Deformation of the Ising model and its ultraviolet completion

Abstract

Pure 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 supersymmetry which is broken in the renormalization group flow to low energies. We also consider multiple 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 → −∞. Finally we also study pure deformations of the off-critical Ising model.