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Universal dynamics of heavy operators in CFT2

Collier, Scott, Maloney, Alexander, Maxfield, Henry, Tsiares, Ioannis·2020·Journal of High Energy Physics
DOI 10.1007/jhep07(2020)074
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Abstract

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with c > 1. This formula is valid when one or more of the operators has large dimension or — in the presence of a twist gap — has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT2 structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT 2, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs.

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Figure 1 The elementary crossing transformations: sphere four-point crossing between S and T channels, and torus one-point crossing between the τ and ( -1 /τ ) frames.
Figure 1 The elementary crossing transformations: sphere four-point crossing between S and T channels, and torus one-point crossing between the τ and ( -1 /τ ) frames.CC-BY · From the PDF
Figure 3 Example of a crossing transformation on the g = 2 partition function.
Figure 3 Example of a crossing transformation on the g = 2 partition function.CC-BY · From the PDF
Figure 4 A conformal block decomposition of the torus two-point function G 1 , 2 , where the kinematic parameters σ consist of a complex structure τ for the torus, and a separation w between operators. We sum over representations in the internal cuffs; for the yellow cuff i 1 , this corresponds to the operators appearing in the OPE of external operators e 1 , e 2 , and for the blue cuff i 2 , an insertion of a complete set of states in the thermal trace.
Figure 4 A conformal block decomposition of the torus two-point function G 1 , 2 , where the kinematic parameters σ consist of a complex structure τ for the torus, and a separation w between operators. We sum over representations in the internal cuffs; for the yellow cuff i 1 , this corresponds to the operators appearing in the OPE of external operators e 1 , e 2 , and for the blue cuff i 2 , an insertion of a complete set of states in the thermal trace.CC-BY · From the PDF
Figure 5 The elementary crossing moves relate different pair-of-pants decompositions of the fourpunctured sphere and the once-punctured torus, or more generally anywhere that these appear as pieces of any decomposition of a surface. The associated crossing kernels relate Virasoro conformal blocks in the corresponding channels. The fusion kernel (top) relates sphere four-point Virasoro blocks in the S- and T-channels, and the modular kernel (middle) relates torus one-point blocks in modular S -transformed frames. In the final line, we show an example relating two channels in the torus two-point function G 1 , 2 by composing these moves.
Figure 5 The elementary crossing moves relate different pair-of-pants decompositions of the fourpunctured sphere and the once-punctured torus, or more generally anywhere that these appear as pieces of any decomposition of a surface. The associated crossing kernels relate Virasoro conformal blocks in the corresponding channels. The fusion kernel (top) relates sphere four-point Virasoro blocks in the S- and T-channels, and the modular kernel (middle) relates torus one-point blocks in modular S -transformed frames. In the final line, we show an example relating two channels in the torus two-point function G 1 , 2 by composing these moves.CC-BY · From the PDF
Figure 6 The sequence of Moore-Seiberg moves to express the OPE channel torus two-point block in terms of necklace channel blocks: a modular S, followed by a fusion move.
Figure 6 The sequence of Moore-Seiberg moves to express the OPE channel torus two-point block in terms of necklace channel blocks: a modular S, followed by a fusion move.CC-BY · From the PDF
Figure 7 The sequence of moves expressing a genus 2 'dumbell' channel block in terms of 'sunset' channel blocks.
Figure 7 The sequence of moves expressing a genus 2 'dumbell' channel block in terms of 'sunset' channel blocks.CC-BY · From the PDF
Figure 8 The sequence of crossing moves applied to the sphere five-point conformal block leading to the pentagon identity (5.4) for the fusion kernel.
Figure 8 The sequence of crossing moves applied to the sphere five-point conformal block leading to the pentagon identity (5.4) for the fusion kernel.CC-BY · From the PDF

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Metadata

Authors
Collier, S., Maloney, A., Maxfield, H., & Tsiares, I.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2020-07
Volume
2020
Issue
7
ISSN
1029-8479
DOI
10.1007/jhep07(2020)074
Cited by
94
References
4
Language
en
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Collier, S., Maloney, A., Maxfield, H., & Tsiares, I. (2020). Universal dynamics of heavy operators in CFT2. Journal of High Energy Physics. https://doi.org/10.1007/jhep07(2020)074