Dynamically restoring conformal invariance in (integrable) $\sigma$-models

TL;DR

This paper explores how making deformation parameters in integrable sigma models dynamical can restore conformal invariance by solving associated differential equations and analyzing RG flow fixed points, with extensions to Yang-Baxter models.

Contribution

It introduces a method to dynamically restore conformal invariance in integrable sigma models by promoting deformation parameters to functions of time and analyzing their RG flow.

Findings

Deformation parameters obey non-linear second-order differential equations.

Explicit solutions at RG fixed points demonstrate conformal invariance restoration.

Interpolation between fixed points is possible by choosing initial conditions.

Abstract

Integrable $\lambda$-deformed $\sigma$-models are characterized by an underlying current algebra/coset model CFT deformed, at the infinitesimal level, by current/parafermion bilinears. We promote the deformation parameters to dynamical functions of time introduced as an extra coordinate. It is conceivable that by appropriately constraining them, the beta-functions vanish and consequently the $\sigma$-model stays conformal. Remarkably, we explicitly materialize this scenario in several cases having a single and even multiple deformation parameters. These generically obey a system of non-linear second-order ordinary differential equations. They are solved by the fixed points of the RG flow of the original $\sigma$-model. Moreover, by appropriately choosing initial conditions we may even interpolate between the RG fixed points as the time varies from the far past to the far future.Finally, we present an extension of our analysis to the Yang--Baxter deformed PCMs.