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

Abstract

Abstract It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant T T ¯ $$ T\overline{T} $$ 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 T ¯ $$ T\overline{T} $$ ] 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 c UV > c IR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [ T T ¯ $$ T\overline{T} $$ ] 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 c IR = 1 2 $$ \frac{1}{2} $$ we find 3 UV completions based on a single massless Majorana fermion description with c UV = 7 10 $$ \frac{7}{10} $$ and 3 2 $$ \frac{3}{2} $$ , which both have N $$ \mathcal{N} $$ = 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 E 8 appropriate to a magnetic perturbation with c UV = 21 22 , 15 12 $$ \frac{21}{22},\frac{15}{12} $$ , and 31 2 $$ \frac{31}{2} $$ . We argue that it is likely there are more cases for this E 8 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 c IR = 4 5 $$ \frac{4}{5} $$ and we find 3 completions with 4 5 $$ \frac{4}{5} $$ < c UV ≤ 16 5 $$ \frac{16}{5} $$ . For the su(4) case, which has 3 particles and c IR = 1, and we find 11 UV completions with 1 < c UV ≤ 5, most of which were previously unknown.

Keywords

• Integrable Field Theories
• Field Theories in Lower Dimensions
• Renormalization Group