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arXiv:1804.09658 · Physical review. D/Physical review. D.

Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories

Yasuaki Hikida A, Yuya Kusuki, Tadashi Takayanagi

Published Jul 3, 2018• 66 citations
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Abstract

We study properties of heavy-light-heavy three-point functions in two-dimensional conformal field theories by using the modular invariance of two-point functions on a torus. We show that the result is nontrivially consistent with the condition of eigenstate thermalization hypothesis. We also study the open-closed duality of cylinder amplitudes and derive behaviors of disk one-point functions.

DOI
10.1103/physrevd.98.026003

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Modular covariance of torus one-point functions constrains the three point function coefficients of a two dimensional CFT. This leads to an asymptotic formula for the average value of light-heavy-heavy three point coefficients, generalizing Cardy’s formula for the high energy density of states. The derivation uses certain asymptotic properties of one-point conformal blocks on the torus. Our asymptotic formula matches a dual AdS3 computation of one point functions in a black hole background. This is evidence that the BTZ black hole geometry emerges upon course-graining over a suitable family of heavy microstates.
Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter t, that have the additional property that the energy of a state at finite t is a function only of t and of the energy and momentum of the corresponding state at t = 0, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at t = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in t, to be that of a TTˉT\bar{T}TTˉ deformed CFT. Non-perturbatively, we find that for one sign of t (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.