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I am reading the Jerusalem Lectures by Harlow. On page 44 he calculates the thermal partition function using the path integral with no matter fields, $$ Z(\beta) = \int \mathcal{D}[g] e^{-I_E[g]}. $$ The calculation is done using saddle-point techniques $$ Z(\beta) \approx \sum_{g_\text{cl}} e^{-I_E[g_\text{cl}]}, $$ where $g_\text{cl}$ are classical solutions of the Einstein equations. I am confused by this sum as I have not seen a similar case in which one expands around multiple saddles in ordinary QFT.

I have read several articles where the approximation is taken around a single saddle $g_\text{cl}$. This is achieved by writing $g = g_\text{cl} + \delta g$ and expanding the action as $$ I_E[g] = I_E[g_\text{cl}] + I_{E,2}[\delta g] + \dots $$ to obtain $$ \log Z = -I_E[g_\text{cl}] + \log\int \mathcal{D}[\delta g] e^{-I_{E,2}[\delta g]} + \dots $$ If we include only the first term, we get $$ Z(\beta) \approx e^{-I_E[g_{cl}]}, $$ which agrees with the sum formula for one saddle. How can we show that the sum formula holds for multiple saddles?

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1 Answer 1

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  1. Ref. 1 uses the method of steepest descent with multiple stationary/saddle points/instanton sectors in the semiclassical limit $\hbar\to 0$, cf. e.g. this related Phys.SE post.

  2. More precisely, the presence xor absence of each saddle point in the sum may depend on the integration contour. See e.g. my Math.SE answer here for how this works in practice for an Airy function toy model.

References:

  1. D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, arXiv:1409.1231.
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