Relaxing unimodularity for Yang-Baxter deformed strings

TL;DR

This paper investigates the conditions under which Yang-Baxter deformations of string sigma-models maintain Weyl invariance beyond the known unimodularity requirement, revealing weaker conditions in degenerate cases and extending invariance to two loops.

Contribution

It introduces weaker conditions than unimodularity for Weyl invariance in Yang-Baxter deformations, especially when the background matrix is degenerate, and demonstrates invariance extension to two loops.

Findings

Weaker conditions replace unimodularity in degenerate cases.

Non-unimodular deformations can preserve Weyl invariance at two loops.

Simplified calculations using $O(D,D)$-covariant doubled formulation.

Abstract

We consider so-called Yang-Baxter deformations of bosonic string sigma-models, based on an $R$-matrix solving the (modified) classical Yang-Baxter equation. It is known that a unimodularity condition on $R$ is sufficient for Weyl invariance at least to two loops (first order in $\alpha'$). Here we ask what the necessary condition is. We find that in cases where the matrix $(G+B)_{mn}$, constructed from the metric and $B$-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions. We further show that for non-unimodular deformations satisfying the one-loop conditions the Weyl invariance extends at least to two loops (first order in $\alpha'$). The calculations are simplified by working in an $O(D,D)$-covariant doubled formulation.