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

The first-order kinetic term makes ${\eta }^a$ and ${\bar{\eta }}_b$ canonically conjugate Grassmann variables. Canonical quantization gives the canonical anticommutation relations

The Hilbert space is the fermionic Fock spaceIn a polarization where ${\hat{\eta }}^a$ acts by exterior multiplication and ${\hat{\bar{\eta }}}_a$ by contraction, a general state isThe fermion number operator $N={\hat{\eta }}^a{\hat{\bar{\eta }}}_a$ returns the exterior degree, and therefore

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.

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.