TT deformation of the Ising model and its ultraviolet completion

Pure T (T) over bar 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 N=1 T (T) over bar 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 -> -infinity. Finally we also study pure T (T) over bar deformations of the off-critical Ising model.

Automated summary

Pure T (T) over bar deformations of conformal field theories are often incomplete in the ultra-violet due to square-root singularities in the ground state energy. This article demonstrates how the theory can be completed by including an infinite number of additional irrelevant perturbations, with specific examples provided for the Ising model at c(IR) = 1/2 in the infra-red. The UV completions for these cases involve N=1 T (T) over bar deformations of a free massless boson.