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Supersymmetric field theories and the elliptic index theorem with complex coefficients

Daniel Berwick-Evans

Geometry & Topology 25 (2021) 2287–2384

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2–dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector-bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushforward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric σ–model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and 2–dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai–Quillen Thom form in complexified KO–theory and a cocycle representative of the  –class for a family of oriented manifolds.

elliptic cohomology, topological modular forms, supersymmetric field theories, Witten genus, Mathai-Quillen forms
Mathematical Subject Classification 2010
Primary: 55N34, 81T60
Received: 18 April 2019
Revised: 11 August 2020
Accepted: 3 October 2020
Published: 3 September 2021
Proposed: Ralph Cohen
Seconded: Stefan Schwede, Haynes R Miller
Daniel Berwick-Evans
Department of Mathematics
University of Illinois at Urbana–Champaign
Urbana, IL
United States