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

TL;DR

This paper investigates an identity between Siegel Theta-constants and Jacobi theta-functions related to computing Renyi entropies of free fermions, providing a proof for n=2 and evidence for higher n.

Contribution

It offers an elementary proof for the identity at n=2 and extends the evidence for its validity at n>2, connecting Riemann surface properties with quantum entropy calculations.

Findings

Proof of the identity for n=2 based on Fay's result.

Evidence supporting the identity for n>2 through zero matching.

Explanation of the challenges in generalizing the proof for n>2.

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 \nth \Renyi 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 $n>2$ 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 $n\ge 3$ it gives new evidence for it. We explain why the existence of additional zeros renders the general proof difficult.