Fermions on replica geometries and the $\Theta$ - $\theta$ relation

In arXiv:1706.09426 we conjectured and provided evidence for an identity between Siegel $\Theta$-constants for special Riemann surfaces of genus $n$ and products of Jacobi $\theta$-functions. This arises by comparing two different ways of computing the $n$^th Rényi entropy of free fermions at finite temperature. Here we show that for $n=2$ the identity is a consequence of an old result due to Fay for doubly branched Riemann surfaces. For $n \gt 2$ we provide a detailed matching of certain zeros on both sides of the identity. This amounts to an elementary proof of the identity for $n = 2$, while for $n \geq 3$ it gives new evidence for it. We explain why the existence of additional zeros renders the general proof difficult.

Keywords

entanglement entropy, Rényi entropy, conformal field theory

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The work of SM1 was partially supported by a J.C. Bose Fellowship, Government of India and that of SM2 was supported by the ERC Consolidator Grant N. 681908, “Quantum black holes: A macroscopic window into the microstructure of gravity”, and by the STFC grant ST/P000258/1.

Received 27 July 2018

Published 29 April 2019