Quantum Spectral Curve for AdS$_3$/CFT$_2$: a proposal

Abstract

We conjecture the Quantum Spectral Curve equations for string theory on $AdS_3 \times S^3 \times T^4$ with RR charge and its CFT$_2$ dual. We show that in the large-length regime, under additional mild assumptions, the QSC reproduces the Asymptotic Bethe Ansatz equations for the massive sector of the theory, including the exact dressing phases found in the literature. The structure of the QSC shares many similarities with the previously known AdS$_5$ and AdS$_4$ cases, but contains a critical new feature - the branch cuts are no longer quadratic. Nevertheless, we show that much of the QSC analysis can be suitably generalised producing a self-consistent system of equations. While further tests are necessary, particularly outside the massive sector, the simplicity and self-consistency of our construction suggests the completeness of the QSC.

Figures (6)

• Figure 1: Grading used in the Asymptotic Bethe equations.
• Figure 2: Standard analytic structure of ${\bf P}$'s with one branch cut. As a consequence of this, ${\bf Q}$ functions will have an infinite ladder of cuts separated by $i$ in the lower or upper half of the analytic plane.
• Figure 3: Two ${\bf Q}$'s from different Q-systems are glued together.
• Figure 4: Two contours we use for analytic continuation
• Figure 5: Periodicity of $\mu$ as a function with long cuts is identical to the property $\mu^{++}=\mu^\gamma$ for a section with short cuts.