Local symplectic field theory (Q2854052)

From the author's introduction: In this paper we define a local version of Eliashberg-Givental-Hofer's symplectic field theory (SFT). It provides a topological quantum theory approach to local Gromov-Witten theory in the same way as standard SFT provides a topological quantum field theory approach to standard Gromov-Witten theory. While in local Gromov-Witten theory one counts multiple covers over a fixed closed holomorphic curve, in local SFT we count multiple covers over punctured holomorphic curves. Instead of getting invariants for contact manifolds, we now get the invariants for closed Reeb orbits that were already studied in the author previous articles. Note that for the orbit curves we used an infinitesimal energy estimate to show that multiple covers of orbit cylinders are isolated in the moduli space of holomorphic curves.NEWLINENEWLINENEWLINEIn this paper we show that the dimension bounds on the kernel of the linearized Cauchy-Riemann operator established in [\textit{C. Wendl}, Comment. Math. Helv. 85, No. 2, 347--407 (2010; Zbl 1207.32021)] (using positivity of intersections in dimension four) can be used to obtain the required isolatedness result for rational multiple covers when the underlying simple (rational) curve is sufficiently nice.