Supersymmetric field theories and the elliptic index theorem with complex coefficients

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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 yA–class for a family of oriented manifolds.

Original languageEnglish (US)
Pages (from-to)2287-2384
Number of pages98
JournalGeometry and Topology
Issue number5
StatePublished - 2021


  • Elliptic cohomology
  • Mathai-Quillen forms
  • Supersymmetric field theories
  • Topological modular forms
  • Witten genus

ASJC Scopus subject areas

  • Geometry and Topology


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