TY - JOUR

T1 - Supersymmetric field theories and the elliptic index theorem with complex coefficients

AU - Berwick-Evans, Daniel

N1 - Publisher Copyright:
© 2021, Mathematical Science Publishers. All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Elliptic cohomology

KW - Mathai-Quillen forms

KW - Supersymmetric field theories

KW - Topological modular forms

KW - Witten genus

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U2 - 10.2140/gt.2021.25.2287

DO - 10.2140/gt.2021.25.2287

M3 - Article

AN - SCOPUS:85122291189

SN - 1465-3060

VL - 25

SP - 2287

EP - 2384

JO - Geometry and Topology

JF - Geometry and Topology

IS - 5

ER -