Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 2 / c / Solution

The coherent-state representation of an ordinary fermionic trace identifies the endpoints with a minus sign, producing antiperiodic fields. Inserting fermion parity $(-1{)}^N$ supplies a second minus sign. The resulting fermionic path integral therefore hasand its time-sliced coherent-state action is precisely $S[\eta ,\bar{\eta }]$. This proves the stated supertrace representation.