Perturbative sigma models, elliptic cohomology and the Witten genus

Abstract

We provide a differential cocycle model for elliptic cohomology with complex coefficients and use analytic methods to construct a cocycle representative for the Witten class in this language. Our motivation stems from the conjectural connection between 2-dimensional field theories and elliptic cohomology originally due to G. Segal and E. Witten. The specifics of our constructions are informed by the work of S. Stolz and P. Teichner on super Euclidean field theories and K. Costello's construction of the Witten genus using perturbative quantization. As a warm-up, we prove analogous results for supersymmetric quantum mechanics and K-theory with complex coefficients.

Changes made from referees suggestions: statements of main theorems were sharpened, physical motivations were separated from the main mathematical discussion, and the relation to Witten's original construction was clarified

References in corpus (2)

• Superconnections and Parallel Transport
• A geometric view of the Chern character

Cited by in corpus (3)

• A model for complex analytic equivariant elliptic cohomology from quantum field theory
• Topological $q$-expansion and the supersymmetric sigma model
• The (anti-)holomorphic sector in $\mathbb{C}/\Lambda$-equivariant cohomology, and the Witten class