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

Abstract

It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant $T\bar{T}$ 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 $[T\bar{T}{]}_{s>1}$ irrelevant operators with tuned couplings, this singularity can be resolved and the theory becomes UV complete with UV central charge $c_{\mathrm{UV}}$. We propose an approach to classifying the possible UV completions that are integrable. For the Ising model with $c_{\mathrm{IR}}=\frac{1}{2}$ we find 3 UV completions based on a single massless Majorana fermion description with $c_{\mathrm{UV}}=\frac{7}{10}$ and $\frac{3}{2}$, which both have $\mathcal{N}=1$ supersymmetry 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 $c_{\mathrm{UV}}=\frac{21}{22},\frac{15}{2}$ and $\frac{31}{2}$. We argue that it is likely there are more cases for this $E_8$ spectrum. We also study simpler cases based on $\mathrm{su}(3)$ and $\mathrm{su}(4)$ where we can propose complete classifications. For $\mathrm{su}(3)$ the infra-red (IR) theory is the 3-state Potts model with $c_{\mathrm{IR}}=\frac{4}{5}$ and we find 3 completions with $\frac{4}{5}<c_{\mathrm{UV}}\le \frac{16}{5}$. For the $\mathrm{su}(4)$ case, which has $3$ particles and $c_{\mathrm{IR}}=1$, and we find $11$ UV completions with $1<c_{\mathrm{UV}}\le 5$, most of which were previously unknown.

21 pages, 3 figures; references added, several minor corrections made, matches version accepted by JHEP

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Cited by in corpus (2)

• Mingling of the infrared and ultraviolet and the "cosmological constant" for interacting QFT in 2d
• Complete Minimal Form Factors for Irrelevant Deformations of Integrable Quantum Field Theory