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arXiv:2205.10905 · Journal of High Energy Physics

On the classification of UV completions of integrable $$ T\overline{T} $$ deformations of CFT

Changrim Ahn, André LeClair

Published Aug 18, 2022• 14 citations
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Abstract

Abstract It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant $$ T\overline{T} $$ T T ¯ perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher [ $$ T\overline{T} $$ T T ¯ ]s>1 irrelevant operators of positive integer scaling dimension 2(s+1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge cUV> cIR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [ $$ T\overline{T} $$ T T ¯ ]s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with cIR = $$ \frac{1}{2} $$ 1 2 we find 3 UV completions based on a single massless Majorana fermion description with cUV = $$ \frac{7}{10} $$ 7 10 and $$ \frac{3}{2} $$ 3 2 , which both have $$ \mathcal{N} $$ N = 1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E8 appropriate to a magnetic perturbation with cUV = $$ \frac{21}{22},\frac{15}{12} $$ 21 22 , 15 12 , and $$ \frac{31}{2} $$ 31 2 . We argue that it is likely there are more cases for this E8 spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infrared (IR) theory is the 3-state Potts model with cIR = $$ \frac{4}{5} $$ 4 5 and we find 3 completions with $$ \frac{4}{5} $$ 4 5 < cUV≤ $$ \frac{16}{5} $$ 16 5 . For the su(4) case, which has 3 particles and cIR = 1, and we find 11 UV completions with 1 < cUV ≤ 5, most of which were previously unknown.

DOI
10.1007/jhep08(2022)179

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On the classification of UV completions of integrable TT‾ T\overline{T} TT deformations of CFT
Abstract It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant TT‾T\overline{T}TT T T ¯ perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher [ TT‾T\overline{T}TT T T ¯ ]s>1 irrelevant operators of positive integer scaling dimension 2(s+1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge cUV> cIR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [ TT‾T\overline{T}TT T T ¯ ]s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with cIR = 12\frac{1}{2}21​ 1 2 we find 3 UV completions based on a single massless Majorana fermion description with cUV = 710\frac{7}{10}107​ 7 10 and 32\frac{3}{2}23​ 3 2 , which both have N\mathcal{N}N N = 1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E8 appropriate to a magnetic perturbation with cUV = 2122,1512\frac{21}{22},\frac{15}{12}2221​,1215​ 21 22 , 15 12 , and 312\frac{31}{2}231​ 31 2 . We argue that it is likely there are more cases for this E8 spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infrared (IR) theory is the 3-state Potts model with cIR = 45\frac{4}{5}54​ 4 5 and we find 3 completions with 45\frac{4}{5}54​ 4 5 < cUV≤ 165\frac{16}{5}516​ 16 5 . For the su(4) case, which has 3 particles and cIR = 1, and we find 11 UV completions with 1 < cUV ≤ 5, most of which were previously unknown.
We present a full classification of UV complete QFTs that RG flow to the 2D Ising CFT by solving the bootstrap equations for massless right--left S-matrices. For the Ising model with E_8 spectrum, we find exactly four completions, arising from higher-T\bar T-type deformations, including a previously unknown ``golden flow'' whose UV fixed point is a diagonal su(2) coset CFT (c=25/14) along Δ_{\rm rel}=2/7. A universal Fibonacci/E_8 structure governs the R--L adjacency matrices and the Y-system periods, so that the E_8 symmetry persists across all RG scales.
The finite-volume spectrum of an integrable massive perturbation of a rational conformal field theory interpolates between massive multiparticle states in infinite-volume (IR limit) and conformal states, which are approached at zero volume (UV limit). Each state is labeled in the IR by a set of “Bethe-ansatz quantum numbers,” while in the UV limit it is characterized primarily by the conformal dimensions of the conformal field creating it. We present explicit conjectures for the UV conformal dimensions corresponding to any IR state in the ϕ1, 3-perturbed minimal models ℳ(2, 5) and ℳ(3, 5). The conjectures, which are based on a combinatorial interpretation of the Rogers-Ramanujan-Schur identities, are consistent with numerical results obtained previously for low-lying energy levels.
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T T ¯ TT‾ \mathrm{T}\overline{\mathrm{T}} TT -deformed 2D quantum field theories
It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator TT‾ \mathrm{T}\overline{\mathrm{T}} TT , built with the components of the energy-momentum tensor, enjoys very special properties in 2D quantum field theories, and can be regarded as a peculiar kind of integrable perturbation. Novel interesting features of this operator have recently emerged from the study of effective string theory models. In this paper we study further properties of this distinguished perturbation. We discuss how it affects the energy levels and one-point functions of a general 2D QFT in finite volume through a surprising relation with a simple hydrodynamic equation. In the case of the perturbation of CFTs, adapting a result by Lüscher and Weisz we give a compact expression for the partition function on a finite-length cylinder and make a connection with the exact g-function method. We argue that, at the classical level, the deformation naturally maps the action of N massless free bosons into the Nambu-Goto action in static gauge, in N + 2 target space dimensions, and we briefly discuss a possible interpretation of this result in the context of effective string models.