Contents Online

# Communications in Number Theory and Physics

## Volume 13 (2019)

### Number 1

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

Pages: 225 – 251

DOI: https://dx.doi.org/10.4310/CNTP.2019.v13.n1.a8

#### Authors

#### Abstract

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

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