Gluing Quantum Spectral Curves: A Two-Copy osp(4|2) Construction

TL;DR

This work extends the Quantum Spectral Curve framework to planar strings on ${AdS}_3\times S^3\times S^3\times S^1$ with pure RR flux by exploiting the ${\mathfrak{osp}}(4|2)$ symmetry. The authors construct a two-copy QQ-system linked by analytic gluing, derive the $\mathbf{Q}\tau$- and $\mathbf{P}\nu$-systems, and analyze the large-volume limit to recover the Asymptotic Bethe Ansatz, including the dispersion relation and dressing phases. They confirm crossing equations for the dressing phases and demonstrate consistency in the symmetric subsector, while highlighting a puzzle about braiding unitarity for individual phases in non-symmetric sectors. The results suggest the QSC could enable non-perturbative, exact spectral studies of this holographic background in the planar limit, with massless-mode extensions and potential further insights required beyond the symmetric sector. Overall, the work provides a concrete, symmetry-driven non-perturbative framework for AdS$_3$ integrability in a richer background than previously solved cases, paving the way for further tests against explicit string/CFT data.

Abstract

We propose a Quantum Spectral Curve for planar string theory on AdS3*S3*S3*S1 supported by pure Ramond-Ramond flux. Our proposal is built on symmetry considerations and integrability-based functional relations. To test our construction, we consider the large volume limit and successfully reproduce the cross- ing equations and the correct structure of the Bethe equations found in the literature. In a symmetric subsector, we find agreement with previously known results and furthermore extend the Asymptotic Bethe Ansatz to include massless modes. Beyond this sector, we identify an interesting puzzle regarding the compatibility of crossing equations with braiding unitarity for individual dressing phases, which warrants further investigation and may require additional physical insights or novel structures not previously encountered in related systems. As we expect the QSC to be exact in the planar limit, our proposal may open the way for non-perturbative analysis of this holographic system.

Figures (7)

• Figure 1: Two contours we use for analytic continuation. For the ABJM case, there is no difference between the two, since all cuts are quadratic there. For AdS$_3$, however, one has to distinguish between the two.
• Figure 2: The Dynkin diagram for $\mathfrak{osp}({6|4})$ (on the left) and its subdiagram $\mathfrak{osp}({4|2})$ (on the right). Each node has a Q-function associated with it. The diagram encodes relations between the functions, forming a closed $QQ$-system.
• Figure 3: Dynkin diagrams for $\mathfrak{osp}({4|2})$ and its Cartan matrix, encoding the structure of the Bethe equations.
• Figure 4: The analytic structure of the functions $\mathbf{P}$ and $\mathbf{Q}^{\downarrow/\uparrow}$.
• Figure 5: The analytic structure of the functions $\mathbf{P}_{a},\mathbf{P}^{a},\mathbf{Q}^{\downarrow/\uparrow}_i$ and $(\mathbf{Q}^{\downarrow/\uparrow})^i$