Topological $q$-expansion and the supersymmetric sigma model

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.