O($D$,$D$)-covariant two-loop $\beta$-functions and Poisson-Lie T-duality

TL;DR

This paper demonstrates that the one- and two-loop $eta$-functions of bosonic string theory can be expressed in an O(D,D)-covariant way, proving two-loop renormalizability of Poisson-Lie symmetric models and their invariance under Poisson-Lie T-duality.

Contribution

It introduces an O(D,D)-covariant formulation of $eta$-functions, proving two-loop renormalizability and invariance under Poisson-Lie T-duality, with a new scheme simplifying calculations.

Findings

$eta$-functions are O(D,D)-covariant

Poisson-Lie symmetric models are two-loop renormalisable

$eta$-functions are invariant under Poisson-Lie T-duality

Abstract

We show that the one- and two-loop $\beta$-functions of the closed, bosonic string can be written in a manifestly O($D$,$D$)-covariant form. Based on this result, we prove that 1) Poisson-Lie symmetric $\sigma$-models are two-loop renormalisable and 2) their $\beta$-functions are invariant under Poisson-Lie T-duality. Moreover, we identify a distinguished scheme in which Poisson-Lie symmetry is manifest. It simplifies the calculation of two-loop $\beta$-functions significantly and thereby provides a powerful new tool to advance into the quantum regime of integrable $\sigma$-models and generalised T-dualities. As an illustrating example, we present the two-loop $\beta$-functions of the integrable $\lambda$- and $\eta$-deformation.