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

Diagonalize $T$ with eigenvalues ${\lambda }_a$ and use the periodic Fourier seriesEach Berezin integral contributes its quadratic coefficient, soPairing $k$ with $-k$ and using the infinite product for the hyperbolic sine gives, up to the local regularization factor,The product of the prefactors is $e^{\mathrm{tr}T/2}=1$ because $T$ is traceless. ThusThis agrees with the direct identity that the supertrace of an induced linear map on an exterior algebra is its characteristic determinant.