副本关联函数和模块化ETH的解析性
Analyticity of replica correlators and Modular ETH
摘要：
研究了共形场论真空态下对应于半空间的$n$ -片复流形上的定域算子的两点关联函数。与逆拉普拉斯变换类似，我们定义了该相关函数的Renyi变换，该相关函数是一个复变量$w$的函数，与Renyi参数$n$对偶。受Caron-Huot反演公式的启发，我们证明了如果Renyi变换$f(w)$在复$w$平面上无穷远处有界，则Renyi变换圆盘$f(w)$的不连续性提供了原始副本相关函数在$n$中的唯一解析延拓.我们检查我们的公式明确计算的Renyi变换的一个特定的副本相关器在一个大的$N$全息CFT $_d$尺寸$d>2$。我们还发现Renyi变换的不连续性与模哈密顿量的两个不同本征态之间的局部算子的矩阵元有关。我们在$2d$共形场论中计算Renyi变换，并使用它来提取（模）ETH的非对角元素。我们认为，在$2d$，这是相当于非对角的CFT的OPE系数，并表明我们的技术完全再现最近的结果在文献中。
英文摘要：
We study the two point correlation function of a local operator on an $n$-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable $w$, dual to the Renyi parameter $n$. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform $f(w)$ has bounded behavior at infinity in the complex $w$ plane, the discontinuity of the Renyi transform disc $f(w)$ provides the unique analytic continuation in $n$ of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large $N$ holographic CFT$_d$ in dimensions $d>2$. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in $2d$ conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in $2d$, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.
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