Universal 1-loop divergences for integrable sigma models

TL;DR

This paper introduces a universal method for calculating 1-loop divergences in integrable sigma models, simplifying previous case-by-case approaches and demonstrating renormalizability for various models.

Contribution

It develops a theory-independent formula for 1-loop divergences using Lax currents, applicable to a broad class of integrable sigma models.

Findings

Derived universal 1-loop divergence formulas for integrable sigma models.

Showed $Z_T$ coset models and deformations are 1-loop renormalizable.

Identified conditions for 1-loop scale invariance in these models.

Abstract

We present a simple, new method for the 1-loop renormalization of integrable $\sigma$-models. By treating equations of motion and Bianchi identities on an equal footing, we derive 'universal' formulae for the 1-loop on-shell divergences, generalizing case-by-case computations in the literature. Given a choice of poles for the classical Lax connection, the divergences take a theory-independent form in terms of the Lax currents (the residues of the poles), assuming a 'completeness' condition on the zero-curvature equations. We compute these divergences for a large class of theories with simple poles in the Lax connection. We also show that $Z_T$ coset models of 'pure-spinor' type and their recently constructed $\eta$- and $\lambda$-deformations are 1-loop renormalizable, and 1-loop scale-invariant when the Killing form vanishes.