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Pure $T\bar{T}$ deformations of conformal field theories are generallyasymptotically incomplete in the ultra-violet (UV) due to square-rootsingularities 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 aninfinite number of additional irrelevant perturbations. This is fullydemonstrated 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 theseUV completions have ${\cal N}=1$ supersymmetry which is broken in therenormalization group flow to low energies. We also consider multiple$T\bar{T}$ deformations of a free massless boson, where we cannot find a UVcompletion 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 theoff-critical Ising model.

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