Sigma models with local couplings: a new integrability-RG flow connection

Abstract

We consider several classes of σ-models (on groups and symmetric spaces, η-models, ⋋-models) with local couplings that may depend on the 2d coordinates, e.g. on time τ . We observe that (i) starting with a classically integrable 2d σ-model, (ii) formally promoting its couplings hα to functions hα(τ ) of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that hα(τ ) must solve the 1-loop RG equations of the original theory with τ interpreted as RG time. This provides a novel example of an ‘integrability-RG flow’ connection. The existence of a Lax connection suggests that these time-dependent σ-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such σ-models with D-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a (D + 2)-dimensional conformal σ-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.

RG flows of integrable σ-models and the twist function

“…The results of this article bring new evidences of the deep relation between integrable properties of σ-models and their renormalization group flow. A connection between these two aspects was also recently put forward in [62] for integrable σ-models on group manifolds and symmetric spaces and their Yang-Baxter and λ-deformations, relating the renormalization group flow of these models to the existence of a Lax connection for the theory obtained by promoting their parameters to local coupling (depending on the two-dimensional spacetime coordinates). It would be interesting to investigate whether a similar relation holds for the integrable coupled σ-model whose renormalization group flow we determine in the present article.…”

Integrability vs. RG flow in G × G and G × G/H sigma models

“…Fixing the light-cone gauge on u, one then gets back the original σ-model with "local" couplings depending on 2d time according to the RG equations. It would be interesting to study whether the connection between the classical Lax integrability of such local-coupling models and the RG evolution of couplings observed in [61] applies also to the models discussed in this paper.…”

Yang–Baxter deformations of the principal chiral model plus Wess–Zumino term

“…A related direction is to investigate, along similar lines, the YB deformation of Z 4 permutation supercosets with WZ term [45], generalising the case without WZ term of [46]. It would also be interesting to study if the new YB deformations of the PCM constructed in [47] can be applied to the PCM plus WZ term, as well as the classical integrability of the action (3.14) with local couplings [48].…”