Relaxing unimodularity for Yang-Baxter deformed strings

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 ${\alpha }^{\prime }$). 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 ${\alpha }^{\prime }$). The calculations are simplified by working in an $O(D,D)$-covariant doubled formulation.

15 pages; v2: Now covers also inhomogeneous deformations. Important clarifications in section 3 regarding symmetric spaces