Chern characters for supersymmetric field theories

Berwick-Evans, Daniel

doi:10.2140/gt.2023.27.1947

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Chern characters for supersymmetric field theories

Daniel Berwick-Evans

Geometry & Topology 27 (2023) 1947–1986

DOI: 10.2140/gt.2023.27.1947

Abstract

We construct a map from $d | 1$ –dimensional Euclidean field theories to complexified K–theory when $d = 1$ and complex-analytic elliptic cohomology when $d = 2$ . This provides further evidence for the Stolz–Teichner program, while also identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parametrized generalization of Fei Han’s realization of the Chern character in K–theory as dimensional reduction for $1 | 1$ –dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of $2 | 1$ –dimensional tori and the derived geometry of complex-analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of $𝒩 = (0, 1)$ supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.

Keywords

elliptic cohomology, topological modular forms, supersymmetric field theories, partition function

Mathematical Subject Classification

Primary: 55N34, 81T60

References

Publication

Received: 13 November 2020

Revised: 26 August 2021

Accepted: 27 November 2021

Published: 27 July 2023

Proposed: Mark Behrens

Seconded: Haynes R Miller, Nathalie Wahl

Authors

Daniel Berwick-Evans
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

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Volume 27, issue 5 (2023)