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

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 $\mathfrak{d}(2,1;\alpha)^{\oplus 2}$. We focus mainly on the case when the radii of the two spheres are the same, i.e. $\alpha = 1/2$, 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 $\alpha\neq 1/2$, corresponding to the superalgebra $\mathfrak{d}(2,1;\alpha)$. This novel algebraic structure represents a significant step towards understanding the Quantum Spectral Curve of the entire theory.