Massless flows between minimal W models

Abstract

We study the renormalisation group flows between minimal W models by means of a new set of nonlinear integral equations which provide access to the effective central charge of both unitary and nonunitary models. We show that the scaling function associated to the nonunitary models is a nonmonotonic function of the system size.1. A recent study of the renormalisation group flows between nonunitary minimal models revealed an unexpected behaviour for the groundstate energy E 0 (R), in that it was a nonmonotonic function of the system size R [1]. The nonmonotonicity was illustrated using the finite-size scaling function c eff (r), which up to the bulk term is proportional to the groundstate energywhere M is the so-called crossover scale (the mass in massive theories). As the system size goes to zero c eff (r) becomes the effective central chargeWe denote the actual central charge by c while ∆ 0 is the conformal dimension of the lowest primary field of the UV CFT. The effective central charge and the central charge of the unitary minimal models coincide, and according to Zamolodchikov's c-theorem [2] there exists a function c which is monotonic. However, apart from the UV and IR points at which c equals the central charge of the relevant CFT, it is not clear if there is any connection with E 0 (r). Nevertheless the groundstate energy of the unitary models is always monotonic. Analogously it had been thought that the groundstate energy of the nonunitary models would also be monotonic, but the results of [1] and [3-6] provide a number of counter examples. In this letter we study a further set of perturbed conformal field theories, demonstrating that c eff (r) behaves nonmonotonically for the majority of nonunitary models.We consider the minimal models W G p,q N based on one of the simply laced Lie algebras. The models are specified by two coprime integers p and q with p > h, in terms of which the central charge and the effective central charge are c = N 1 − h(h + 1)(p − q) 2 pq , c eff = N 1 − h(h+1) pq .

T T ¯ $$ \mathrm{T}\overline{\mathrm{T}} $$ -deformed 2D quantum field theories

“…In the same paper, the setup was generalised to describe the deformation, via the same CDD factor, of an arbitrary 2D conformal field theory (CFT). A tight link was observed to previous studies on massless RG flows with leading IR attracting operator given by the TT composite operator built with the chiral and anti-chiral components of the CFT stress-energy tensor (see [10][11][12][13][14]). Earlier observations on the special rôle played by the TT operator in integrable perturbations of CFTs were recorded in a few occasions, mainly in the framework of Form Factors [15,16] for correlation functions, and hints on possible links with CDD factors and TBA may be envisaged in [17].…”

“…Let us also briefly mention that there are other NLIEs describing integrable CFTs, as well as massless flows between minimal models [11,13,14]. The analysis of this section could be repeated without essential modifications to study the t-deformation of these systems as well.…”

Finite size effects and the supersymmetric sine-Gordon models

“…Furthermore, it is curious that the TBA driving term trick to find equations for the massless flows also works for the NLIEs; though only for the unitary coset models. A rather different method was used in [39] for the sine-Gordon model, also [43,58], to obtain NLIEs describing massless flows at any value of β 2 . It remains to be seen whether a similar idea can be used on the new NLIEs.…”

ODE/IM correspondence and the Argyres-Douglas theory

“…This is the non-unitary W-minimal model W A m (p, q) for a pair of coprime integers p = m + 1 and q = m + n + 2. The effective central charge of this CFT is given by [41]…”