  For $G$ a finite group, 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 induced character formula of
Hopkins, Kuhn, and Ravenel. For the homomorphism $G\to *$ we obtain Vafa's
formula for 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.
