  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.
