1-loop renormalisability of integrable sigma-models from 4d Chern-Simons theory

TL;DR

This paper proves 1-loop renormalisability of a broad class of integrable 2d sigma-models constructed via 4d Chern-Simons theory, confirming their quantum consistency and extending previous conjectures.

Contribution

It establishes the 1-loop renormalisability of integrable sigma-models from 4d Chern-Simons theory and derives the flow of the twist 1-form, confirming earlier conjectures.

Findings

Classically integrable sigma-models are renormalisable at 1-loop.

Derived the flow of the twist 1-form, a key 4d coupling.

Results apply to rational, trigonometric, and elliptic models.

Abstract

Large families of integrable 2d sigma-models have been constructed at the classical level, partly motivated by the utility of integrability on the string worldsheet. It is natural to ask whether these theories are renormalisable at the quantum level, and whether they define quantum integrable field theories. By considering examples, a folk theorem has emerged: the classically integrable sigma-models always turn out to be renormalisable, at least at 1-loop order. We prove this theorem for a large class of models engineered on surface defects in the 4d Chern-Simons theory by Costello and Yamazaki. We derive the flow of the 'twist 1-form' (a 4d coupling constant that distinguishes different 2d models), proving earlier conjectures and extending previous results. Our approach is general, using the 'universal' form of 2d integrable models' UV divergences in terms of their Lax connection and reinterpreting the result in the language of 4d Chern-Simons. These results apply equally to rational, trigonometric and elliptic models.