Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 307 / 3 / d

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- Solution d

Solution

The insertion of

(- 1)^{F}

makes the fermions periodic, while the isometry

f

twists both fields by its action on

M

and its differential on the tangent bundle. Thus

x (β) = f (x (0)), ψ (β) = d f_{x (0)} ψ (0) .

The path integral with these boundary conditions represents the equivariant supertrace

Tr_{H} ((- 1)^{F} f e^{- β H})

.

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