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arXiv:1902.01434 · Journal of Physics A Mathematical and Theoretical

On correlation functions in J T ¯ -deformed CFTs

Monica Guica

Published Mar 12, 2019• 59 citations
arXivHTMLPDF

Abstract

Abstract The deformation, built from the components of the stress tensor and of a current, is a universal irrelevant deformation of two-dimensional CFTs that preserves the left-moving conformal symmetry, while breaking locality on the right-moving side. Operators in the -deformed CFT are naturally labeled by the left-moving position and right-moving momentum and transform in representations of the one-dimensional extended conformal group. We derive an all-orders formula for the spectrum of conformal dimensions and charges of the deformed CFT, which we cross-check at leading order using conformal perturbation theory. We also compute the linear corrections to the one-dimensional OPE coefficients and comment on the extent to which the correlation functions in -deformed CFTs can be obtained from field-dependent coordinate transformations.

DOI
10.1088/1751-8121/ab0ef3

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Abstract The deformation, built from the components of the stress tensor and of a current, is a universal irrelevant deformation of two-dimensional CFTs that preserves the left-moving conformal symmetry, while breaking locality on the right-moving side. Operators in the -deformed CFT are naturally labeled by the left-moving position and right-moving momentum and transform in representations of the one-dimensional extended conformal group. We derive an all-orders formula for the spectrum of conformal dimensions and charges of the deformed CFT, which we cross-check at leading order using conformal perturbation theory. We also compute the linear corrections to the one-dimensional OPE coefficients and comment on the extent to which the correlation functions in -deformed CFTs can be obtained from field-dependent coordinate transformations.
It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator T \bar T T T ‾ , built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that possess a conserved U(1) U ( 1 ) current, J J . The deformation takes the schematic form J \bar T J T ‾ and is interesting because it preserves an SL(2,\mathbb{R}) \times U(1) S L ( 2 , ℝ ) × U ( 1 ) subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions.
TT‾ T\overline{T} TT deformation of correlation functions
Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the λTT‾ \lambda T\overline{T} λTT λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the TT‾ T\overline{T} TT T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.
Abstract We analyse the TT‾ T\overline{T} TT T T ¯ deformation of 2d CFTs in a special double-scaling limit, of large central charge and small deformation parameter. In particular, we derive closed formulae for the deformation of the product of left and right moving CFT characters on the torus. It is shown that the 1/c contribution takes the same form as that of a CFT, but with rescalings of the modular parameter reflecting a state-dependent change of coordinates. We also extend the analysis for more general deformations that involve TT‾ T\overline{T} TT T T ¯ , JT‾ J\overline{T} JT J T ¯ and TJ‾ T\overline{J} TJ T J ¯ simultaneously. We comment on the implications of our results for holographic proposals of irrelevant deformations.
Modular covariance and uniqueness of JT‾ J\overline{T} JT deformed CFTs
A bstract We study families of two dimensional quantum field theories, labeled by a dimensionful parameter μ , that contain a holomorphic conserved U(1) current J ( z ). We assume that these theories can be consistently defined on a torus, so their partition sum, with a chemical potential for the charge that couples to J , is modular covariant. We further require that in these theories, the energy of a state at finite μ is a function only of μ , and of the energy, momentum and charge of the corresponding state at μ = 0, where the theory becomes conformal. We show that under these conditions, the torus partition sum of the theory at μ = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in μ , to be that of a μJT‾ \mu J\overline{T} μJT μ J T ¯ deformed conformal field theory (CFT). We derive a flow equation for the JT‾ J\overline{T} JT J T ¯ deformed partition sum, and use it to study non-perturbative effects. We find non-perturbative ambiguities for any non-zero value of μ , and comment on their possible relations to holography.
In this paper, we investigate the correlation functions of the conformal field theory (CFT) with the TT‾T\overline{T}TT deformation on a torus in terms of the perturbative CFT approach, which is the extension of the previous investigations on correlation functions defined on a plane. We systematically obtain the first-order correction to the correlation functions of the CFTs with a TT‾T\overline{T}TT deformation in both operator formalism and path integral language. As a consistency check, we compute the deformed partition function, namely, the zero-point correlation function, up to the first order, which is consistent with results in the literature. Moreover, we obtain a new recursion relation for correlation functions with multiple TTT's and T‾\overline{T}T's inserted in generic CFTs on a torus. Based on the recursion relations, we study some correlation functions of stress tensors up to the first order under TT‾T\overline{T}TT deformation.
In this paper, we consider the energy and momentum transport in (1+11+11+1)-dimension conformal field theories (CFTs) that are deformed by an irrelevant operator TT‾T\overline{T}TT, using the integrability based generalized hydrodynamics, and holography. The two complementary methods allow us to study the energy and momentum transport after the in-homogeneous quench, derive the exact nonequilibrium steady states, and calculate the Drude weights. Our analysis reveals that all of these quantities satisfy universal formulae regardless of the underlying CFT, thereby generalizing the universal formulas for these quantities in pure CFTs. We also compute the exact momentum diffusion constant using the integrability-based method and confirm that it agrees with the conformal perturbation. These fundamental physical insights have important consequences for our understanding of the TT‾T\overline{T}TT-deformed CFTs. First of all, they provide the first check of the TT‾T\overline{T}TT-deformed AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​ correspondence from the dynamical standpoint. And second, we are able to identify a remarkable connection between the TT‾T\overline{T}TT-deformed CFTs and reversible cellular automata.