RG flows of integrable $σ$-models and the twist function

Abstract

In the study of integrable non-linear σ-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central rôle. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function.

RG flow of integrable $\mathcal{E}$-models

“…controlling the one-loop RG flow. Note that this function differs from the one presented in [16] for the λdeformation. It only has one simple pole instead of two.…”

“…In particular, we calculate the one-and two-loop RG flow of the integrable E -models they propose based on twist functions with 2N simple poles. At one loop these models are renormalisable and the running of the twist function is completely encoded in a meromorphic function f (z), like conjectured by [16] after analysing several examples. At two loops only models with N = 1 are renormalisable.…”

“…which was already conjectured in [16]. Taking furthermore into account that K A and l ∞ are invariant, the complete one-loop RG flow of the twist function is summarised by the compact equation…”

Hamiltonian integrability of the webs of integrable theories

“…More recently, the systematic and powerful construction of classically integrable theories that is based on their interpretation as explicit realisations of affine Gaudin models [11] having an arbitrary number of cites was presented in [12,13] (see also [14]). The one and two loop renormalisation group (RG) flows of some of these models were derived in [15] and [16], respectively.…”

Duality and Integrability in String Theory

“…Additionally, some of these authors, together with collaborators, pointed out the relevance of affine Gaudin models for our comprehension of two-dimen sional integrability [66,187,188]. Broadly speaking, the twist function has emerged as a central actor out of which new models can be easily built and quantum corrections explored [67,68].…”