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.

中文翻译：

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复制几何上的费米子和 $\Theta$–$\theta$ 关系

在 arXiv:1706:09426 中，我们推测并提供了证据，证明 $n$ 属的特殊黎曼曲面的 Siegel $\Theta$-常数与 Jacobi$\theta$-函数的乘积之间的同一性。这是通过比较计算有限温度下自由费米子的 \nth \Renyi 熵的两种不同方法而产生的。在这里，我们表明，对于 $n=2$，身份是双分支黎曼曲面的 Fay 旧结果的结果。对于 $n>2$，我们提供身份两侧某些零的详细匹配。这相当于 $n=2$ 的基本身份证明，而 $n\ge 3$ 则为它提供了新的证据。我们解释了为什么额外零的存在使一般证明变得困难。