Counting trivial curves in symplectic field theory

Branched covers of trivial cylinders over closed Reeb orbits are the trivial examples for (non-constant) punctured holomorphic curves studied in symplectic field theory. Unlike trivial cylinders themselves, the branched covers might come with right Fredholm index to contribute to the SFT differential. However, since in these cases the actual dimension for the moduli spaces is strictly greater than the virtual dimension expected by the Fredholm index, one has add coherent (!) abstract perturbations to the Cauchy-Riemann operator for counting these curves. In this paper we show that for any chosen coherent perturbation we get a trivial algebraic count, which proves that branched covers of trivial cylinders do not enter the SFT algebra. For this we show that we can inductively define an Euler number for gluing-compatible sections in the cokernel bundle over the compactified moduli space of branched covers, which is zero. As main application it follows that the differential is strictly decreasing with respect to the action filtration not only for cylindrical homology, but for the full contact homology. This allows us e.g. to compute the full contact homology of the three-torus.