Equivariant elliptic cohomology, gauged sigma models, and discrete torsion

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 → ∗, 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.