Breaking of integrability in elliptic deformations of AdS3 strings with mixed flux

We construct elliptic deformations of the $AdS_3 \times S^3 \times T^4$ superstring supported by mixed NSNS and RR fluxes in type IIB supergravity and investigate their integrability. The NSNS sector includes a Wess-Zumino term, and imposing that the theory preserves 8 supersymmetries also requires the presence of a B-field coupling the deformed $AdS_3$ and $S^3$ factors. We explore integrability from two different perspectives: through the construction of a flat Lax connection and the computation of the tree-level worldsheet S-matrix. For generic elliptic parameters and NSNS flux, we find inelastic $2 \rightarrow 2$ amplitudes, violating the elasticity required by integrability. In the integrable cases as identified by the Lax analysis, these inelastic amplitudes vanish and the tree-level S-matrix satisfies the classical Yang-Baxter equation. In particular, the addition of a Wess-Zumino term, which preserves integrability in the rational and trigonometric models, breaks integrability for elliptic deformations.