Communications in Number Theory and Physics

Volume 13 (2019)

Number 1

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

Pages: 225 – 251



Sunil Mukhi (Indian Institute of Science Education and Research, Pashan, India)

Sameer Murthy (Department of Mathematics, King’s College London, United Kingdom)


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.


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