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arXiv · 1610.00747

Supersymmetric field theories and the elliptic index theorem with complex coefficients

Daniel Berwick-Evans
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Abstract

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 pushfoward 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 ${\rm KO}$-theory and a cocycle representative of the $\hat{A}$-class for a family of oriented manifolds.

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BibTeXRIS

Daniel Berwick-Evans. 2020-10-06. Supersymmetric field theories and the elliptic index theorem with complex coefficients. https://doi.org/10.2140/gt.2021.25.2287

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