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 $c_{ \rm UV}$. We propose an approach to classifying the possible UV completions that are integrable. For the Ising model with $c_{ \rm IR} = \tfrac{ 1}{ 2}$ we find 3 UV completions based on a single massless Majorana fermion description with $c_{ \rm UV} = \tfrac{ 7}{ 10}$ and $ \tfrac{ 3}{ 2}$, which both have ${ \cal N}=1$ 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 $c_{ \rm UV} = \tfrac{ 21}{ 22} , \tfrac{ 15}{ 2}$ and $\tfrac{ 31}{ 2}$. We argue that it is likely there are more cases for this $E_8$ spectrum. We also study simpler cases based on ${ \rm su(3)}$ and ${ \rm su(4)}$ where we can propose complete classifications. For ${ \rm su(3)}$ the infra-red (IR) theory is the 3-state Potts model with $c_{ \rm IR} = \tfrac{ 4}{ 5}$ and we find 3 completions with $\tfrac{ 4}{ 5} < c_{ \rm UV} \leq \tfrac{ 16}{ 5}$. For the ${ \rm su(4)}$ case, which has $3$ particles and $c_{ \rm IR} =1$, and we find $11$ UV completions with $1 < c_{ \rm UV} \leq 5$, most of which were previously unknown.
Categories: hep-th cond-mat.stat-mech math-ph math.MP