Polyfolds of Lagrangian Floer Theory in All Genera

Abstract

This dissertation describes a polyfold approach to the construction of an integration

theory on moduli spaces of pseudo-holomorphic curves with Lagrangian boundary conditions.

Polyfold theory was introduced by Hofer, Wysocki, and Zehnder in a series

of articles. Polyfolds have been employed to study closed Gromov-Witten theory and

Symplectic Field Theory. From its initial introduction, polyfold theory has seen generalizations

which have allowed for a recyclable and modular construction of moduli spaces

arising in symplectic geometry.

The present work sets the beginning of a geometric realization of the theory of Lagrangian

Floer theory in all genera, generalizing the work of Fukaya, Oh, Ohta, and Ono. Similarly

to the construction of Lagrangian Floer theory, construction of Lagrangian Floer theory

in all genera separates into two pieces:

1. The construction of an integration theory on the moduli spaces of pseudo-holomorphic

maps with Lagrangian boundary conditions

2. An ordered perturbation scheme on these moduli spaces which relates geometry (in

particular integration) along the boundary of a given moduli space in terms of geometry

of moduli spaces of lower order

The present article treats the first of these pieces. The second piece is left to subsequent

work.