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 TT 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.
We demonstrate the presence of modular properties in partition functions of TT deformed conformal field theories. These properties are verified explicitly for the deformed free boson. The modular features facilitate a derivation of the asymptotic density of states in these theories, which turns out to interpolate between Cardy and Hagedorn behaviours. We also point out a sub-sector of the spectrum that remains undeformed under the TT flow. Finally, we comment on the deformation of the CFT vacuum character and its implications for the holographic dual.
We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential µ for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in µ in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) × SL(3, R) Chern-Simons theory. Our result suggests that the order µ 2 correction to the entanglement entropy may be universal for W-algebra CFTs with spinthree chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS 3 .
We investigate the off-diagonal sector of eigenstate thermalization using both local and non-local probes in 2-dimensional conformal field theories. A novel analysis of the asymptotics of OPE coefficients via the modular bootstrap is performed to extract the behaviour of the off-diagonal matrix elements. We also probe this sector using semi-classical heavy-light Virasoro blocks. The results demonstrate signatures of thermality and confirms the entropic suppression of the off-diagonal elements as necessitated by the eigenstate thermalization hypothesis.
Abstract:We consider conformal blocks of two heavy operators and an arbitrary number of light operators in a (1+1)-d CFT with large central charge. Using the monodromy method, these higher-point conformal blocks are shown to factorize into products of 4-point conformal blocks in the heavy-light limit for a class of OPE channels. This result is reproduced by considering suitable worldline configurations in the bulk conical defect geometry. We apply the CFT results to calculate the entanglement entropy of an arbitrary number of disjoint intervals for heavy states. The corresponding holographic entanglement entropy calculated via the minimal area prescription precisely matches these results from CFT. Along the way, we briefly illustrate the relation of these conformal blocks to Riemann surfaces and their associated moduli space.
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