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

TL;DR

This paper reveals a novel link between integrability and RG flow in 2D sigma models with local couplings, showing that time-dependent models satisfying integrability conditions follow RG equations, suggesting they may be integrable.

Contribution

It demonstrates that promoting couplings to time-dependent functions in integrable sigma models leads to RG flow equations, establishing a new connection between integrability and renormalization group dynamics.

Findings

Time-dependent sigma models obey RG equations when integrability is preserved.

Existence of Lax connection indicates potential integrability of these models.

Models relate to string theory backgrounds with specific geometric and dilaton configurations.

Abstract

We consider several classes of $\sigma$-models (on groups and symmetric spaces, $\eta$-models, $\lambda$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $\tau$. We observe that (i) starting with a classically integrable 2d $\sigma$-model, (ii) formally promoting its couplings $h_\alpha$ to functions $h_\alpha(\tau)$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_\alpha(\tau)$ must solve the 1-loop RG equations of the original theory with $\tau$ 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 $\sigma$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such $\sigma$-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 $\sigma$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.