Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 354 / 2 / b / ii

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- Solution ii

Solution

The divergence is local in the induced boundary metric and is cancelled by the leading holographic renormalization counterterm

I_{ct} = - \frac{1}{8 π GR} \int_{\partial M} d^{2} y - h .

On the regulated cylinder this is

I_{ct} = - \frac{R Δ t}{4 G} \frac{c o s ϵ}{s i n ^{2} ϵ} = - \frac{R Δ t}{4 G} (ϵ^{- 2} - \frac{1}{6} + O (ϵ^{2})),

which cancels the

ϵ^{- 2}

I_{GR}

. An overall sign changes if one uses the oppositely signed Euclidean generating functional; the local counterterm changes with it.

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