Bordered contact instantons and their Fredholm theory and generic transversalities

TL;DR

This paper develops the Fredholm theory for bordered contact instantons on punctured Riemann surfaces, proves generic transversality results under Legendrian boundary perturbations, and lays groundwork for their moduli space construction.

Contribution

It introduces the Fredholm framework for bordered contact instantons and establishes their generic transversality properties, advancing the understanding of their moduli spaces.

Findings

Fredholm theory for bordered contact instantons established

Generic transversality under Legendrian boundary perturbations proven

Moduli space construction foundations laid

Abstract

In this article, we first establish the Fredholm theory for the bordered contact instantons defined on the punctured Riemann surfaces with prescribed asymptotic condition near the boundary punctures. We then prove the generic mapping transversality under the perturbation of Legendrian boundary condition. We also establish their generic (0-jet) evaluation transversality results of their moduli space under the perturbations of CR almost complex structures and of Legendrian boundary conditions. These are fundamental ingredients of the construction of the moduli space of bordered contact instantons and their applications.