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Chern characters for supersymmetric field theories

Daniel Berwick-Evans

Geometry & Topology 27 (2023) 1947–1986
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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