Construction of general symplectic field theory

Abstract

We construct symplectic field theory in general case completely. We use Kuranishi theory for the construction. For the construction of the Kuranishi neighborhood of a holomorphic building of genus $>0$, we introduce a new space which parametrizes the deformations of both of the domain curve and the target space. We also improve the theory of Kuranishi structure and introduce the new notion of pre-Kuranishi structure and its weakly good coordinate system. Although the product of good coordinate systems is not a good coordinate system, weakly good coordinate system is closed with respect to product, and we can use their product directly for the product of pre-Kuranishi spaces. We also explain a new way to prove the smoothness of pre-Kuranishi structure by using the estimates of the differentials of implicit functions. We can obtain the estimate of the implicit functions by direct calculations using appropriate coordinates. We treat symplectic field theory of Bott-Morse case by using a triangulation of the space of periodic orbits.