The complete AdS_3 x S^3 x T^4 worldsheet S-matrix

TL;DR

The authors derive the non-perturbative worldsheet S-matrix for Type IIB strings on AdS$_3\times$S$^3\times$T$^4$ with RR flux, incorporating massless excitations by carefully constructing the off-shell symmetry algebra $\mathcal{A}$ and its representations. They build the S-matrix from tensor products of short $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ representations, constrained by unitarity, Yang–Baxter, and crossing symmetry, and they express the full matrix in terms of five dressing factors tied to the massive, massless, and mixed sectors. The work provides explicit S-matrix blocks, their normalization, and crossing equations, enabling an integrable description of the AdS$_3$/CFT$_2$ correspondence that includes massless modes. This framework lays the groundwork for solving the spectrum via Bethe-Yang/quantum spectral curve approaches and paves the way for generalizations to mixed fluxes and related AdS$_3$ backgrounds, with potential links to black-hole physics and orbifold constructions.

Abstract

We derive the non-perturbative worldsheet S matrix for fundamental excitations of Type IIB superstring theory on AdS_3 x S^3 x T^4 with Ramond-Ramond flux. To this end, we study the off-shell symmetry algebra of the theory and its representations. We use these to determine the S matrix up to scalar factors and we derive the crossing equations that these scalar factors satisfy. Our treatment automatically includes fundamental massless excitations, removing a long-standing obstacle in using integrability to study the AdS_3/CFT_2 correspondence. The present paper contains a detailed derivation of results first announced in arXiv:1403.4543.

Figures (2)

• Figure 1: The left and right $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ multiplets consists of two bosons $Y^{\hbox{\tiny L},\hbox{\tiny R}}$, $Z^{\hbox{\tiny L},\hbox{\tiny R}}$ and of two fermions $\eta_{\ \dot{a}}^{\hbox{\tiny L},\hbox{\tiny R}}$, corresponding to transverse directions on $\textup{AdS}_3\times\textup{S}^3$. The fermions carry an index $\dot{a}$ of the fundamental representation of $\mathfrak{su}(2)_{\bullet}$. Note that off-shell any excitation is charged under all supercharges, whereas on-shell left (respectively right) excitations are charged only under left (respectively right) supercharges. We indicate the supercharges whose action corresponds to the outermost arrows of the diagram. The innermost ones follow by Hermitian conjugation.
• Figure 2: The eight massless excitations form two $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ multiplets. The four bosons $T^{\dot{a}a}$ are charged both under $\mathfrak{su}(2)_{\bullet}$ and $\mathfrak{su}(2)_{\circ}$, while the four fermions $\chi^{a}, \widetilde{\chi}^{a}$ are in the fundamental representation of $\mathfrak{su}(2)_{\circ}$ only. Again we indicate the charges whose action corresponds to the outermost arrows of the diagram, while the innermost ones follow by Hermitian conjugation. Note how $\mathfrak{su}(2)_{\circ}$ relates the two $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ modules, yielding a single irreducible representation of $\mathcal{A}$, denoted by a box.