Solution
= 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. Thus
$$
\boxed{x(\beta)=f(x(0)),\qquad
\psi(\beta)=df_{x(0)}\psi(0)}.
$$
The path integral with these boundary conditions represents the <equivariant supertrace> $\operatorname{Tr}_{\mathcal H}((-1)^Ff e^{-\beta H})$.