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 has
$$
\boxed{\eta(1)=\eta(0),\qquad\bar\eta(1)=\bar\eta(0)},
$$
and its time-sliced coherent-state action is precisely $S[\eta,\bar\eta]$. This proves the stated supertrace representation.

Solution

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

A fermionic path integral integrates Grassmann-valued histories. Its Gaussian integral is a determinant, in contrast with the inverse determinant produced by a bosonic Gaussian integral.


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