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Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes

Belin, Alexandre, de Boer, Jan, Liska, Diego·2022·Journal of High Energy Physics
DOI 10.1007/jhep06(2022)116
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Abstract

The Eigenstate Thermalization Hypothesis makes a prediction for the statistical distribution of matrix elements of simple operators in energy eigenstates of chaotic quantum systems. As a leading approximation, off-diagonal matrix elements are described by Gaussian random variables but higher-point correlation functions enforce non-Gaussian corrections which are further exponentially suppressed in the entropy. In this paper, we investigate non- Gaussian corrections to the statistical distribution of heavy-heavy-heavy OPE coefficients in chaotic two-dimensional conformal field theories. Using the Virasoro crossing kernels, we provide asymptotic formulas involving arbitrary numbers of OPE coefficients from modular invariance on genus-g surfaces. We find that the non-Gaussianities are further exponentially suppressed in the entropy, much like the ETH. We discuss the implication of these results for products of CFT partition functions in gravity and Euclidean wormholes. Our results suggest that there are new connected wormhole geometries that dominate over the genus-two wormhole.

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Figures

Figure 1 On the left the trivalent graph associated with the configuration of OPE coefficients in the skyline channel. On the right, the comb channel.
Figure 1 On the left the trivalent graph associated with the configuration of OPE coefficients in the skyline channel. On the right, the comb channel.CC-BY · From the PDF
Figure 2 The two conformal block decompositions of the four-point function with four different primary operators. The fusion kernel is the change-of-basis matrix relating the two bases of conformal blocks in the s- and t-channel.
Figure 2 The two conformal block decompositions of the four-point function with four different primary operators. The fusion kernel is the change-of-basis matrix relating the two bases of conformal blocks in the s- and t-channel.CC-BY · From the PDF
Figure 3 The modular kernel relating the two different conformal block decompositions of the torus one-point function.
Figure 3 The modular kernel relating the two different conformal block decompositions of the torus one-point function.CC-BY · From the PDF
Figure 4 The crossing kernels relating the different pair-of-pants decompositions or channels of the genus-three Riemann surface. From left to right the names of these channels are: skyline, sunset, necklace, and comb. The trivalent graphs associated to each decomposition are shown below each surface. Note that the fusion kernel may change the topology of the graph while the modular kernel always leaves the diagram invariant.
Figure 4 The crossing kernels relating the different pair-of-pants decompositions or channels of the genus-three Riemann surface. From left to right the names of these channels are: skyline, sunset, necklace, and comb. The trivalent graphs associated to each decomposition are shown below each surface. Note that the fusion kernel may change the topology of the graph while the modular kernel always leaves the diagram invariant.CC-BY · From the PDF
Figure 5 The sequence of transformations we use to solve for the statistics of the OPE coefficients associated with the skyline channel.
Figure 5 The sequence of transformations we use to solve for the statistics of the OPE coefficients associated with the skyline channel.CC-BY · From the PDF
Figure 6 A summary of the process we use to simplify the skyline channel statistics. This is a relationship between two different OPE densities. Here, ¯ K corresponds to the antiholomorphic counterpart of the crossing kernel. Corrections to this relation are exponentially suppressed at large values of the internal momenta P 3 , ¯ P 3 .
Figure 6 A summary of the process we use to simplify the skyline channel statistics. This is a relationship between two different OPE densities. Here, ¯ K corresponds to the antiholomorphic counterpart of the crossing kernel. Corrections to this relation are exponentially suppressed at large values of the internal momenta P 3 , ¯ P 3 .CC-BY · From the PDF
Figure 7 The different pair-of-pants decompositions of the genus-two Riemann surface and the crossing transformations that relate them. The leftmost cutting corresponds to the dumbbell channel and the rightmost to the sunset channel.
Figure 7 The different pair-of-pants decompositions of the genus-two Riemann surface and the crossing transformations that relate them. The leftmost cutting corresponds to the dumbbell channel and the rightmost to the sunset channel.CC-BY · From the PDF
Figure 8 The sequence of transformations we use to find the statistics of the OPE coefficients in the comb channel.
Figure 8 The sequence of transformations we use to find the statistics of the OPE coefficients in the comb channel.CC-BY · From the PDF
Image 9
CC-BY · From the PDF
Figure 11 Analytic structure of the integrand in the fusion kernel with s = σP s and P s → ∞ . There are eight semi-infinite lines of poles extending to the left and to the right of the integration contour. These families of poles are represented in the plot by black wavy lines. In this example, several of these families overlap so only four wavy lines appear in the image. The dotted line is the integration contour we use to compute the asymptotics of the integral. In the diagram, lighter shading corresponds to a larger absolute value of the integrand in each section.
Figure 11 Analytic structure of the integrand in the fusion kernel with s = σP s and P s → ∞ . There are eight semi-infinite lines of poles extending to the left and to the right of the integration contour. These families of poles are represented in the plot by black wavy lines. In this example, several of these families overlap so only four wavy lines appear in the image. The dotted line is the integration contour we use to compute the asymptotics of the integral. In the diagram, lighter shading corresponds to a larger absolute value of the integrand in each section.CC-BY · From the PDF
Figure 12 Analytic structure of the integrand in the fusion kernel in the limit where all operators are heavy. The contour of integration is shown with a dotted line. The integral is dominated by the two poles at the origin.
Figure 12 Analytic structure of the integrand in the fusion kernel in the limit where all operators are heavy. The contour of integration is shown with a dotted line. The integral is dominated by the two poles at the origin.CC-BY · From the PDF
Figure 13 Analytic structure of the integrand in the fusion kernel in the limit P s -δ s = P 1 -δ 1 = P 2 -δ 2 = P →∞ and light momenta not equal to the identity. This time integral is dominated by its saddle point at σ = -i/ 2 .
Figure 13 Analytic structure of the integrand in the fusion kernel in the limit P s -δ s = P 1 -δ 1 = P 2 -δ 2 = P →∞ and light momenta not equal to the identity. This time integral is dominated by its saddle point at σ = -i/ 2 .CC-BY · From the PDF

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Metadata

Authors
Belin, A., de Boer, J., & Liska, D.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2022-06-20
Volume
2022
Issue
6
ISSN
1029-8479
DOI
10.1007/jhep06(2022)116
Cited by
48
References
3
Language
en
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Belin, A., de Boer, J., & Liska, D. (2022). Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes. Journal of High Energy Physics. https://doi.org/10.1007/jhep06(2022)116