# A two-category of Hamiltonian manifolds, and a (1+1+1) field theory

## TL;DR

This paper constructs a novel extended field theory in 1+1+1 dimensions using a strict 2-category framework, extending previous Floer theories and inspired by gauge theory, with potential for higher-dimensional generalizations.

## Contribution

It introduces a new quasi 2-functor field theory in 3D, based on a completed 2-category of Hamiltonian manifolds, extending prior Floer and symplectic instanton theories.

## Key findings

- Defines a strict 2-category of Hamiltonian manifolds
- Constructs a quasi 2-functor extending Floer field theory
- Lays groundwork for higher-dimensional gauge theory extensions

## Abstract

We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category' $\mathcal{H}am$, notions which we define. Our construction extends Wehrheim and Woodward's Floer Field theory, and is inspired by Manolescu and Woodward's construction of symplectic instanton homology. It can be seen, in dimensions $1+1$, as a real analog of a construction by Moore and Tachikawa. Our construction is motivated by instanton gauge theory in dimensions 3 and 4: we expect to promote $\widehat{\mathcal{H}am}$ to a (sort of) 3-category via equivariant Lagrangian Floer homology, and extend our quasi 2-functor to dimension 4, via equivariant analogues of Donaldson polynomials.