The all-loop integrable spin-chain for strings on AdS_3 x S^3 x T^4: the massive sector

TL;DR

This work constructs the all-loop dynamic S-matrix for the massive sector of strings on AdS$_3\times S^3\times T^4$ by bootstrapping from the centrally extended $\mathfrak{psu}(1,1|2)^2$ symmetry, resulting in a tensor-product structure of two $\mathfrak{su}(1|1)^2$ S-matrices and two antisymmetric dressing phases. The S-matrix is shown to satisfy the Yang–Baxter equation and crossing symmetry, enabling a consistent nesting-based Bethe Ansatz that yields all-loop Bethe equations for the massive sector, with explicit weak-coupling and finite-gap limits. Perturbative comparisons demonstrate agreement with near-BMN and near-flat-space results (HT, SW, and certain BLMMT predictions) while highlighting tensions with some finite-gap-derived dressing phases. The framework lays a solid groundwork for exploring massless modes and mixed-flux backgrounds in AdS$_3$/CFT$_2$, and for extending the bootstrap to a full all-loop description of the AdS$_3$ system.

Abstract

We bootstrap the all-loop dynamic S-matrix for the homogeneous psu(1,1|2)^2 spin-chain believed to correspond to the discretization of the massive modes of string theory on AdS_3 x S^3 x T^4. The S-matrix is the tensor product of two copies of the su(1|1)^2 invariant S-matrix constructed recently for the d(2,1;alpha)^2 chain, and depends on two antisymmetric dressing phases. We write down the crossing equations that these phases have to satisfy. Furthermore, we present the corresponding Bethe Ansatz, which differs from the one previously conjectured, and discuss how our construction matches several recent perturbative calculations.

Figures (3)

• Figure 1: Three Dynkin diagrams for $\mathfrak{psu}(1,1|2)$.
• Figure 2: The action of the supercharges $\mathfrak{Q}_i$ and $\mathfrak{S}_i$ on the bi-fundamental representation \ref{['eq:bi-fund-fields']}.
• Figure 3: The Dynkin diagram for $\mathfrak{psu}(1,1|2)^2$ with the various interaction terms appearing in the Bethe ansatz indicated. The label $\pm 1$ inside the middle Dynkin nodes indicate the $\mathfrak{su}(2)$ and $\mathfrak{sl}(2)$ gradings of the left- and right-moving sectors.