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Analyticity of replica correlators and Modular ETH
High Energy Physics - Theory · Submitted 2 Sep 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.74 | 0.84 | 0.04 | asserted 0.84 |
| s1 | 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 $n$. | method 0.66 | 0.91 | 0.05 | asserted 0.99 |
| s2 | 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. | result 0.41 | 0.67 | 0.63 | hedged possibility 0.64 |
| s3 | We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large $N$ holographic CFT$_d$ in dimensions $d>2$. | method 0.53 | 0.96 | 0.06 | asserted 1.00 |
| s4 | 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. | result 0.94 | 0.93 | 0.15 | asserted 0.99 |
| s5 | We calculate the Renyi transform in $2d$ conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. | contribution 0.53 | 0.96 | 0.18 | asserted 1.00 |
| s6 | 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. | result 0.53 | 0.86 | 0.21 | asserted 0.99 |
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The paper
Original abstract
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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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 36 sentences without a confident label
Uses the method or tool of 4 papers
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In the following sections, we use the methods recently popularised by Caron-Huot in deriving the inversion formula [ref] to find the correct analytic continuation.
Analyticity in Spin in Conformal Theories · Definition · 1.00
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In the following sections, we use the methods recently popularised by Caron-Huot in deriving the inversion formula [ref] to find the correct analytic continuation.
A spacetime derivation of the Lorentzian OPE inversion formula · Definition · 1.00
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We will follow the prescription of Faulkner et al. [ref] where it was argued that the correlator of the form ([ref]) can be obtained by computing the length of the geodesic that extends from the boundary to the bulk $Z_N$ fixed point – the horizon.
A modular toolkit for bulk reconstruction · Higher $d$ Holographic CFT · 1.00
Compares with 4 papers
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We find that they match exactly with the results on the averaged off-diagonal OPE coefficients $|C_E_nOE_m|^2$ in any modular invariant $2d$ CFT [ref] at asymptotically large energies.
Probing thermality beyond the diagonal · Introduction · 1.00
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We find that they match exactly with the results on the averaged off-diagonal OPE coefficients $|C_E_nOE_m|^2$ in any modular invariant $2d$ CFT [ref] at asymptotically large energies.
A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled · Introduction · 1.00
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Comparing ([ref]) with ([ref]) we obtain for nonzero $δE$, [equation] Up to a prefactor that arises due to conventions involving $c_h,$, this result is identical to the recent results of [ref] about the off-diagonal OPE coefficients in any modular invariant $2d$ CFT on flat space.
A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots · Off-diagonal elements of modular ETH · 1.00
Cites as support 2 papers
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For the half line in a 2d CFT, it is a fact that $c_n$ is indeed independent of $n$ [ref] .
Entanglement entropy and conformal field theory · 2d CFT · 0.98
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For the half line in a 2d CFT, it is a fact that $c_n$ is indeed independent of $n$ [ref] .
Geometric and Renormalized Entropy in Conformal Field Theory · 2d CFT · 0.98
Cites as background 10 papers
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In particular, the entanglement entropy of a quantum field theory associated to the half space has been widely studied in the literature [ref] , and will serve as our starting point.
Entanglement entropy and conformal field theory · Introduction · 0.94
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In particular, the entanglement entropy of a quantum field theory associated to the half space has been widely studied in the literature [ref] , and will serve as our starting point.
Towards a derivation of holographic entanglement entropy · Introduction · 0.94
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In particular, the entanglement entropy of a quantum field theory associated to the half space has been widely studied in the literature [ref] , and will serve as our starting point.
The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT · Introduction · 0.94
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