Equivariant elliptic cohomology, gauged sigma models, and discrete torsion

Berwick-Evans, Daniel

doi:10.1090/tran/8527

Equivariant elliptic cohomology, gauged sigma models, and discrete torsion
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Trans. Amer. Math. Soc. 375 (2022), 369-427 Request permission

Abstract:

For a finite group $G$, we show that functions on fields for the 2-dimensional supersymmetric sigma model with background $G$-symmetry determine cocycles for complex analytic $G$-equivariant elliptic cohomology. Similar structures in supersymmetric mechanics determine cocycles for equivariant K-theory with complex coefficients. The path integral for gauge theory with a finite group constructs wrong-way maps associated to group homomorphisms. When applied to an inclusion of groups, we obtain the height 2 induced character formula of Hopkins, Kuhn, and Ravenel. For the homomorphism $G\to *$, we recover Vafa’s gauging with discrete torsion. The image of equivariant Euler classes under gauging constructs modular form-valued invariants of representations that depend on a choice of string structure. We illustrate nontrivial dependence on the string structure for a 16-dimensional representation of the Klein 4-group.

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