## Abstract

In arXiv:1706.09426 we conjectured and provided evidence for an identity between Siegel Θ-constants for special Riemann surfaces of genus n and products of Jacobi θ-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 > 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 ≥ 3 it gives new evidence for it. We explain why the existence of additional zeros renders the general proof difficult.

Original language | English |
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Pages (from-to) | 225-251 |

Number of pages | 27 |

Journal | Communications in Number Theory and Physics |

Volume | 13 |

Issue number | 1 |

Early online date | 29 Apr 2019 |

DOIs | |

Publication status | Published - 29 Apr 2019 |

## Keywords

- Conformal field theory
- Entanglement entropy
- Rényi entropy