Gluing theories of contact instantons and of pseudoholomoprhic curves in SFT

TL;DR

This paper develops a gluing theory for contact instantons in open and closed string contexts, enabling the construction of Floer cohomology and Legendrian contact homology, and relates it to holomorphic buildings in SFT.

Contribution

It introduces a unified gluing framework for contact instantons and pseudoholomorphic curves, facilitating new invariants in contact and symplectic topology.

Findings

Constructed contact instanton Floer cohomology.

Established Legendrian contact instanton homology.

Linked contact instantons to holomorphic buildings in SFT.

Abstract

We develop the gluing theory of contact instantons in the context of open strings and in the context of closed strings \emph{with vanishing charge}, for example in the symplectization context. This is one of the key ingredients for the study of (virtually) smooth moduli space of (bordered) contact instantons needed for the construction of contact instanton Floer cohomology and more generally for the construction of Fukaya-type category of Legendrian submanifolds in contact manifold $(M,\xi)$. As an application, we apply the gluing theorem to give the construction of (cylindrical) Legendrian contact instanton homology that enters in our solution to Sandon's question for the nondegenerate case. We also apply this gluing theory to that of moduli spaces of holomorphic buildings arising in Symplectic Field Theory (SFT) by canonically lifting the former to that of the latter.