The Quantum Spectral Curve (QSC) conjectured in 2021 for the planar superstring on AdS3×S3×T4 with pure Ramond–Ramond flux reproduces the massive-sector Asymptotic Bethe Ansatz at large volume; work in 2022–2026 showed that its unmodified equations also encode massless worldsheet excitations in that asymptotic limit, and yielded massive-state energies at finite coupling from weak to strong coupling. What remains open is the completion of this program: constructing finite-coupling QSC solutions for string states built on massless excitations — whose wrapping effects are not exponentially suppressed and whose Q-functions require non-quadratic branch cuts and h1h^1 weak-coupling scaling — computing their exact planar spectrum and validating it against the mirror Thermodynamic Bethe Ansatz, and completing the rigorous derivation of the QSC equations themselves from the mirror TBA, whose main steps have now been carried out in the left-right-symmetric sector while its extension beyond that sector and the proof of its analyticity assumptions remain open. Extensions to mixed-flux and related backgrounds are suggested by the 2021 proposal as further directions but lie outside this open core.

Background

In the planar (large-NN) limit, several AdS/CFT dual pairs are integrable, and their exact spectrum of string energies / scaling dimensions can be computed by the Quantum Spectral Curve (QSC): a finite closed system of functional equations for Q-functions of a spectral parameter uu, supplemented by analyticity conditions. The QSC resums all finite-size ('wrapping') corrections that the Asymptotic Bethe Ansatz (ABA) — the large-volume Bethe-equation description — misses. This machinery solved the spectral problem of planar N=4N=4 super Yang–Mills theory (strings on AdS5×S5) and of ABJM theory.

String theory on AdS3×S3×T4 supported by Ramond–Ramond (RR) three-form flux is dual to a two-dimensional conformal field theory with small N=(4,4)N=(4,4) superconformal symmetry, conjecturally a symmetric-product orbifold of T4T^4 — the one integrable AdS/CFT pair whose dual CFT is still unidentified. Unlike its higher-dimensional cousins, its worldsheet spectrum contains, besides massive modes, massless excitations originating from the compactification torus. The exact two-body worldsheet S-matrix, the ABA for the massive sector with its exact dressing phases (the scalar S-matrix factors fixed by crossing symmetry), and the mirror Thermodynamic Bethe Ansatz (TBA) — infinitely many integral equations encoding the exact finite-volume spectrum — were established by Borsato, Ohlsson Sax, Sfondrini and Stefański and by Frolov and Sfondrini. Massless modes make wrapping corrections non-exponentially suppressed at any volume, so no asymptotic method can determine the spectrum: an exact finite-volume formulation is required.

In 2021, Cavaglià, Gromov, Stefański and Torrielli (and, independently and simultaneously, Ekhammar and Volin) conjectured the QSC for this background: two psu(1,1|2) Q-systems — supersymmetric ladders of Q-functions tied by QQ-relations — glued through their branch cuts, with a new structural feature: the branch points of the Q-functions are no longer of square-root type, i.e. the cuts are non-quadratic. Under a working assumption that certain intermediate ('mu-') functions become quadratic at large string length, the conjecture reproduces the massive-sector ABA including the exact dressing phases. The authors explicitly listed what remained open: whether massless modes and their wrapping effects are captured; the unproven mu-monodromy assumption; finite-coupling completeness checks (weak-coupling expansions, near-BPS solutions — solutions close to supersymmetry-protected states — and high-precision numerics); possible modifications of the QQ-relations (e.g. the condition Q12∣12=1Q_{12|12}=1) or gluing; and, as suggested further directions, extensions to AdS3×S3×S3×S1, mixed RR/NSNS (Ramond–Ramond / Neveu–Schwarz–Neveu–Schwarz) flux, and AdS2/CFT1.

Since then the conjecture has passed growing tests. It was solved at weak coupling and numerically at finite coupling for massive-sector states — the analogues of the Konishi operator of N=4N=4 SYM, i.e. the simplest unprotected states — and most recently for short strings from weak to strong coupling. A wider class of solutions was shown to reduce, at large volume, to Bethe equations for both massive and massless particles, so the unmodified QSC does encode massless modes asymptotically, and its analyticity fixes the dressing phases more restrictively than the S-matrix bootstrap. Non-trivial exchange relations of massless modes were found to modify the crossing equations. Finally, the derivation of the QSC from the mirror TBA has been carried out in the left-right-symmetric sector: the Y-system (functional relations on two T-hook-shaped lattices of Y-functions) and its discontinuity relations were followed by a derivation of the conjectured QSC in a compact form, at the price of analyticity and minimality assumptions that remain unproven. No work has yet computed the finite-coupling spectrum of states containing massless excitations from the QSC — the open core stated below.

Problem Statement

For the planar superstring on AdS3×S3×T4\mathrm{AdS}_3\times S^3\times T^4 supported by pure Ramond–Ramond flux, complete the Quantum Spectral Curve program that the 2021 proposal initiated:

(a) Construct solutions of the conjectured QSC, at finite values of the 't Hooft coupling hh (the effective string tension of the planar theory), that describe string states built on massless worldsheet excitations — implementing the required modified analytic properties (non-quadratic branch cuts, h1h^1 weak-coupling scaling) — and compute their exact planar energies as functions of hh and the charges (Δ,S,J,K)(\Delta, S, J, K): the conformal dimension Δ\Delta, the spin SS, and the two R-symmetry charges JJ and KK, i.e. the two S3S^3 angular momenta of the dual CFT. The relevant states live in the sector of zero winding and momentum on T4T^4 in which the conjectured QSC is formulated. This would demonstrate concretely that the QSC resums the non-exponentially-suppressed massless and mixed-mass wrapping corrections.

(b) Validate these massless-state spectra against the mirror Thermodynamic Bethe Ansatz, completing the QSC–TBA cross-validation that currently exists only for the massive sector.

(c) Extend the rigorous derivation of the QSC equations themselves (QQ-relations, analyticity assumptions, gluing conditions) from the mirror TBA / Y-system beyond the scope of arXiv:2610.07147: the derivation existing to date holds in the left-right-symmetric sector and relies on unproven analyticity, minimality and monodromy assumptions, so the open part is to lift that restriction and prove those assumptions, thereby turning the conjecture into a theorem and settling the status of the analyticity and large-length monodromy assumptions used in the original asymptotic match.

The common objective uniting these parts is a single exact, validated QSC description of the planar AdS3/CFT2 string spectrum for the pure-RR background. Parts (a)–(c) are jointly required: they are three independently needed components of one program, and resolving any one of them — or any subset — does not resolve the task, since partial progress on one part leaves the others open. Part (a) is a program-level request covering the massless-excitation sector of the states described there rather than a single state; it is met at the standard of evidence already attained for massive-sector states (finite-coupling numerical solutions together with weak-coupling expansions), not by any isolated numerical point. The extension directions suggested by the source — the mixed RR/NSNS-flux AdS3×S3×T4\mathrm{AdS}_3\times S^3\times T^4 background, AdS3×S3×S3×S1\mathrm{AdS}_3\times S^3\times S^3\times S^1, and AdS2/CFT1\mathrm{AdS}_2/\mathrm{CFT}_1 — are natural generalisations beyond this problem and are not required for its completion. The asymptotic encoding of massless modes and the non-necessity of QQ/gluing modifications are settled and lie outside this task, while everything asked in (a)–(c) above remains open.

References

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