FJ(q)j FQ(and)e FJ(D)45 b FQ(=)e FL(f)p FQ(0)p FL(g)p FQ(.)71 b(The)40 b(general)e(case)h(can)g (b)r(e)h(treated)f(quite)456 4577 y(similarly;)33 b(ho)n(w)n(ev)n(er,)e (it)i(is)f(not)g(ev)n(en)f(necessary)g(to)g(do)h(that)h(due)f(to)g (Lemma)g(6.3.13.)48 b(W)-7 b(e)456 4676 y(will)28 b(c)n(ho)r(ose)e(co)r (ordinates)h FJ(t)1348 4688 y FM(1)1385 4676 y FJ(;)14 b(t)1452 4688 y FM(2)1517 4676 y FQ(in)29 b(the)f(neigh)n(b)r(orho)r(o) r(d)f(of)h(the)g(double)g(p)r(oin)n(t)g FJ(a)23 b FL(2)h FJ(C)3208 4688 y FI(S)3285 4676 y FQ(suc)n(h)456 4776 y(that)j FJ(t)665 4788 y FM(1)703 4776 y FJ(t)733 4788 y FM(2)793 4776 y FQ(=)c FJ(q)30 b FQ(\(this)f(is)e(alw)n(a)n(ys)f(p)r (ossible\).)605 4879 y(By)f(Lemma)g(7.6.2,)f(it)h(su\016ces)g(to)g(pro) n(v)n(e)e(that)j(for)e(ev)n(ery)g FJ(n)f FL(\025)f FQ(0,)k(the)f(mo)r (dule)g FJ(\034)3171 4848 y FM(\()p FI(n)p FM(\))3294 4879 y FQ(o)n(v)n(er)456 4990 y FL(O)524 4947 y FM(\()p FI(n)p FM(\))522 5014 y FI(D)649 4990 y FQ(de\014ned)j(b)n(y)g (\(7.6.4\))f(for)g(our)g(family)h(of)f(curv)n(es)g(is)g(free)g(of)h (\014nite)g(rank.)605 5109 y(In)34 b(order)e(to)h(pro)n(v)n(e)f(that)i FJ(\034)1506 5079 y FM(\()p FI(n)p FM(\))1637 5109 y FQ(is)f(free)h(o)n(v)n(er)d FL(O)2143 5066 y FM(\()p FI(n)p FM(\))2141 5134 y FI(D)2241 5109 y FQ(,)k(let)f(us)f(construct)g (another)f(family)456 5216 y FJ(C)521 5185 y FE(_)603 5216 y FQ(of)i(curv)n(es)e(o)n(v)n(er)g FJ(D)1219 5185 y FM(\()p FI(n)p FM(\))1316 5216 y FQ(.)55 b(Namely)-7 b(,)35 b(tak)n(e)e FJ(C)1973 5185 y FE(_)1967 5236 y FM(0)2056 5216 y FQ(to)h(b)r(e)g(the)g(normalization)e(of)h FJ(C)3123 5228 y FM(0)3161 5216 y FQ(;)k(this)d(is)p eop %%Page: 189 23 189 192 bop 967 226 a FM(7.8.)29 b(BUNDLE)g(OF)g(COINV)-7 b(ARIANTS)30 b(F)n(OR)f(A)g(SINGULAR)g(F)-7 b(AMIL)i(Y)389 b(189)456 425 y FQ(a)35 b(nonsingular)f(curv)n(e)h(with)h(the)g(same)f (mark)n(ed)g(p)r(oin)n(ts)g(as)h FJ(C)2514 437 y FM(0)2551 425 y FQ(,)i(plus)e(t)n(w)n(o)f(more)g(mark)n(ed)456 525 y(p)r(oin)n(ts)28 b(whic)n(h)h(w)n(e)g(denote)f FJ(a)1381 495 y FE(0)1404 525 y FJ(;)14 b(a)1485 495 y FE(00)1528 525 y FQ(.)40 b(The)29 b(c)n(hoice)f(of)h(lo)r(cal)f(co)r(ordinates)f FJ(t)2776 537 y FM(1)2814 525 y FJ(;)14 b(t)2881 537 y FM(2)2947 525 y FQ(on)28 b FJ(C)3122 537 y FI(S)3199 525 y FQ(de\014nes)456 624 y(lo)r(cal)f(co)r(ordinates)f FJ(t)1123 636 y FM(1)1160 624 y FJ(;)14 b(t)1227 636 y FM(2)1292 624 y FQ(in)28 b(the)g(neigh)n(b)r(orho)r(o)r(d)e(of)i FJ(a)2192 594 y FE(0)2238 624 y FL(2)23 b FJ(C)2381 594 y FE(_)2458 624 y FQ(\(resp)r(ectiv)n(ely)-7 b(,)27 b FJ(a)3003 594 y FE(00)3046 624 y FQ(\).)605 740 y(No)n(w,)20 b(let)f(us)f(de\014ne)h(the)g(sheaf)f FL(O)1652 697 y FM(\()p FI(n)p FM(\))1650 765 y FI(C)1702 748 y Fx(_)1768 740 y FQ(as)g(follo)n(ws.)33 b(Let)19 b FJ(U)31 b FQ(=)23 b FJ(C)2544 710 y FE(_)2538 761 y FM(0)2593 740 y FL(nf)p FJ(a)2721 710 y FE(0)2744 740 y FJ(;)14 b(a)2825 710 y FE(00)2867 740 y FL(g)23 b FQ(=)f FJ(C)3078 752 y FM(0)3116 740 y FL(nf)p FJ(a)p FL(g)p FQ(.)33 b(By)456 860 y(de\014nition,)c(let) g FL(O)1038 817 y FM(\()p FI(n)p FM(\))1036 884 y FI(C)1088 867 y Fx(_)1135 860 y FL(j)1158 872 y FI(U)1239 860 y FQ(=)24 b FL(O)1396 817 y FM(\()p FI(n)p FM(\))1394 884 y FI(C)1494 860 y FL(j)1517 872 y FI(U)1572 860 y FQ(.)41 b(T)-7 b(o)28 b(extend)h(it)g(to)f(the)h(p)r(oin)n(ts)g FJ(a)2655 830 y FE(0)2678 860 y FJ(;)14 b(a)2759 830 y FE(00)2801 860 y FQ(,)29 b(de\014ne)g(the)g(stalks)456 979 y FL(O)524 936 y FM(\()p FI(n)p FM(\))522 1003 y FI(a)558 987 y Fx(0)644 979 y FQ(=)23 b FL(O)r FQ(\()p FJ(t)862 991 y FM(1)900 979 y FQ(\))7 b FL(\012)g(O)1079 936 y FM(\()p FI(n)p FM(\))1077 1003 y FI(D)1176 979 y FQ(,)23 b(where)e FL(O)r FQ(\()p FJ(t)1586 991 y FM(1)1624 979 y FQ(\))h(is)f(the)i(ring)e(of)g(germs)g(of)h(analytic)f(functions) h(in)g FJ(t)3253 991 y FM(1)3312 979 y FQ(in)g(a)456 1098 y(neigh)n(b)r(orho)r(o)r(d)i(of)h FJ(t)1097 1110 y FM(1)1158 1098 y FQ(=)d(0,)k(and)f(similarly)g(for)g FJ(a)2002 1068 y FE(00)2044 1098 y FQ(.)37 b(Ob)n(viously)-7 b(,)24 b(eac)n(h)h FJ(f)32 b FL(2)23 b(O)2912 1055 y FM(\()p FI(n)p FM(\))2910 1123 y FI(a)2946 1106 y Fx(0)3035 1098 y FQ(also)h(de\014nes)456 1218 y(a)e(section)g(of)h FL(O)950 1175 y FM(\()p FI(n)p FM(\))948 1242 y FI(C)1000 1226 y Fx(_)1047 1218 y FL(j)1070 1230 y FI(U)1148 1218 y FQ(on)g(some)f(punctured)h(neigh)n(b)r(orho)r(o)r(d)e(of)i FJ(a)2504 1188 y FE(0)2549 1218 y FQ(b)n(y)g FJ(t)2690 1230 y FM(1)2750 1218 y FL(7!)g FJ(t)2886 1230 y FM(1)2923 1218 y FJ(;)14 b(q)26 b FL(7!)d FJ(t)3159 1230 y FM(1)3197 1218 y FJ(t)3227 1230 y FM(2)3264 1218 y FQ(,)h(and)456 1337 y(th)n(us)j(w)n(e)g(can)g(glue)g(the)h(sheaf)f FL(O)1508 1294 y FM(\()p FI(n)p FM(\))1506 1362 y FI(C)1558 1345 y Fx(_)1633 1337 y FQ(from)g(its)g(restriction)g(to)g FJ(U)36 b FQ(and)27 b(stalks)g(at)g FJ(a)3076 1307 y FE(0)3099 1337 y FJ(;)14 b(a)3180 1307 y FE(00)3223 1337 y FQ(.)37 b(This)456 1444 y(de\014nes)20 b(on)h FJ(C)895 1414 y FE(_)965 1444 y FQ(a)f(structure)h(of)f(a)h(family)g(of)g(curv)n (es)e(o)n(v)n(er)g FJ(D)2351 1414 y FM(\()p FI(n)p FM(\))2448 1444 y FQ(;)24 b(this)d(family)g(is)f(non-singular.)605 1543 y(No)n(w)j(let)g(us)g(assign)f(the)i(mo)r(dules)f FJ(L)1765 1513 y FI(k)1765 1567 y(\025)1808 1543 y FJ(;)14 b(D)r(L)1973 1513 y FI(k)1973 1567 y(\025)2039 1543 y FQ(to)23 b(the)g(p)r(oin)n(ts)g FJ(a)2563 1513 y FE(0)2587 1543 y FJ(;)14 b(a)2668 1513 y FE(00)2733 1543 y FQ(and)23 b(tak)n(e)f(direct)h(sum)456 1662 y(o)n(v)n(er)i(all)j FJ(\025)23 b FL(2)h FJ(P)964 1632 y FI(k)952 1682 y FM(+)1007 1662 y FQ(.)37 b(By)27 b(Prop)r(osition)f(7.6.9,)h(this)h(de\014nes)f (a)g(lisse)g(mo)r(dule)h FJ(\034)2883 1632 y FE(_)p FM(\()p FI(n)p FM(\))3054 1662 y FQ(o)n(v)n(er)d FL(O)3299 1619 y FM(\()p FI(n)p FM(\))3297 1686 y FI(D)3397 1662 y FQ(.)605 1880 y FP(Pr)n(oposition)31 b FQ(7.8.2)p FP(.)40 b FO(The)30 b(map)1263 2138 y FJ(\036)9 b FQ(:)29 b FJ(V)1440 2104 y FM(\()p FI(n)p FM(\))1560 2138 y FL(!)23 b FJ(V)1733 2104 y FM(\()p FI(n)p FM(\))1494 2293 y FJ(v)j FL(7!)1666 2214 y Fy(X)1685 2392 y FI(\025;i)1800 2293 y FJ(q)1840 2258 y FE(\000)11 b FM(deg)h FI(e)2046 2267 y FG(\025;i)2130 2293 y FJ(v)22 b FL(\012)c FJ(e)2314 2305 y FI(\025;i)2419 2293 y FL(\012)g FJ(e)2541 2258 y FE(\003)2541 2313 y FI(\025;i)2627 2293 y FJ(;)456 2257 y FQ(\(7.8.1\))456 2646 y FO(wher)l(e)28 b FJ(e)727 2658 y FI(\025;i)841 2646 y FO(is)g(a)g(homo)l(gene)l(ous)g(b)l(asis)h(in)e FJ(L)1848 2616 y FI(k)1848 2670 y(\025)1891 2646 y FO(,)i(and)f FJ(e)2143 2616 y FE(\003)2143 2670 y FI(\025;i)2257 2646 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(that)i(in)f(fact)g(the)g(image)456 4142 y(of)c(this)h(em)n(b)r(edding) g(is)f(analytic)g(functions.)605 4241 y(It)h(is)f(also)g(easy)g(to)g (sho)n(w)g(b)n(y)g(explicit)h(calculation)f(that)h(the)g(v)n(ector)1117 4508 y FJ(w)1176 4520 y FI(\025)1243 4508 y FQ(=)1331 4430 y Fy(X)1379 4606 y FI(i)1464 4508 y FJ(q)1504 4474 y FE(\000)12 b FM(deg)f FI(e)1710 4483 y FG(\025;i)1795 4508 y FJ(e)1834 4520 y FI(\025;i)1939 4508 y FL(\012)18 b FJ(e)2061 4474 y FE(\003)2061 4529 y FI(\025;i)2170 4508 y FL(2)24 b FQ(\()p FJ(L)2338 4474 y FI(k)2338 4529 y(\025)2400 4508 y FL(\012)18 b FJ(D)r(L)2611 4474 y FI(k)2611 4529 y(\025)2654 4508 y FQ(\))2686 4474 y FM(\()p FI(n)p FM(\))456 4508 y FQ(\(7.8.2\))456 4836 y(is)27 b(in)n(v)-5 b(arian)n(t)27 b(under)g(the)h(image)f(of)g(the)h(em)n(b)r (edding)995 5137 y FA(g)p FQ([[)p FJ(t)1113 5149 y FM(1)1150 5137 y FJ(;)14 b(t)1217 5149 y FM(2)1254 5137 y FQ(]])p FJ(=)p FQ(\()p FJ(t)1404 5149 y FM(1)1441 5137 y FJ(t)1471 5149 y FM(2)1508 5137 y FQ(\))1540 5102 y FI(n)p FM(+1)1693 5137 y FL(!)1799 5020 y Fy(\022)1860 5137 y FA(g)p FQ(\(\()p FJ(t)1996 5149 y FM(1)2034 5137 y FQ(\)\)[)p FJ(q)s FQ(])19 b FL(\010)f FA(g)p FQ(\(\()p FJ(t)2422 5149 y FM(2)2460 5137 y FQ(\)\)[)p FJ(q)s FQ(])2610 5020 y Fy(\023)2671 5137 y FJ(=q)2753 5102 y FI(n)p FM(+1)2882 5137 y FJ(:)p eop %%Page: 190 24 190 193 bop 456 226 a FM(190)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 456 425 y FQ(Indeed,)34 b(it)f(su\016ces)f(to)h(sho)n(w)e(this)i(for)f FJ(xt)1823 395 y FI(n)1823 446 y FM(1)1869 425 y FJ(t)1899 395 y FI(m)1899 446 y FM(2)1962 425 y FJ(;)14 b(x)32 b FL(2)g FA(g)p FQ(.)51 b(In)33 b(this)g(case,)g(it)g(follo)n(ws)f (from)g(the)456 525 y(follo)n(wing)26 b(sequence)h(of)h(iden)n(tities:) 507 661 y(\()p FJ(x)p FQ([)p FJ(n)19 b FL(\000)f FJ(m)p FQ(])p FJ(q)897 627 y FI(m)979 661 y FL(\012)g FQ(1+)o(1)g FL(\012)g FJ(x)p FQ([)p FJ(m)h FL(\000)f FJ(n)p FQ(])p FJ(q)1669 627 y FI(n)1714 661 y FQ(\))p FJ(w)1805 673 y FI(\025)1191 816 y FQ(=)23 b(\()p FJ(x)p FQ([)p FJ(n)c FL(\000)f FJ(m)p FQ(])p FJ(q)1669 781 y FI(m)1750 816 y FL(\012)g FQ(1)g(+)g(1)g FL(\012)g FJ(x)p FQ([)p FJ(m)h FL(\000)f FJ(n)p FQ(])p FJ(q)2477 781 y FI(n)2522 816 y FQ(\)\()p FJ(q)2626 781 y FE(\000)p FI(d)2736 816 y FL(\012)g FQ(1\))2907 737 y Fy(X)2955 914 y FI(i)3040 816 y FJ(e)3079 828 y FI(\025;i)3184 816 y FL(\012)g FJ(e)3306 781 y FE(\003)3306 836 y FI(\025;i)1191 1042 y FQ(=)23 b(\()p FJ(q)1351 1008 y FE(\000)p FI(d)1460 1042 y FL(\012)18 b FQ(1\)\()p FJ(x)p FQ([)p FJ(n)h FL(\000)f FJ(m)p FQ(])p FJ(q)2007 1008 y FI(n)2071 1042 y FL(\012)g FQ(1)g(+)g(1)g FL(\012)g FJ(x)p FQ([)p FJ(m)h FL(\000)f FJ(n)p FQ(])p FJ(q)2798 1008 y FI(n)2843 1042 y FQ(\))2889 963 y Fy(X)2937 1140 y FI(i)3023 1042 y FJ(e)3062 1054 y FI(\025;i)3166 1042 y FL(\012)g FJ(e)3288 1008 y FE(\003)3288 1063 y FI(\025;i)1191 1256 y FQ(=)23 b(\()p FJ(q)1351 1222 y FE(\000)p FI(d)p FM(+)p FI(n)1552 1256 y FL(\012)18 b FQ(1\)\()p FJ(x)p FQ([)p FJ(n)h FL(\000)f FJ(m)p FQ(])g FL(\012)g FQ(1)g(+)g(1)g FL(\012)g FJ(x)p FQ([)p FJ(n)h FL(\000)f FJ(m)p FQ(]\)1)1191 1381 y(=)23 b(0)p FJ(;)456 1530 y FQ(where)29 b(1)c(=)856 1468 y Fy(P)944 1555 y FI(i)985 1530 y FJ(e)1024 1542 y FI(\025;i)1130 1530 y FL(\012)19 b FJ(e)1253 1500 y FE(\003)1253 1554 y FI(\025;i)1369 1530 y FQ(is)30 b(considered)e(as)h(a)g(v)n(ector)f(in)i(a)f(certain)g (completion)g(of)g FJ(L)3317 1500 y FI(k)3317 1554 y(\025)3380 1530 y FL(\012)456 1644 y FQ(\()p FJ(L)545 1614 y FI(k)545 1667 y(\025)588 1644 y FQ(\))620 1614 y FE(\003)658 1644 y FQ(.)66 b(Note)37 b(that)g(in)g(the)h(last)e(line)i(w)n(e)e(replaced) g FJ(D)r(L)2334 1614 y FI(k)2334 1667 y(\025)2414 1644 y FQ(b)n(y)h(\()p FJ(L)2628 1614 y FI(k)2628 1667 y(\025)2672 1644 y FQ(\))2704 1614 y FE(\003)2742 1644 y FQ(,)j(whic)n(h)c (resulted)h(in)456 1744 y(replacing)23 b FJ(x)p FQ([)p FJ(m)12 b FL(\000)g FJ(n)p FQ(])24 b(b)n(y)g FJ(x)p FQ([)p FJ(n)12 b FL(\000)g FJ(m)p FQ(]|see)25 b(\(7.1.4\))o(.)36 b(W)-7 b(e)24 b(lea)n(v)n(e)g(it)g(to)h(the)f(reader)f(to)i(c)n(hec)n (k)e(that)-3827 b Fq(?!)456 1843 y FQ(the)33 b(fact)g(that)g(1)g(do)r (es)g(not)g(lie)g(in)g FJ(L)1654 1813 y FI(k)1654 1867 y(\025)1719 1843 y FL(\012)22 b FQ(\()p FJ(L)1895 1813 y FI(k)1895 1867 y(\025)1938 1843 y FQ(\))1970 1813 y FE(\003)2041 1843 y FQ(but)34 b(only)e(in)i(some)e(completion)h(do)r (es)f(not)456 1943 y(cause)26 b(an)n(y)h(problems.)605 2042 y(Therefore,)k(if)h FJ(f)37 b FL(2)30 b FA(g)1294 2012 y FM(\()p FI(n)p FM(\))1391 2042 y FQ(\()p FJ(C)1482 2054 y FI(S)1551 2042 y FL(\000)21 b FJ(~)-43 b(p)p FQ(\))p FJ(;)14 b(v)33 b FL(2)c FJ(V)19 b FQ(,)32 b(then)g FJ(\036)p FQ(\()p FJ(f)9 b FQ(\()p FJ(v)s FQ(\)\))30 b(=)f FJ(A)p FQ(\()p FJ(f)9 b FQ(\))p FJ(\036)p FQ(\()p FJ(v)s FQ(\))33 b(and)e(th)n(us)g(the)456 2142 y(map)24 b FJ(\036)h FQ(descends)f(to)h (the)g(space)e(of)i(coin)n(v)-5 b(arian)n(ts;)24 b(w)n(e)g(will)h (denote)f(the)h(corresp)r(onding)e(map)456 2242 y(also)j(b)n(y)h FJ(\036)p FQ(.)605 2353 y(No)n(w)35 b(the)g(pro)r(of)g(of)g(prop)r (osition)f(is)h(easy)-7 b(.)59 b(Indeed,)38 b(w)n(e)d(ha)n(v)n(e)f(a)g (morphism)h(of)g FL(O)3319 2310 y FM(\()p FI(n)p FM(\))3317 2377 y FI(D)3417 2353 y FQ(-)456 2459 y(mo)r(dules)e FJ(\036)h FQ(:)g FJ(\034)972 2429 y FM(\()p FI(n)p FM(\))1103 2459 y FL(!)g FJ(\034)1265 2429 y FE(_)p FM(\()p FI(n)p FM(\))1407 2459 y FQ(.)56 b(By)34 b(Theorem)f(7.7.1,)h FJ(\036)g FQ(induces)g(an)g(isomorphism)f(on)g(the)456 2570 y(\014b)r(ers)g(at)g(zero)g FJ(\034)1020 2540 y FM(\()p FI(n)p FM(\))1118 2570 y FJ(=q)s(\034)1245 2540 y FM(\()p FI(n)p FM(\))1403 2523 y FE(\030)1375 2570 y FL(\000)-40 b(!)33 b FJ(\034)1561 2540 y FE(_)p FM(\()p FI(n)p FM(\))1704 2570 y FJ(=q)s(\034)1831 2540 y FE(_)p FM(\()p FI(n)p FM(\))1973 2570 y FQ(.)55 b(Since)33 b FJ(\034)2318 2540 y FE(_)p FM(\()p FI(n)p FM(\))2495 2570 y FQ(is)g(free)g(o)n(v)n(er)f FL(O)3001 2527 y FM(\()p FI(n)p FM(\))2999 2595 y FI(D)3098 2570 y FQ(,)k(this)d(im-)456 2670 y(mediately)c(implies)h(that)g FJ(\036)g FQ(is)g(surjectiv)n(e.)42 b(T)-7 b(o)29 b(pro)n(v)n(e)f(that)i FJ(\036)h FQ(is)e(injectiv)n(e,)i (c)n(ho)r(ose)d(a)h(basis)456 2770 y FJ(v)496 2782 y FM(1)533 2770 y FJ(;)14 b(:)g(:)g(:)g(;)g(v)758 2782 y FI(k)826 2770 y FQ(in)27 b FJ(\034)967 2740 y FM(\()p FI(n)p FM(\))1065 2770 y FJ(=q)s(\034)1192 2740 y FM(\()p FI(n)p FM(\))1289 2770 y FQ(.)37 b(Since)27 b FJ(\034)1610 2740 y FE(_)p FM(\()p FI(n)p FM(\))1780 2770 y FQ(is)f(free,)h(this)h (implies)f(that)g FJ(v)2706 2782 y FM(1)2744 2770 y FJ(;)14 b(:)g(:)g(:)f(;)h(v)2968 2782 y FI(k)3036 2770 y FQ(are)26 b(linearly)456 2885 y(indep)r(enden)n(t)k(o)n(v)n(er)d FL(O)1175 2842 y FM(\()p FI(n)p FM(\))1173 2910 y FI(D)1273 2885 y FQ(.)42 b(On)29 b(the)h(other)f(hand,)h(it)g(follo)n(ws)e(from)h (the)h(de\014nition)f(that)h(the)456 2992 y(mo)r(dule)22 b FJ(K)28 b FQ(=)23 b FJ(\034)975 2962 y FM(\()p FI(n)p FM(\))1073 2992 y FJ(=)p FL(h)p FJ(v)1187 3004 y FM(1)1224 2992 y FJ(;)14 b(:)g(:)g(:)g(;)g(v)1449 3004 y FI(k)1490 2992 y FL(i)22 b FQ(satis\014es)f FJ(q)s(K)29 b FQ(=)22 b FJ(K)6 b FQ(;)24 b(since)e FJ(q)2433 2962 y FI(n)p FM(+1)2585 2992 y FQ(=)g(0,)h(this)f(implies)h FJ(K)28 b FQ(=)23 b(0.)456 3095 y(Th)n(us,)j FJ(\034)733 3065 y FM(\()p FI(n)p FM(\))857 3095 y FQ(is)g(freely)g(generated)f(b)n(y)h FJ(v)1695 3107 y FM(1)1732 3095 y FJ(;)14 b(:)g(:)g(:)g(;)g(v)1957 3107 y FI(k)1998 3095 y FQ(.)36 b(Therefore,)25 b FJ(\036)i FQ(is)f(an)g(isomorphism,)g(whic)n(h)456 3194 y(completes)h(the)h(pro)r (of)f(of)g(the)h(prop)r(osition.)p 3384 3194 4 57 v 3388 3142 50 4 v 3388 3194 V 3437 3194 4 57 v 605 3346 a(Since)e(b)n(y)f (Prop)r(osition)g(7.6.9)f(the)i(sheaf)g FJ(\034)1970 3316 y FE(_)p FM(\()p FI(n)p FM(\))2138 3346 y FQ(is)g(lisse,)f(this)h (prop)r(osition)f(implies)h(that)456 3446 y(the)i(same)f(holds)g(for)g FJ(\034)1196 3416 y FM(\()p FI(n)p FM(\))1321 3446 y FQ(and)h(th)n(us)f(completes)h(the)g(pro)r(of)f(of)g(Theorem)g(7.8.1.)p 3384 3446 V 3388 3393 50 4 v 3388 3446 V 3437 3446 4 57 v 1314 3621 a FK(7.9.)47 b(Pro)s(of)31 b(of)h(the)g(gluing)e(axiom) 605 3770 y FQ(In)39 b(this)h(section)f(w)n(e)g(giv)n(e)f(a)h(pro)r(of)f (of)h(the)h(gluing)f(axiom)f(for)h(the)g(WZW)h(mo)r(dular)456 3870 y(functor.)34 b(Recall)19 b(that)h(this)g(axiom)f(describ)r(es)g (the)h(b)r(eha)n(viour)f(of)g(the)i(bundle)f(of)f(coin)n(v)-5 b(arian)n(ts)456 3969 y(in)36 b(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)h (the)h(b)r(oundary)e(of)h(the)g(mo)r(duli)h(space;)i(in)e(particular,)f (it)h(claims)456 4069 y(that)c(the)h(connection)f(has)g(\014rst)g (regular)e(singularities)i(at)g(the)h(b)r(oundary)-7 b(,)34 b(and)f(describ)r(es)456 4168 y(the)28 b(sp)r(ecialization)e(of) i(this)g(connection.)605 4268 y(Recall)c(that)g(the)h(b)r(oundary)e(of) h(the)h(mo)r(duli)f(space)f(consists)h(of)g(the)g(stable)g(curv)n(es)f (with)456 4368 y(ordinary)i(double)h(p)r(oin)n(ts)h(\(see)f(Section)h (6.2\))f(and)h(that)g(it)g(su\016ces)f(to)h(c)n(hec)n(k)e(the)i (regularit)n(y)456 4467 y(condition)e(for)h(an)f(op)r(en)h(part)f(of)h (the)h(b)r(oundary)-7 b(.)35 b(Th)n(us,)26 b(w)n(e)g(need)g(to)f(pro)n (v)n(e)g(regularit)n(y)f(and)456 4567 y(calculate)j(sp)r(ecialization)g (of)i(the)f(connection)g(in)g FJ(\034)2105 4579 y FI(S)2154 4567 y FQ(,)g(where)g FJ(S;)14 b(C)2593 4579 y FI(S)2641 4567 y FJ(;)g(D)r(;)g(:)g(:)g(:)42 b FQ(are)27 b(same)h(as)f(in)456 4667 y(the)e(b)r(eginning)f(of)g(the)h(previous)f(section.)35 b(By)25 b(the)f(construction)g(of)h(the)g(previous)e(sections,)456 4766 y FJ(\034)492 4778 y FI(S)571 4766 y FQ(carries)29 b(a)h(natural)g(pro)5 b(jectiv)n(ely)29 b(\015at)h(connection)h(o)n(v)n (er)d FJ(S)d FL(n)20 b FJ(D)r FQ(.)46 b(Also,)32 b(w)n(e)e(ha)n(v)n(e)f (sho)n(wn)456 4866 y(in)e(the)g(previous)e(section)h(that)h FJ(\034)1514 4878 y FI(S)1590 4866 y FQ(is)f(lisse,)h(i.e.,)g(is)f(a)h (sheaf)f(of)h(sections)f(of)g(a)h(v)n(ector)e(bundle)456 4965 y(on)i FJ(S)5 b FQ(.)605 5116 y FP(Theorem)32 b FQ(7.9.1)p FP(.)40 b FO(Under)29 b(the)h(assumptions)f(ab)l(ove,)i(the) f(c)l(onne)l(ction)f(in)h FJ(\034)3049 5128 y FI(S)3126 5116 y FO(has)h(lo)l(ga-)456 5216 y(rithmic)f(singularities)h(at)f FJ(D)r FO(.)p eop %%Page: 191 25 191 194 bop 1352 226 a FM(7.9.)29 b(PR)n(OOF)g(OF)g(THE)g(GLUING)h (AXIOM)774 b(191)605 425 y FP(Pr)n(oof.)41 b FQ(As)33 b(b)r(efore,)i(c)n(ho)r(ose)d(a)h(lo)r(cal)g(co)r(ordinate)f FJ(q)37 b FQ(in)d(a)f(neigh)n(b)r(orho)r(o)r(d)f(of)h FJ(D)j FQ(suc)n(h)456 525 y(that)26 b FJ(q)g FQ(=)d(0)i(is)h(the)h (equation)e(of)h FJ(D)r FQ(.)37 b(Recall)26 b(\(see)g(\(6.3.5\))o(\))g (that)h FL(D)2589 495 y FM(0)2587 548 y FI(S)2658 525 y FL(\032)22 b(D)2809 537 y FI(S)2884 525 y FQ(b)r(e)k(the)h(subsheaf) 456 624 y(generated)f(\(as)h(sheaf)h(of)f(algebras\))f(b)n(y)h FL(O)1810 636 y FI(S)1886 624 y FQ(and)h(v)n(ector)e(\014elds)i(whic)n (h)f(are)g(tangen)n(t)g(to)g FJ(D)r FQ(.)605 779 y FP(Pr)n(oposition)k FQ(7.9.2)p FP(.)40 b FO(The)30 b(she)l(af)h FJ(\034)40 b FO(has)30 b(a)g(natur)l(al)g(structur)l(e)e(of)i(a)g FL(D)2922 748 y FM(0)2920 801 y FI(S)2969 779 y FO(-mo)l(dule.)605 933 y FQ(This)22 b(prop)r(osition)f(is)h(a)f(generalization)f(of)i (Theorem)f(7.4.1,)h(and)g(is)g(pro)n(v)n(ed)e(in)j(the)f(same)456 1032 y(w)n(a)n(y)-7 b(.)35 b(The)27 b(only)g(c)n(hange)e(is)i(that)g (instead)g(of)g(claiming)f(that)h(an)n(y)g(v)n(ector)e(\014eld)i(on)g FJ(S)32 b FQ(can)26 b(b)r(e)456 1132 y(lifted)i(to)g(a)f(v)n(ector)f (\014eld)i(on)f FJ(C)1444 1144 y FI(S)1511 1132 y FL(\000)19 b FJ(~)-43 b(p)p FQ(\()p FJ(S)5 b FQ(\),)28 b(w)n(e)f(use)h(the)g (follo)n(wing)e(lemma.)605 1286 y FP(Lemma)31 b FQ(7.9.3)p FP(.)40 b FO(L)l(et)33 b FJ(\022)j FO(b)l(e)d(ve)l(ctor)h(\014eld)g(on) f FJ(S)38 b FO(which)d(is)f(tangent)f(to)g FJ(D)r FO(.)50 b(Then)34 b(lo)l(c)l(al)t(ly)456 1386 y(in)26 b FJ(S)5 b FO(,)28 b(such)f(a)g(\014eld)g(c)l(an)f(b)l(e)h(lifte)l(d)h(to)f(a)g (ve)l(ctor)g(\014eld)g(on)g FJ(C)2302 1398 y FI(S)2377 1386 y FO(which)h(has)f(p)l(oles)h(at)f(the)f(marke)l(d)456 1485 y(p)l(oints.)605 1640 y FP(Example)31 b FQ(7.9.4)p FP(.)40 b FQ(Let)f FJ(S)45 b FQ(b)r(e)40 b(a)f(neigh)n(b)r(orho)r(o)r (d)f(of)i(zero)e(in)i FH(C)15 b FQ(,)49 b(with)40 b(co)r(ordinate)e FJ(q)s FQ(,)456 1739 y FJ(D)26 b FQ(=)e FL(f)p FQ(0)p FL(g)p FQ(.)38 b(As)29 b(b)r(efore,)f(in)n(tro)r(duce)g(co)r(ordinates) f FJ(t)2066 1751 y FM(1)2103 1739 y FJ(;)14 b(t)2170 1751 y FM(2)2236 1739 y FQ(near)27 b(the)i(double)g(p)r(oin)n(t)f(in)h FJ(C)3208 1751 y FI(S)3285 1739 y FQ(suc)n(h)456 1839 y(that)20 b FJ(q)26 b FQ(=)c FJ(t)808 1851 y FM(1)846 1839 y FJ(t)876 1851 y FM(2)913 1839 y FQ(.)34 b(Then)20 b(in)g(the)g(neigh)n(b)r(orho)r(o)r(d)f(of)g(the)h(double)g(p)r(oin)n (t,)i(the)e(lifting)g(of)g(the)g(v)n(ector)456 1938 y(\014eld)27 b FJ(q)s(@)719 1950 y FI(q)784 1938 y FQ(m)n(ust)h(b)r(e)g(of)f(the)h (form)f FJ(\013t)1619 1950 y FM(1)1657 1938 y FJ(@)1701 1950 y FI(t)1726 1958 y FF(1)1781 1938 y FQ(+)18 b FJ(\014)t(t)1945 1950 y FM(2)1983 1938 y FJ(@)2027 1950 y FI(t)2052 1958 y FF(2)2116 1938 y FQ(for)27 b(some)g FJ(\013;)14 b(\014)33 b FQ(satisfying)27 b FJ(\013)18 b FQ(+)g FJ(\014)28 b FQ(=)22 b(1.)605 2093 y(This)30 b(prop)r(osition,)f(along)g(with)h(the) h(fact)f(that)g FJ(\034)2204 2105 y FI(S)2282 2093 y FQ(is)g(lisse,)g(immediately)g(implies)g(the)456 2192 y(statemen)n(t)d(of)h(the)g(theorem.)p 3384 2192 4 57 v 3388 2139 50 4 v 3388 2192 V 3437 2192 4 57 v 605 2353 a FP(Example)j FQ(7.9.5)p FP(.)40 b FQ(Let)f FJ(S)45 b FQ(b)r(e)40 b(a)f(neigh)n(b)r(orho)r(o)r(d)f(of)i(zero)e(in)i FH(C)15 b FQ(,)49 b(with)40 b(co)r(ordinate)e FJ(q)s FQ(.)456 2453 y(De\014ne)28 b(the)g(family)f FJ(C)1168 2465 y FI(S)1240 2453 y FL(\032)c FH(C)14 b FJ(P)1447 2423 y FM(2)1508 2453 y FL(\002)k FJ(S)32 b FQ(b)n(y)c(the)g(equation) 1210 2596 y FJ(uv)e FQ(=)d FJ(q)s(w)1513 2561 y FM(2)1551 2596 y FJ(;)180 b FQ(\()p FJ(u)23 b FQ(:)g FJ(v)j FQ(:)d FJ(w)r FQ(\))h FL(2)g FH(C)14 b FJ(P)2330 2561 y FM(2)2373 2596 y FJ(;)96 b(q)27 b FL(2)c FJ(S)456 2743 y FQ(with)28 b(the)h(mark)n(ed)e(p)r(oin)n(ts)i FJ(p)1378 2755 y FM(1)1415 2743 y FQ(\()p FJ(q)s FQ(\))c(=)f(\(1)g(:)g(0)g(:)g(0\))p FJ(;)14 b(p)2044 2755 y FM(2)2081 2743 y FQ(\()p FJ(q)s FQ(\))25 b(=)f(\(0)g(:)g(1)g(:)h(0\),)j(and)g(lo)r(cal)g(parameters)456 2843 y(at)38 b(these)h(p)r(oin)n(ts)g FJ(t)1083 2855 y FM(1)1162 2843 y FQ(=)j FJ(w)r(=u;)14 b(t)1487 2855 y FM(2)1565 2843 y FQ(=)42 b FJ(w)r(=v)s FQ(.)71 b(The)39 b(same)f(argumen)n(t)g(as)h(in)g(Example)f(6.2.4)456 2943 y(sho)n(ws)e(that)i(for)g FJ(q)43 b FL(6)p FQ(=)d(0,)g(the)e(curv) n(e)f FJ(C)1766 2955 y FI(q)1841 2943 y FQ(is)h(isomorphic)e(to)i(a)f (sphere)h FH(P)2878 2913 y FM(1)2914 2943 y FQ(,)i(with)f(mark)n(ed)456 3042 y(p)r(oin)n(ts)c FJ(p)755 3054 y FM(1)828 3042 y FQ(=)h(0)p FJ(;)14 b(p)1050 3054 y FM(2)1122 3042 y FQ(=)36 b FL(1)f FQ(and)h(lo)r(cal)e(parameters)g FJ(z)t(;)14 b FQ(1)p FJ(=z)37 b FQ(resp)r(ectiv)n(ely)-7 b(.)60 b(F)-7 b(or)34 b FJ(q)39 b FQ(=)d(0,)h(the)456 3142 y(\014b)r(er)27 b FJ(C)706 3154 y FM(0)770 3142 y FQ(consists)g(of)f(t)n(w)n(o)h(comp)r (onen)n(ts,)f(eac)n(h)h(of)f(them)i(isomorphic)e(to)h(a)f(sphere)h FH(P)3197 3112 y FM(1)3233 3142 y FQ(,)g(with)456 3242 y(co)r(ordinates)j FJ(z)944 3211 y FE(0)995 3242 y FQ(=)f FJ(u=w)r(;)14 b(z)1320 3211 y FE(0)o(0)1391 3242 y FQ(=)29 b FJ(v)s(=w)34 b FQ(resp)r(ectiv)n(ely)-7 b(,)31 b(whic)n(h)g(ha)n(v)n (e)g(one)g(common)f(p)r(oin)n(t)i FJ(z)3328 3211 y FE(0)3380 3242 y FQ(=)456 3341 y FJ(z)499 3311 y FE(0)o(0)563 3341 y FQ(=)23 b(0.)35 b(The)24 b(mark)n(ed)f(p)r(oin)n(ts)h FJ(p)1498 3353 y FM(1)1559 3341 y FQ(and)f FJ(p)1758 3353 y FM(2)1819 3341 y FQ(are)g(the)h(p)r(oin)n(ts)g FL(1)2422 3311 y FE(0)2446 3341 y FJ(;)14 b FL(1)2566 3311 y FE(00)2632 3341 y FQ(|in\014nite)24 b(p)r(oin)n(ts)g(of)g(the) 456 3441 y(\014rst)d(and)h(the)g(second)f(spheres)g(resp)r(ectiv)n(ely) -7 b(,)22 b(with)h(lo)r(cal)e(co)r(ordinates)f FJ(t)2765 3453 y FM(1)2826 3441 y FQ(=)i(1)p FJ(=z)3040 3411 y FE(0)3062 3441 y FJ(;)14 b(t)3129 3453 y FM(2)3189 3441 y FQ(=)23 b(1)p FJ(=z)3404 3411 y FE(0)n(0)456 3540 y FQ(resp)r(ectiv)n(ely)-7 b(.)605 3640 y(It)28 b(is)f(easy)g(to)h(see)f (that)h(an)n(y)f(v)n(ector)f(\014eld)i(of)f(the)h(form)1292 3783 y(~)-45 b FJ(v)26 b FQ(=)d FJ(\013u@)1588 3795 y FI(u)1650 3783 y FQ(+)18 b FJ(\014)t(v)s(@)1871 3795 y FI(v)1929 3783 y FQ(+)g FJ(q)s(@)2096 3795 y FI(q)2133 3783 y FJ(;)97 b(\013)19 b FQ(+)f FJ(\014)27 b FQ(=)c(1)456 3926 y(de\014nes)g(a)g(v)n(ector)f(\014eld)i(on)f FJ(C)1380 3938 y FI(S)1452 3926 y FQ(whic)n(h)h(is)f(a)g(lifting)h(of)f(the)h(v)n (ector)e(\014eld)i FJ(q)s(@)2802 3938 y FI(q)2862 3926 y FQ(on)g FJ(S)5 b FQ(.)35 b(Rewriting)459 4026 y(~)-45 b FJ(v)32 b FQ(in)e(terms)f(of)g(co)r(ordinates)f FJ(t)1429 4038 y FM(1)1466 4026 y FJ(;)14 b(q)s FQ(,)30 b(w)n(e)f(get)j(~)-45 b FJ(v)30 b FQ(=)25 b FL(\000)p FJ(\013t)2168 4038 y FM(1)2205 4026 y FJ(@)2249 4038 y FI(t)2274 4046 y FF(1)2331 4026 y FQ(+)19 b FJ(q)s(@)2499 4038 y FI(q)2536 4026 y FQ(,)30 b(and)f(th)n(us)g FJ(\015)2979 4038 y FI(p)3013 4046 y FF(1)3050 4026 y FQ(\()s(~)-45 b FJ(v)s FQ(\))27 b(=)e FJ(\013L)3384 4038 y FM(0)3421 4026 y FQ(.)456 4125 y(Similarly)-7 b(,)33 b(expansion)e(near)h FJ(p)1453 4137 y FM(2)1522 4125 y FQ(giv)n(es)f FJ(\015)1773 4137 y FI(p)1807 4145 y FF(2)1844 4125 y FQ(\()s(~)-45 b FJ(v)s FQ(\))32 b(=)e FJ(\014)t(L)2186 4137 y FM(0)2223 4125 y FQ(.)51 b(Therefore,)33 b(the)f(action)g(of)h FJ(q)s(@)3288 4137 y FI(q)3357 4125 y FQ(on)456 4225 y(coin)n(v)-5 b(arian)n(ts)25 b(is)j(giv)n(en)e(b)n(y)i FJ(\013)p FQ(\()p FJ(L)1473 4237 y FM(0)1510 4225 y FQ(\))1542 4237 y FI(p)1576 4245 y FF(1)1632 4225 y FQ(+)18 b FJ(\014)t FQ(\()p FJ(L)1855 4237 y FM(0)1892 4225 y FQ(\))1924 4237 y FI(p)1958 4245 y FF(2)1995 4225 y FQ(.)605 4385 y(This)30 b(statemen)n(t)g(also)e(has) h(an)h(in\014nitesimal)g(analogue.)42 b(Recall)29 b(the)h(notation)f FJ(\034)3255 4355 y FM(\()p FI(n)p FM(\))3380 4385 y FQ(=)456 4496 y FJ(\034)492 4508 y FI(S)540 4496 y FJ(=q)622 4466 y FI(n)p FM(+1)751 4496 y FJ(\034)787 4508 y FI(S)865 4496 y FQ(\(see)g(the)h(previous)e(section\).)43 b(This)29 b(is)g(a)h(lisse)f FL(O)2452 4453 y FM(\()p FI(n)p FM(\))2450 4521 y FI(D)2549 4496 y FQ(-mo)r(dule.)42 b(It)30 b(immediately)456 4603 y(follo)n(ws)21 b(from)h(Prop)r(osition)f(7.9.2)g(that)i FJ(\034)1769 4573 y FM(\()p FI(n)p FM(\))1890 4603 y FQ(has)f(a)g(natural)f(action)h(of)h(the)g(sheaf)f(of)h(algebras)456 4702 y FL(D)522 4672 y FM(0)520 4733 y FI(D)576 4717 y FF(\()p FG(n)p FF(\))689 4702 y FQ(=)f FL(D)842 4672 y FM(0)840 4725 y FI(S)888 4702 y FJ(=q)970 4672 y FI(n)p FM(+1)1099 4702 y FL(D)1165 4672 y FM(0)1163 4725 y FI(S)1211 4702 y FQ(.)605 4815 y(Similar)30 b(result)g(also)f(holds)h(for)g(the)g (sheaf)g FJ(\034)2050 4785 y FE(_)p FM(\()p FI(n)p FM(\))2223 4815 y FQ(describ)r(ed)g(in)h(the)g(previous)e(section:)456 4915 y(it)f(follo)n(ws)g(from)g(Prop)r(osition)f(7.6.9)g(that)i FJ(\034)1883 4885 y FE(_)p FM(\()p FI(n)p FM(\))2054 4915 y FQ(has)f(a)g(natural)f(structure)h(of)h(a)f(pro)5 b(jectiv)n(e)456 5014 y FL(D)522 4984 y FM(0)520 5045 y FI(D)576 5029 y FF(\()p FG(n)p FF(\))666 5014 y FQ(-mo)r(dule.)41 b(Let)29 b(us)g(t)n(wist)g(this)h(action,)f(de\014ning)g(a)g(new)g (action)f(of)h FJ(q)s(@)2914 5026 y FI(q)2980 5014 y FQ(b)n(y)g(adding)g(to)456 5116 y(the)j(old)g(action)f(the)i(constan)n (t)e(\001)1555 5128 y FI(\025)1599 5116 y FQ(,)i(de\014ned)f(b)n(y)h (\(7.4.7\))e(\(cf.)51 b(Example)31 b(7.4.11\).)49 b(W)-7 b(e)32 b(will)456 5216 y(denote)27 b(this)h(new)g(action)f(b)n(y)g FL(r)1489 5185 y FE(_)1538 5216 y FQ(.)p eop %%Page: 192 26 192 195 bop 456 226 a FM(192)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 605 425 y FQ(Note)23 b(that)h(a)e(lifting)i(of)f(the)h(v)n(ector)e (\014eld)h FJ(q)s(@)2012 437 y FI(q)2072 425 y FQ(to)g FJ(C)2234 395 y FE(_)2228 456 y FI(D)2284 439 y FF(\()p FG(n)p FF(\))2398 425 y FQ(can)g(b)r(e)g(explicilt)n(y)g(describ)r(ed)g (as)456 538 y(follo)n(ws:)33 b(lift)25 b FJ(q)s(@)970 550 y FI(q)1030 538 y FQ(to)f(a)f(deriv)-5 b(ation)26 b(~)-45 b FJ(v)27 b FQ(of)c FL(O)1804 508 y FM(\()p FI(n)p FM(\))1902 538 y FQ(\()p FJ(C)17 b FL(\000)11 b FJ(~)-43 b(p)o FQ(\);)26 b(as)c(w)n(as)h(discussed)g(in)h(Example)f(7.9.4,)456 638 y(this)i(lifting)i(in)e(a)g(neigh)n(b)r(orho)r(o)r(d)g(of)g(the)h (double)g(p)r(oin)n(t)f(has)g(the)h(form)f FJ(\013t)2814 650 y FM(1)2852 638 y FJ(@)2896 650 y FI(t)2921 658 y FF(1)2972 638 y FQ(+)14 b FJ(\014)t(t)3132 650 y FM(2)3169 638 y FJ(@)3213 650 y FI(t)3238 658 y FF(2)3275 638 y FJ(;)g(\013)h FQ(+)456 737 y FJ(\014)33 b FQ(=)c(1.)48 b(De\014ne)31 b FJ(v)1046 707 y FE(_)1127 737 y FQ(b)n(y)g FJ(v)1289 707 y FE(_)1367 737 y FQ(=)h(~)-45 b FJ(v)35 b FQ(on)c FJ(C)1720 707 y FE(_)1790 737 y FL(n)20 b(f)p FJ(a)1938 707 y FE(0)1961 737 y FJ(;)14 b(a)2042 707 y FE(00)2084 737 y FL(g)29 b FQ(=)f FJ(C)2307 749 y FM(0)2366 737 y FL(n)20 b(f)p FJ(a)p FL(g)p FQ(,)32 b(and)f FJ(v)2819 707 y FE(_)2897 737 y FQ(=)e FJ(\013t)3074 749 y FM(1)3112 737 y FJ(@)3156 749 y FI(t)3181 757 y FF(1)3238 737 y FQ(+)21 b FJ(q)s(@)3408 749 y FI(q)456 837 y FQ(at)29 b FJ(a)603 807 y FE(0)626 837 y FQ(;)h(similarly)-7 b(,)29 b(let)j(~)-45 b FJ(v)29 b FQ(=)c FJ(\014)t(t)1398 849 y FM(2)1436 837 y FJ(@)1480 849 y FI(t)1505 857 y FF(2)1561 837 y FQ(+)19 b FJ(q)s(@)1729 849 y FI(q)1795 837 y FQ(at)29 b FJ(a)1942 807 y FE(00)1984 837 y FQ(.)42 b(It)29 b(is)g(easy)g(to)g (c)n(hec)n(k)f(that)h(this)h(de\014nes)f(an)456 943 y(elemen)n(t)e(of)h (\002)922 913 y FM(\()p FI(n)p FM(\))1019 943 y FQ(\()p FJ(C)1116 913 y FE(_)1183 943 y FL(\000)19 b FJ(~)-43 b(p)p FQ(\).)605 1095 y FP(Example)31 b FQ(7.9.6)p FP(.)40 b FQ(Under)d(the)g(assumptions)f(of)h(Example)f(7.9.5,)i(the)g(lifting) f(of)g(the)456 1195 y(v)n(ector)g(\014eld)j FJ(q)s(@)992 1207 y FI(q)1067 1195 y FQ(is)f(giv)n(en)g(b)n(y)f FJ(v)1560 1165 y FE(_)1651 1195 y FQ(=)k FJ(\013z)1854 1165 y FE(0)1877 1195 y FJ(@)1921 1207 y FI(z)1955 1191 y Fx(0)2008 1195 y FQ(+)26 b FJ(q)s(@)2183 1207 y FI(q)2258 1195 y FQ(on)39 b(the)h(\014rst)e(comp)r(onen)n(t,)k(and)d(b)n(y)456 1294 y FJ(v)499 1264 y FE(_)590 1294 y FQ(=)i FJ(\014)t(z)790 1264 y FE(00)833 1294 y FJ(@)877 1306 y FI(z)911 1290 y Fx(00)982 1294 y FQ(+)25 b FJ(q)s(@)1156 1306 y FI(q)1232 1294 y FQ(on)39 b(the)g(second)g(one.)70 b(Therefore,)41 b(its)e(action)g(on)g(the)g(bundle)h(of)456 1394 y(coin)n(v)-5 b(arian)n(ts)25 b(is)j(giv)n(en)e(b)n(y)955 1534 y FL(r)1024 1500 y FE(_)1024 1554 y FI(q)r(@)1091 1562 y FG(q)1153 1534 y FQ(=)c FJ(q)s(@)1324 1546 y FI(q)1380 1534 y FQ(+)c FJ(\013)1516 1467 y Fy(\000)1554 1534 y FQ(\()p FJ(L)1643 1546 y FM(0)1680 1534 y FQ(\))1712 1546 y FI(p)1746 1554 y FF(1)1802 1534 y FL(\000)g FQ(\()p FJ(L)1974 1546 y FM(0)2011 1534 y FQ(\))2043 1546 y FI(a)2079 1530 y Fx(0)2106 1467 y Fy(\001)2162 1534 y FQ(+)g FJ(\014)2296 1467 y Fy(\000)2335 1534 y FQ(\()p FJ(L)2424 1546 y FM(0)2461 1534 y FQ(\))2493 1546 y FI(p)2527 1554 y FF(2)2582 1534 y FL(\000)g FQ(\()p FJ(L)2754 1546 y FM(0)2791 1534 y FQ(\))2823 1546 y FI(a)2859 1530 y Fx(0)q(0)2904 1467 y Fy(\001)2961 1534 y FQ(+)g(\001)3113 1546 y FI(\025)3157 1534 y FJ(:)-2724 b FQ(\(7.9.1\))605 1706 y FP(Pr)n(oposition)31 b FQ(7.9.7)p FP(.)40 b FO(The)34 b(isomorphism)i FJ(\036)9 b FQ(:)30 b FJ(\034)2200 1675 y FM(\()p FI(n)p FM(\))2327 1706 y FL(!)g FJ(\034)2485 1675 y FE(_)p FM(\()p FI(n)p FM(\))2627 1706 y FO(,)35 b(de\014ne)l(d)f(by)41 b FQ(\(7.8.1\))o FO(,)35 b(is)456 1805 y(an)29 b(isomorphism)j(of)f FL(D)1227 1775 y FM(0)1225 1836 y FI(D)1281 1819 y FF(\()p FG(n)p FF(\))1371 1805 y FO(-mo)l(dules.)605 1958 y FP(Pr)n(oof.)41 b FQ(It)24 b(su\016ces)f(to)g(c)n(hec)n(k)g(that)h FJ(\036)g FQ(comm)n(utes)f(with)h(the)g(action)f(of)g(the)h(v)n(ector)e(\014eld) 456 2057 y FJ(q)s(@)540 2069 y FI(q)576 2057 y FQ(.)37 b(T)-7 b(o)28 b(pro)n(v)n(e)e(this,)h(it)h(su\016ces)g(to)f(c)n(hec)n (k)g(that)1408 2197 y FL(r)1477 2163 y FE(_)1477 2217 y FI(q)r(@)1544 2225 y FG(q)1582 2197 y FQ(\()p FJ(v)22 b FL(\012)c FJ(w)1818 2209 y FI(\025)1862 2197 y FQ(\))24 b(=)e(\()p FL(r)2106 2209 y FI(q)r(@)2173 2217 y FG(q)2212 2197 y FJ(v)s FQ(\))d FL(\012)f FJ(w)2448 2209 y FI(\025)456 2345 y FQ(where)27 b FJ(w)755 2357 y FI(\025)826 2345 y FQ(w)n(as)g(de\014ned)h(in)g(\(7.8.2\))o(.)37 b(But)28 b(this)g(is)f(immediate)h(from)f(the)h(de\014nition)g(of)g FL(r)3352 2315 y FE(_)3401 2345 y FQ(:)650 2484 y FL(r)719 2450 y FE(_)719 2505 y FI(q)r(@)786 2513 y FG(q)824 2484 y FQ(\()p FJ(v)22 b FL(\012)c FJ(w)1060 2496 y FI(\025)1104 2484 y FQ(\))h FL(\000)f FQ(\()p FL(r)1339 2496 y FI(q)r(@)1406 2504 y FG(q)1444 2484 y FJ(v)s FQ(\))h FL(\012)f FJ(w)1680 2496 y FI(\025)1747 2484 y FQ(=)23 b FJ(v)f FL(\012)c FQ(\()p FJ(q)s(@)2096 2496 y FI(q)2151 2484 y FL(\000)g FJ(\013)p FQ(\()p FJ(L)2376 2496 y FM(0)2414 2484 y FQ(\))2446 2496 y FI(a)2482 2480 y Fx(0)2527 2484 y FL(\000)g FJ(\014)t FQ(\()p FJ(L)2750 2496 y FM(0)2788 2484 y FQ(\))2820 2496 y FI(a)2856 2480 y Fx(00)2919 2484 y FQ(+)g(\001)3071 2496 y FI(\025)3115 2484 y FQ(\))p FJ(w)3206 2496 y FI(\025)1747 2617 y FQ(=)23 b FJ(v)f FL(\012)c FQ(\()p FL(\000)p FJ(d)g FQ(+)g(\001)2290 2629 y FI(\025)2352 2617 y FL(\000)g FJ(\013)p FQ(\()p FJ(L)2577 2629 y FM(0)2615 2617 y FQ(\))2647 2629 y FI(a)2683 2613 y Fx(0)2728 2617 y FL(\000)g FJ(\014)t FQ(\()p FJ(L)2951 2629 y FM(0)2989 2617 y FQ(\))3021 2629 y FI(a)3057 2613 y Fx(00)3102 2617 y FQ(\))p FJ(w)3193 2629 y FI(\025)1747 2741 y FQ(=)23 b(0)p FJ(:)p 3384 2881 4 57 v 3388 2828 50 4 v 3388 2881 V 3437 2881 4 57 v 605 3038 a FQ(No)n(w)32 b(let)h(us)g(calculate)f(the)h(sp)r (ecialization)f(of)h(the)g(connection)f(in)h FJ(\034)2861 3050 y FI(S)2910 3038 y FQ(.)52 b(Let)33 b(us)f(recall)456 3138 y(the)d(de\014nition)g(of)g(the)h(sp)r(ecialization)e(functor,)h (sligh)n(tly)g(mo)r(difying)g(it)h(for)e(our)g(needs.)41 b(As)456 3237 y(in)27 b(Chapter)g(6,)g(assume)f(that)i(\()p FJ(F)r(;)14 b FL(r)p FQ(\))28 b(is)f(\015at)g(connection)g(with)h (\014rst)f(order)f(p)r(oles)g(at)i FJ(D)r FQ(.)36 b(As)456 3340 y(b)r(efore,)f(w)n(e)e(denote)h(b)n(y)g FL(F)42 b FQ(the)34 b(sheaf)g(of)g(sections)f(of)h FJ(F)12 b FQ(,)35 b(and)f FL(F)2605 3310 y FM(\(0\))2728 3340 y FQ(=)f FL(F)8 b FJ(=q)s FL(F)g FQ(.)55 b FL(F)3190 3310 y FM(\(0\))3313 3340 y FQ(is)34 b(a)456 3456 y(sheaf)27 b(on)h FJ(D)i FQ(whic)n(h)e(has)g(a)g(natural)f(action)h(of)g(the)g (sheaf)g(of)g(algebras)e FL(D)2813 3413 y FM(\(0\))2811 3480 y FI(D)2926 3456 y FQ(=)e FL(D)3081 3426 y FM(0)3079 3479 y FI(S)3127 3456 y FJ(=q)s FL(D)3275 3426 y FM(0)3273 3479 y FI(S)3321 3456 y FQ(.)38 b(It)456 3562 y(turns)27 b(out)h(that)g(the)g(sp)r(ecialization)e FJ(S)5 b(p)1756 3574 y FI(D)1816 3562 y FJ(F)40 b FQ(can)27 b(b)r(e)h(de\014ned)g (using)f(only)g FL(F)2927 3532 y FM(\(0\))3044 3562 y FQ(as)g(follo)n(ws.)605 3716 y FP(Lemma)k FQ(7.9.8)p FP(.)40 b FO(L)l(et)35 b FQ(\()p FJ(G;)1450 3695 y FQ(~)1436 3716 y FL(r)q FQ(\))h FO(b)l(e)f(a)h(ve)l(ctor)g(bund)t(le)g(on)g(the)f (normal)h(bund)t(le)g FJ(N)9 b(D)38 b FO(with)456 3815 y(a)c(mono)l(dr)l(omic)h FJ(l)r(og)16 b(D)36 b FO(\015at)e(c)l(onne)l (ction,)i(and)e(let)g FJ(i)g FO(b)l(e)g(a)g(home)l(omorphism)j 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FI(S)2390 4942 y FQ(suc)n(h)g(that)g FJ(t)2810 4954 y FM(1)2847 4942 y FJ(t)2877 4954 y FM(2)2956 4942 y FQ(=)j FJ(q)g FQ(giv)n(es)c(an)456 5042 y(iden)n(ti\014cation)27 b(of)h(a)f(neigh)n (b)r(orho)r(o)r(d)f(of)i FJ(D)h FQ(in)f FJ(S)33 b FQ(with)28 b(a)f(neigh)n(b)r(orho)r(o)r(d)f(of)i FJ(D)i FQ(in)e FJ(N)9 b(D)29 b FQ(b)n(y)1482 5182 y FJ(i)9 b FQ(:)27 b(\()p FJ(d;)14 b(q)s FQ(\))24 b FL(7!)f FQ(\()p FJ(d;)14 b(q)s(@)2080 5194 y FI(t)2105 5202 y FF(1)2161 5182 y FL(\012)k FJ(@)2288 5194 y FI(t)2313 5202 y FF(2)2349 5182 y FQ(\))p FJ(;)p eop %%Page: 193 27 193 196 bop 1352 226 a FM(7.9.)29 b(PR)n(OOF)g(OF)g(THE)g(GLUING)h (AXIOM)774 b(193)456 425 y FQ(or,)26 b(passing)h(from)g(v)n(ectors)f (to)i(co)n(v)n(ectors,)1494 600 y FJ(i)9 b FQ(:)27 b(\()p FJ(d;)14 b(q)s FQ(\))24 b FL(7!)f FQ(\()p FJ(d;)2019 544 y(dt)2092 556 y FM(1)2147 544 y FL(\012)18 b FJ(dt)2303 556 y FM(2)p 2019 581 323 4 v 2160 657 a FJ(q)2351 600 y FQ(\))p FJ(:)-1950 b FQ(\(7.9.3\))605 780 y(No)n(w,)28 b(let)h(us)f(de\014ne)h(a)f(family)h(of)f(p)r(oin)n(ted)h(curv)n(es)e (o)n(v)n(er)f FJ(N)9 b(D)31 b FQ(b)n(y)d FJ(C)2792 792 y FI(d;q)2907 780 y FQ(=)c FJ(C)3061 750 y FE(_)3055 804 y FI(d)3139 780 y FQ(with)29 b(the)456 880 y(parameters)d(at)i FJ(a)1032 850 y FE(0)1055 880 y FJ(;)14 b(a)1136 850 y FE(00)1206 880 y FQ(giv)n(en)27 b(b)n(y)h FJ(t)1569 892 y FM(1)1606 880 y FJ(=q)s(;)14 b(t)1755 892 y FM(2)1792 880 y FQ(.)38 b(This)28 b(de\014nes)g(a)g(bundle)g(of)g(coin)n(v)-5 b(arian)n(ts)31 b(~)-47 b FJ(\034)38 b FQ(on)28 b(a)456 979 y(neigh)n(b)r(orho)r(o)r(d)e(of)i FJ(D)h FQ(in)f FJ(S)5 b FQ(.)605 1130 y FP(Theorem)32 b FQ(7.9.9)p FP(.)40 b FO(The)30 b(map)1270 1278 y FL(O)1336 1290 y FI(S)1403 1278 y FL(\012)18 b FJ(V)42 b FL(!)23 b(O)1748 1290 y FI(N)6 b(D)1886 1278 y FL(\012)1969 1199 y Fy(X)2009 1378 y FI(\025)2102 1278 y FJ(V)38 b FL(\012)18 b FJ(L)2328 1244 y FI(k)2328 1298 y(\025)2389 1278 y FL(\012)g FJ(D)r(L)2600 1244 y FI(k)2600 1298 y(\025)1357 1506 y FJ(f)9 b FQ(\()p 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FQ(in)n(tro)r(duced)h(in)g(the)456 2105 y(previous)i(section.)49 b(By)32 b(Prop)r(osition)e(7.9.7,)i(the)g(isomorphism)f FJ(\036)9 b FQ(:)29 b FJ(\034)2717 2075 y FM(\(0\))2837 2105 y FL(!)i FJ(\034)2996 2075 y FE(_)p FM(\(0\))3130 2105 y FQ(,)i(de\014ned)456 2216 y(b)n(y)26 b(\(7.8.1\))f(is)h(an)g (isomorphism)e(of)i FL(D)1669 2173 y FM(\(0\))1667 2241 y FI(D)1758 2216 y FQ(-mo)r(dules.)36 b(On)26 b(the)g(other)f(hand,)h (let)h(us)e(sho)n(w)g(that)456 2319 y(the)j(map)f FJ(V)37 b FL(\012)18 b FJ(L)1008 2289 y FI(k)1008 2343 y(\025)1070 2319 y FL(\012)g FJ(D)r(L)1281 2289 y FI(k)1281 2343 y(\025)1347 2319 y FL(!)23 b FJ(V)37 b FL(\012)18 b FJ(L)1678 2289 y FI(k)1678 2343 y(\025)1740 2319 y FL(\012)g FJ(D)r(L)1951 2289 y FI(k)1951 2343 y(\025)1994 2319 y FQ(,)28 b(giv)n(en)f(b)n(y) 1380 2470 y FJ(v)22 b FL(\012)c FJ(v)1568 2436 y FE(0)1610 2470 y FL(\012)g FJ(v)1736 2436 y FE(00)1802 2470 y FL(7!)23 b FJ(q)1948 2436 y FM(deg)11 b FI(v)2094 2411 y Fx(0)2121 2470 y FJ(v)22 b FL(\012)c 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a(F)-7 b(or)27 b(tec)n(hnical)g(reasons,)f(it)i(is)g(more)e(con)n(v)n(enien)n(t)h(to)g (pass)g(to)h(the)g(dual)f(sheaf)1409 4676 y FJ(\034)1454 4642 y FE(\003)1493 4676 y FQ(\()p FJ(C)q(;)15 b(~)-43 b(p;)14 b(V)1749 4688 y FI(i)1777 4676 y FQ(\))24 b(=)1920 4609 y Fy(\000)1958 4676 y FJ(\034)9 b FQ(\()p FJ(C)q(;)16 b(~)-44 b(p)r(;)14 b(D)r(V)2332 4688 y FI(i)2360 4676 y FQ(\))2392 4609 y Fy(\001)2430 4626 y FE(\003)2468 4676 y FJ(:)456 4817 y FQ(Ob)n(viously)-7 b(,)28 b(the)h(previous)f (theorem)h(immediately)g(implies)g(that)h(the)f(shea)n(v)n(es)f FJ(\034)3089 4787 y FE(\003)3128 4817 y FQ(\()p FJ(C)q(;)15 b(~)-43 b(p;)14 b(V)3384 4829 y FI(i)3412 4817 y FQ(\))456 4917 y(also)29 b(form)i(a)f(mo)r(dular)h(functor)g(with)g(the)g (additiv)n(e)g(cen)n(tral)f(c)n(harge)f FJ(c)p FQ(.)47 b(This)30 b(functor)h(will)456 5016 y(b)r(e)d(called)f FO(Wess-Zumino-Witten)h(mo)l(dular)j(functor)p FQ(.)605 5116 y(As)f(a)g(corollary)-7 b(,)28 b(w)n(e)h(ha)n(v)n(e)g(pro)n(v)n (ed)f(the)j(theorem)e(form)n(ulated)g(in)i(the)f(in)n(tro)r(duction)g (to)456 5216 y(this)d(c)n(hapter.)p eop %%Page: 194 28 194 197 bop 456 226 a FM(194)826 b(7.)30 b(WESS{ZUMINO{WITTEN)e(MODEL) 605 425 y FP(Cor)n(ollar)-6 b(y)33 b FQ(7.9.11)p FP(.)39 b FO(The)k(c)l(ate)l(gory)h FL(O)1956 395 y FM(in)n(t)1954 449 y FI(k)2082 425 y FO(has)f(a)g(structur)l(e)e(of)i(a)g(mo)l(dular)h (tensor)456 537 y(c)l(ate)l(gory,)31 b(with)f FK(1)23 b FQ(=)f FJ(L)1200 507 y FI(k)1200 558 y FM(0)1241 537 y FJ(;)14 b(\022)1317 549 y FI(V)1397 537 y FQ(=)23 b FJ(e)1524 507 y FM(2)p FI(\031)r FM(i)p FI(L)1663 515 y FF(0)1699 537 y FO(,)30 b(and)g(the)g(tensor)f(pr)l(o)l(duct)2623 472 y FI(:)2600 537 y FL(\012)g FO(de\014ne)l(d)i(by)947 681 y FQ(Hom)1120 700 y FE(O)1175 680 y FF(in)o(t)1173 721 y FG(k)1252 681 y FQ(\()p FK(1)p FJ(;)14 b(V)1417 693 y FM(1)1500 616 y FI(:)1478 681 y FL(\012)22 b FJ(:)14 b(:)g(:)1708 616 y FI(:)1685 681 y FL(\012)23 b FJ(V)1821 693 y FI(n)1866 681 y FQ(\))h(=)2009 614 y 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b(p)q(;)14 b(V)3385 1394 y FI(k)3364 1449 y(~)3366 1464 y(\025)3426 1424 y FQ(\))456 1530 y FO(de\014ne)23 b(a)h(genus)f(zer)l (o)h(mo)l(dular)g(functor.)36 b(The)25 b(c)l(orr)l(esp)l(onding)f(ribb) l(on)g(c)l(ate)l(gory)h(is)e(the)h(Drin-)456 1630 y(feld's)31 b(c)l(ate)l(gory.)605 1779 y FP(Pr)n(oof.)41 b FQ(The)32 b(pro)r(of)f(is)h(obtained)f(b)n(y)h(noticing)g(that)g(w)n(e)f(ha)n(v)n (e)g(used)h(in)n(tegrabilit)n(y)f(of)456 1879 y FJ(L)513 1849 y FI(k)513 1902 y(\025)578 1879 y FQ(only)21 b(in)h(t)n(w)n(o)f (places:)34 b(when)22 b(c)n(hec)n(king)f(\014nite-dimensionalit)n(y)g (of)h(the)g(spaces)f(of)h(coin)n(v)-5 b(ari-)456 1978 y(an)n(ts,)27 b(and)h(in)g(the)g(pro)r(of)g(of)g(Theorem)f(7.7.1,)f (iden)n(tifying)j(the)f(coin)n(v)-5 b(arian)n(ts)26 b(for)h(a)h (singular)456 2078 y(curv)n(e)i FJ(C)37 b FQ(and)31 b(its)g (normalization)f FJ(C)1656 2048 y FE(_)1705 2078 y FQ(.)48 b(On)31 b(the)g(other)g(hand,)h(if)f(w)n(e)g(restrict)f(ourselv)n(es)g (to)456 2178 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y(time)f(w)n(e)g(w)n(ere)g(only)g(using)f(the)i(action)f(of)g(the) h(Virasoro)d(algebra)g(on)i(in)n(tegrable)f(mo)r(dules,)456 3429 y(giv)n(en)33 b(b)n(y)i(the)g(Suga)n(w)n(ara)d(construction.)57 b(The)35 b(only)f(places)g(w)n(ere)g(w)n(e)g(actually)g(used)h(the)456 3529 y(de\014nition)h(of)f(coin)n(v)-5 b(arian)n(ts)34 b(and)i(prop)r(erties)f(of)g(in)n(tegrable)g(mo)r(dules)g(w)n(ere)g (the)h(pro)r(of)g(of)456 3629 y(\014nite-dimensionalit)n(y)g(of)g(the)h (v)n(ector)f(spaces)f(of)i(coin)n(v)-5 b(arian)n(ts)34 b(and)j(the)g(pro)r(of)f(of)g(Theo-)456 3728 y(rem)26 b(7.7.1,)f(iden)n(tifying)i(the)f(coin)n(v)-5 b(arian)n(ts)25 b(for)g(a)h(singular)f(curv)n(e)h FJ(C)32 b FQ(and)27 b(its)f(normalization)456 3828 y FJ(C)521 3798 y FE(_)570 3828 y FQ(.)62 b(Th)n(us,)38 b(if)f(w)n(e)e(could)h(rep)r(eat)g(these)g (t)n(w)n(o)f(steps)h(in)h(other)e(setups|for)g(example,)j(re-)456 3927 y(placing)33 b(the)h(category)e FL(O)1306 3897 y FM(in)n(t)1304 3951 y FI(k)1424 3927 y FQ(b)n(y)i(a)f(suitable)h (category)e(of)i(Virasoro)d(mo)r(dules|w)n(e)j(w)n(ould)456 4027 y(again)29 b(get)i(a)f(mo)r(dular)g(functor.)46 b(Indeed,)32 b(it)f(is)g(rather)e(easy)h(to)h(mo)r(dify)g(these)g (argumen)n(ts)456 4127 y(to)23 b(de\014ne)h(the)f(mo)r(dular)g(functor) g(related)g(to)h(the)f(so-called)f(minimal)i(mo)r(dels)f(of)h (Conformal)456 4226 y(Field)g(Theory)-7 b(,)24 b(in)g(whic)n(h)g(the)h (mo)r(dules)f FJ(L)1811 4196 y FI(k)1811 4250 y(\025)1878 4226 y FQ(are)f(replaced)h(b)n(y)f(irreducible)h(unitary)f(mo)r(dules) 456 4326 y(o)n(v)n(er)k FJ(V)19 b(ir)32 b FQ(with)f(a)e(suitable)g(cen) n(tral)g(c)n(harge.)41 b(If)30 b(w)n(e)g(try)f(to)g(pursue)h(this)g (idea)f(as)g(far)g(as)g(w)n(e)456 4426 y(can)34 b(and)h(see)g(what)f (is)h(the)h(most)e(general)g(situation)h(in)g(whic)n(h)g(w)n(e)f(can)h (apply)g(the)g(same)456 4525 y(pro)r(of,)c(w)n(e)f(will)h(arriv)n(e)e (at)i(the)g(notion)g(of)f(Rational)h(Conformal)e(Field)j(Theory)d (\(or,)i(to)g(b)r(e)456 4625 y(more)23 b(precise,)i(the)f(holomorphic)g (\(c)n(hiral\))g(half)g(of)h(R)n(CFT\).)f(The)h(n)n(um)n(b)r(er)f(of)g (references)g(on)456 4724 y(this)33 b(sub)5 b(ject)32 b(is)h(tremendous;)i(some)d(of)h(the)g(more)f(suitable)g(for)g (mathematical)h(audience)456 4824 y(are)g([)p FK(Hua)p FQ(],)i(in\015uen)n(tial)f(but)h(unpublished)g(man)n(uscript)e([)p FK(BFM)p FQ(],)i(and)f([)p FK(Gai)p FQ(].)56 b(F)-7 b(or)33 b(more)456 4924 y(ph)n(ysical)26 b(exp)r(osition)h(and)h(extra)f (references,)f(see)h([)p FK(FMS)p FQ(].)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF