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

Table of contents
- Solution a

Solution

Expand

X^{a} = x^{a} + θ ψ^{a}

,

D X^{a} = ψ^{a} - i θ \overset{x}{˙}^{a}

, and

g_{ab} (X) = g_{ab} (x) + θ ψ^{c} \partial_{c} g_{ab}

. Performing the Berezin integral gives

S = \int d t [\frac{1}{2} g_{ab} \overset{x}{˙}^{a} \overset{x}{˙}^{b} + \frac{i}{2} g_{ab} \dot{ψ}^{a} ψ^{b} + \frac{i}{2} (\partial_{c} g_{ab}) ψ^{c} \overset{x}{˙}^{a} ψ^{b}] .

After a fermionic integration by parts, this is the manifestly covariant expression

S = \frac{1}{2} \int d t [g_{ab} \overset{x}{˙}^{a} \overset{x}{˙}^{b} - i g_{ab} ψ^{a} \frac{D ψ ^{b}}{d t}], \frac{D ψ ^{a}}{d t} = \dot{ψ}^{a} + Γ^{a}_{b c} \overset{x}{˙}^{b} ψ^{c} .

Under a diffeomorphism,

x

is a point of

M

,

ψ \in T_{x} M

is a tangent vector,

D_{t} ψ

is its covariant derivative along a curve, and every index is contracted with the Riemannian metric; the action is therefore invariant.

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