Solution
= 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 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.