Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 3 / d / Solution

The insertion of $(-1{)}^F$ makes the fermions periodic, while the isometry $f$ twists both fields by its action on $M$ and its differential on the tangent bundle. ThusThe path integral with these boundary conditions represents the equivariant supertrace ${\mathrm{Tr}}_{\mathcal{H}}((-1{)}^{F}f{e}^{-\beta H})$.