On the quantum spectral curve for AdS3 × S3 × S3 × S1 strings and the 𝔡(2, 1; α) 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 AdS3 × S3 × S3 × S1, a theory with global symmetry 𝔡(2, 1; α)⊕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 𝔬𝔰𝔭(4|2)⊕2. In this case, our proposal is based on two copies of an 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 ℤ2-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 ≠ 1/2, corresponding to the superalgebra 𝔡(2, 1; α). This novel algebraic structure represents a significant step towards understanding the Quantum Spectral Curve of the entire theory.