$AdS_3/CFT_2$, finite-gap equations and massless modes

TL;DR

This work revisits the finite-gap formulation of string theory on AdS3-based backgrounds and shows that the Virasoro constraints require a generalized residue condition (GRC) rather than the traditional null-residue constraint. The GRC enables the inclusion of massless string modes, which were previously missing in the finite-gap spectrum, and is demonstrated through explicit BMN-limit analyses on AdS3×S3×S^3×S^1 and AdS3×S^3×T^4. By deriving and applying the GRC within the D(2,1;α)^2×U(1)^2 and PSU(1,1|2)^2×(U(1)^4)^2 finite-gap frameworks, the authors reproduce the full BMN spectra, including massless excitations, and show consistency with known AdS5/AdS4 results where the GRC reduces to standard conditions. These results provide a path to a complete finite-gap/Bethe Ansatz description of AdS3/CFT2 with massless modes and suggest avenues for extending the approach to other semi-symmetric cosets. The work thus resolves a long-standing discrepancy and broadens the applicability of integrability to massless sectors in AdS3/CFT2.

Abstract

It is known that string theory on $AdS_3\times M_7$ backgrounds preserving 16 supercharges is classically integrable. This integrability has been previously used to write down a set of integral equations, known as the finite-gap equations. These equations can be solved for the closed string spectrum of the theory. However, it has been known for some time that the $AdS_3\times M_7$ finite-gap equations do not capture the dynamics of the massless modes of the closed string theory. In this paper we re-examine the derivation of the $AdS_3\times M_7$ finite-gap system. We find that the conditions that had previously been imposed on these integral equations in order to implement the Virasoro constraints are too strict, and are in fact not required. We identify the correct implementation of the Virasoro constraints on finite-gap equations and show that this new, less restrictive condition captures the complete closed string spectrum on $AdS_3\times M_7$.