Translator Disclaimer
2021 Supersymmetric field theories and the elliptic index theorem with complex coefficients
Daniel Berwick-Evans
Geom. Topol. 25(5): 2287-2384 (2021). DOI: 10.2140/gt.2021.25.2287

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 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.

Citation

Download Citation

Daniel Berwick-Evans. "Supersymmetric field theories and the elliptic index theorem with complex coefficients." Geom. Topol. 25 (5) 2287 - 2384, 2021. https://doi.org/10.2140/gt.2021.25.2287

Information

Received: 18 April 2019; Revised: 11 August 2020; Accepted: 3 October 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.2287

Subjects:
Primary: 55N34, 81T60

Rights: Copyright © 2021 Mathematical Sciences Publishers

JOURNAL ARTICLE
98 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.25 • No. 5 • 2021
MSP
Back to Top