New families of flows between two-dimensional conformal field theories

Abstract

We present evidence for the existence of infinitely-many new families of renormalisation group flows between the nonunitary minimal models of conformal field theory. These are associated with perturbations by the φ 21 and φ 15 operators, and generalise a family of flows discovered by Martins. In all of the new flows, the finitevolume effective central charge is a non-monotonic function of the system size. The evolution of this effective central charge is studied by means of a nonlinear integral equation, a massless variant of an equation recently found to describe certain massive perturbations of these same models. We also observe that a similar nonmonotonicity arises in the more familiar φ 13 perturbations, when the flows induced are between nonunitary minimal models.

Massless flows between minimal W models

“…Motivated by [1,6], we associate two functions f (a) R and f (a) L to each node of the Dynkin diagram, introduce new kernels and twists such that the functions satisfy…”

“…As explained in [1], we fix the kernel functions by considering the equations in the limits in which r → 0 and r → ∞. In the far infrared, the massless equations coincide with the ultraviolet limit of the massive equations, with kernel ϕ ab (θ) replaced by φ ab (θ).…”

“…After some manipulations [1], the fourier transformed massive and massless equations can be directly compared. With the tilde denoting the fourier transformed functions, the equations will match provided χ ab (k) and α ′ a satisfy…”

“…The expansion coefficients can be calculated by taking the appropriate residues, but we will only need φ (1) 1a and χ (1) 1a , both of which can be written as…”

“…From the NLIE point of view [1] corrections to c eff (r) come from the θ → −∞ asymptotic of χ ab (θ) given by (29), the e hθ/(ξ−1) term generating a a series of the form r −2h/(ξ−1) . Comparing with the CPT expansion (38) leads to the prediction ∆ IR = 1 + h/(ξ−1), which can be identified with the conformal dimension of the primary field Φ adj ′ with weights Ω = Ω id , Ω ′ = Ω adj .…”