$T \bar{T}$ deformation of the Ising model and its ultraviolet completion

TL;DR

This paper explores how adding an infinite set of irrelevant perturbations can complete pure $T\bar{T}$ deformations of the Ising model, resolving UV singularities and revealing new UV fixed points with supersymmetry.

Contribution

It demonstrates the UV completion of $T\bar{T}$ deformed Ising models through irrelevant perturbations, identifying new fixed points with specific central charges and supersymmetry.

Findings

Two UV completions with $c_{UV} = 3/2$ and $c_{UV} = 7/10$ for the Ising model.

Supersymmetry is present in the UV completions but broken in the IR.

For free bosons, certain deformations violate the c-theorem, leading to negative central charges.

Abstract

Pure $T\bar{T}$ deformations of conformal field theories are generally asymptotically incomplete in the ultra-violet (UV) due to square-root singularities in the ground state energy on a cylinder of circumference $R$, such that the theory is ill-defined for distances shorter than some critical $R_*$. In this article we show how a theory can be completed if one includes an infinite number of additional irrelevant perturbations. This is fully demonstrated in the case of the Ising model at $c_{IR}= 1/2$ in the infra-red (IR), where we find two completions with central charges $c_{UV} = 3/2$ and $c_{UV} = 7/10$, the latter being the tri-critical Ising model. Both of these UV completions have ${\cal N}=1$ supersymmetry which is broken in the renormalization group flow to low energies. We also consider multiple $T\bar{T}$ deformations of a free massless boson, where we cannot find a UV completion that is consistent with the c-theorem. For negative coupling $g$, which violates the c-theorem, in both cases we find $c_{UV} = -c_{IR}$ as $g \to -\infty$. Finally we also study pure $T\bar{T}$ deformations of the off-critical Ising model.