Topological $q$-expansion and the supersymmetric sigma model

TL;DR

This paper develops a framework comparing Hamiltonian and Lagrangian formalisms in supersymmetric sigma models to construct topological invariants and models for elliptic and ordinary cohomology, linking physics and topology.

Contribution

It introduces a novel comparison between Hamiltonian and Lagrangian approaches, leading to new models for elliptic cohomology and modular forms in the context of supersymmetric sigma models.

Findings

Constructs a model for elliptic cohomology at the Tate curve over Z.

Develops a model for ordinary cohomology valued in weak modular forms over C.

Establishes a topological q-expansion principle relating the two formalisms.

Abstract

The Hamiltonian and Lagrangian formalisms offer two perspectives on quantum field theory. This paper sets up a framework to compare these approaches for the supersymmetric sigma model. The goal is to use techniques from physics to construct topological invariants. In brief, the Hamiltonian formalism studies positive energy representations of super annuli. This leads to a model for elliptic cohomology at the Tate curve over $\mathbb{Z}$. The Lagrangian approach studies sections of line bundles over a moduli stack of super tori. This leads to a model for ordinary cohomology valued in weak modular forms over $\mathbb{C}$. Compatibility between the two formalisms is a field theory version of the topological $q$-expansion principle. Combining these ingredients constructs a cohomology theory admitting an orientation for string manifolds that is closely related to Witten's Dirac operator on loop space.