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

Table of contents
- Solution d

Solution

Diagonalize

T

with eigenvalues

λ_{a}

and use the periodic Fourier series

η^{a} (t) = \sum_{k \in Z} η_{k}^{a} e^{2 πik t}, \overset{η}{ˉ}^{a} (t) = \sum_{k \in Z} \overset{η}{ˉ}_{k}^{a} e^{- 2 πik t} .

Each Berezin integral contributes its quadratic coefficient, so

Z = \prod_{a = 1}^{n} \prod_{k \in Z} (2 πik + λ_{a}) .

k

with

- k

and using the infinite product for the hyperbolic sine gives, up to the local regularization factor,

\prod_{k \in Z} (2 πik + λ) = 2 sinh \frac{λ}{2} = e^{λ /2} (1 - e^{- λ}) .

The product of the prefactors is

e^{tr T /2} = 1

because

T

is traceless. Thus

Z = a \prod (1 - e^{- λ_{a}}) = det (1 - e^{- T}) .

This agrees with the direct identity that the supertrace of an induced linear map on an exterior algebra is its characteristic determinant.

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