On the Quantum Spectral Curve for $\text{AdS}_3\times \text{S}^3\times \text{S}^3\times \text{S}^1$ strings and the $\mathfrak{d}(2,1;α)$ Q-system

TL;DR

The paper develops a Quantum Spectral Curve for strings on $AdS_3\times S^3\times S^3\times S^1$, based on two copies of a covariant $\mathfrak{osp}(4|2)$ Q-system glued across branch cuts, at the symmetric point $\alpha=1/2$, with a broader proposal to generalize to $\mathfrak{d}(2,1;\alpha)$. In the symmetric sector, the QSC yields Asymptotic Bethe Ansatz equations that constrain worldsheet dressing factors in line with known S-matrix data, while in the non-symmetric sector a tension emerges between the QSC constraints, crossing, and braiding unitarity, suggesting possible refinements of the QSC or crossing relations. The work introduces a novel $\mathfrak{d}(2,1;\alpha)$ Q-system, derives its BAEs in multiple gradings, and provides a structured path toward a fullQSC for generic $\alpha$, including a detailed analysis of a new dressing phase $\Sigma_{\text{new}}$ governed by a linear integral equation and numerous perturbative and numerical checks. The results establish a framework for linking QSC constructions to the AdS$_3$/CFT$_2$ spectrum and dressings, with significant implications for integrability in lower-dimensional holography and for understanding non-quadratic branch structure in QSCs.

Abstract

In this paper, we put forward and discuss a proposal for a Quantum Spectral Curve (QSC) describing the planar spectrum of the holographic CFT dual to strings on AdS$_3\times$ S$^3\times$ S$^3\times$ S$^1$, a theory with global symmetry $\mathfrak{d}(2,1;α)^{\oplus 2}$. We focus mainly on the case when the radii of the two spheres are the same, i.e. $α= 1/2$, where the symmetry reduces to $\mathfrak{osp}(4|2)^{\oplus 2}$. In this case, our proposal is based on two copies of an $\mathfrak{osp}(4|2)$ Q-system, glued through the branch cuts of the Q-functions in a minimal way. We study in detail the ensuing analytic properties of the Q-functions in this proposal. Focusing on purely massive excitations, we consider the large worldsheet limit in which the QSC leads to a set of Asymptotic Bethe Ansatz (ABA) equations, yielding strong constraints on the (so-far unfixed) dressing factors of the worldsheet S-matrix. In a $\mathbb{Z}_2$-symmetric sector, our proposal is consistent with all previous results on the worldsheet S-matrix. However, in the non-symmetric case, we found a subtle incompatibility between the analytic constraints arising from the proposed QSC, the crossing equations present in the literature, and braiding unitarity. We discuss possible explanations for this mismatch: either our minimal QSC proposal does not hold beyond the symmetric sector, or the crossing unitarity equations receive a nontrivial correction that needs to be understood. Finally, we also propose a generalisation of the Q-system for the case of $α\neq 1/2$, corresponding to the superalgebra $\mathfrak{d}(2,1;α)$. This novel algebraic structure represents a significant step towards understanding the Quantum Spectral Curve of the entire theory.

Figures (30)

• Figure 1: Dynkin diagrams for the 3 inequivalent gradings of $\mathfrak{osp}(4|2)$ with the relevant Q-functions drawn on them. The corresponding simple root systems are built from an $\epsilon-\delta$ sequence, i.e. an ordering of 3 orthonormal unit vectors $\delta,\epsilon_1,\epsilon_2$ with $\delta$ being odd and $\epsilon_i$ even. In the gradings, the numbers $\bar{0}$ and $\bar{1}$ indicate whether the corresponding vector in the $\epsilon-\delta$ sequence is even or odd.
• Figure 2: $\mathbf{P}_A$ in the Riemann sheet with a single short cut.
• Figure 3: $\mathbf{Q}_{ij}$ in the Riemann sheet with a single long cut.
• Figure 4: Analytic structure of $\mathbf{S}^{\downarrow}$, $\hat{\mathbf{S}}^{\downarrow}$.
• Figure 5: Analytic structure of $\mathbf{S}^{\uparrow}$, $\hat{\mathbf{S}}^{\uparrow}$.