Local Symplectic Field Theory

Abstract

Generalizing local Gromov-Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is fourdimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and discuss the resulting algebraic structures. Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. O. Fabert for rational multiple covers u = v • ϕ in M v,d (Γ + , Γ − ), then every infinitesimal deformation of u as a holomorphic curve is again a multiple cover of v. Furthermore, the cokernels of the linearized Cauchy-Riemann operator∂ J fit together to a smooth obstruction bundle Coker v∂JUsing these obstruction bundles we can solve the transversality problem for multiple covers of immersed curves with elliptic orbits without employing the polyfold machinery from [7], see [9, Sec. 7.2] for the general approach. We emphasize that the following transversality result is a generalization of the well-known automatic transversality result in dimension four from [10]. Indeed, note that when the multiple cover is still immersed, then the unperturbed moduli space M is discrete and hence the obstruction bundle vanishes.which is extended (using parallel transport and cut-off functions, as described in [5,9,8]) to a section in the full Banach space bundle E → B. Then it holds:• If ν is a transversal section in Coker∂ J , then∂ ν J is a transversal section in E, i.e. M ν is regular. • The linearization of ν at every zero is a compact operator, so that the linearizations of∂ J and∂ ν J belong to the same class of Fredholm operators. 1350041-3 Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. M 0 or M 0 × M − the cokernel bundle Coker∂ J = Coker v,d∂J (Γ + , Γ − ) is given by2 ) denote the cokernel bundle over the moduli space M 0 of multiple covers of the immersed curves with elliptic orbits and the moduli spaces M + , M − of multiple covers of cylinders over positive or negative asymptotic Reeb orbits γ ± of v, respectively. With this we can now give the analogue of the above definition of special sectionsν =ν v,d (Γ + , Γ − ) in obstruction bundles Coker∂ J = Coker v,d∂J (Γ + , Γ − ) 1350041-11 Int. J. Math. 2013.24. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. O. Fabert over moduli spaces M = M v,d (Γ + , Γ − ) of multiple covers of immersed curves with elliptic orbits v. Assume that we have already coherently chosen sections ν ± =ν γ ± ,d (Γ + , Γ − ) in the cokernel bundles Coker ±∂ J = Coker γ ± ,d∂J (Γ + , Γ − ) over all moduli spaces M ± = M γ,d (Γ + , Γ − ) of branched covers of cylinders over positive and negative asymptotic Reeb orbits γ ± of v. Definition 4.1. Assume that we have chosen sectionsν in the cokernel bundles Coker∂ J over all moduli spaces M of multiple covers of the immersed cu...

Generic Transversality for Unbranched Covers of Closed Pseudoholomorphic Curves

“…2 We aim in future work to prove similar results for covers of finite-energy punctured Jholomorphic curves in symplectic cobordisms, which should have interesting applications in Symplectic Field Theory [EGH00] and Embedded Contact Homology [Hut14]. A few special cases of super-rigidity in the punctured case have previously been observed by the second author [Wen10], as well as work of Fabert [Fab13], and unpublished work of Hutchings [Hut]; those examples were restricted to dimension four, but the methods introduced in the present paper have no such restrictions. and a J-holomorphic curve 3 u : (Σ, j) → (M, J), the index of u is the integer (1.1) ind(u) = (n − 3)χ(Σ) + 2c 1 (u),…”