The Quantum Spectral Curve (QSC) conjectured in 2021 for planar type IIB string theory on AdS3×S3×T4AdS_3\times S^3\times T^4 with pure Ramond–Ramond flux is the candidate exact solution of the planar spectral problem of this holographic duality, but it was obtained by bottom-up guesswork rather than derivation, and several of its load-bearing analyticity inputs remain unproven. Later work has confirmed parts of the proposal: the asymptotic large-volume limit now reproduces Bethe equations for both massive and massless excitations, weak- and strong-coupling numerical solutions exist for massive short-string states, and the original dressing-phase incompatibility was resolved by modified dressing factors — with a remaining mismatch for the massless–massless phase. The open problem is to establish the QSC's exactness and completeness from first principles: prove or derive the monodromy properties of the asymptotic μ\mu-functions and the off-diagonal gluing-matrix structure assumed in the construction, resolve the massless–massless dressing-phase mismatch, derive the exact mirror thermodynamic Bethe ansatz (TBA) equations from the QSC (or vice versa), and extract validated finite-coupling spectral predictions for states containing massless excitations.

Background

String theory on AdS3×S3×T4AdS_3\times S^3\times T^4 supported by Ramond–Ramond (R–R) flux, the near-horizon geometry of the D1–D5 brane system, is one of the maximally supersymmetric holographic dualities. In the planar limit (large string angular momentum JJ), this string theory is believed to be integrable: its exact worldsheet 2-to-2 S-matrix is fixed up to dressing phases by the symmetry algebra psu(1,1∣2)2\mathfrak{psu}(1,1|2)^2, and these phases are constrained by crossing equations and were determined by Borsato, Ohlsson Sax, Sfondrini, Stefański jr. and Torrielli (arXiv:1306.2512). A novel feature compared to AdS5/CFT4AdS_5/CFT_4 is the presence of massless worldsheet excitations (the four T4T^4 bosons and their superpartners), whose interactions are nontrivial and whose wrapping corrections are not exponentially suppressed at large volume, so the Asymptotic Bethe Ansatz (ABA) alone cannot capture the finite-size spectrum. Exact finite-size information is instead encoded in the mirror thermodynamic Bethe ansatz (TBA) — integral equations for the ground state and, by analytic continuation, excited states, built on the mirror theory (the worldsheet theory analytically continued to imaginary momenta, whose virtual particles encode finite-size wrapping effects) — which Frolov and Sfondrini constructed for this background (arXiv:2112.08898). The Quantum Spectral Curve (QSC) is a more compact reformulation of such exact spectral equations as a Riemann–Hilbert-type system for a finite set of Q-functions (building blocks of Bethe wavefunctions) tied together by QQ-relations and analyticity (gluing) conditions; it solved the planar spectral problem in AdS5/CFT4AdS_5/CFT_4 and AdS4/CFT3AdS_4/CFT_3. In 2021, Cavaglià, Gromov, Stefański jr. and Torrielli (arXiv:2109.05500) — independently and identically to Ekhammar and Volin's monodromy-bootstrap construction (arXiv:2109.06164) — conjectured a QSC for this model. Its key novelty is that the branch cuts of the Q-functions are no longer of square-root (quadratic) type. The proposal was verified in the large-volume limit, where it reproduces the massive-sector ABA including dressing phases, but only under an unproven technical assumption about the monodromy (mirror ii-periodicity and effectively quadratic cuts) of certain μ\mu-functions — auxiliary functions built from the Q-functions — in the asymptotic limit; the gluing matrices GG, which relate Q-functions across the branch cut and were motivated to be constant and off-diagonal, were likewise imposed rather than derived. A revised version of the paper also showed that the building block χ−(u,v)\chi^-(u,v) of the original dressing phases carries an extra branch point at u∼∞u\sim\infty incompatible with the QSC analyticity, a tension resolved by the modified dressing factors of Frolov and Sfondrini (arXiv:2112.08896). Subsequent work substantially advanced the program: Cavaglià, Ekhammar, Gromov and Ryan (arXiv:2211.07810) developed weak-coupling perturbative and high-precision numerical solutions (for massive-sector twist-two states, i.e. states carrying two units of the S3S^3 angular-momentum charge that plays the role of spin-chain length), finding isolated solutions — evidence of completeness — but also parasitic branch points obstructing strong-coupling numerics; Ekhammar, Gromov and Stefański jr. (arXiv:2412.11915) showed that the large-volume limit of the QSC yields Bethe equations for both massive and massless excitations and that QSC analyticity imposes discontinuity-type constraints on the dressing phases stronger than crossing, in agreement with the Frolov–Sfondrini factors except for the massless–massless phase; and Ekhammar, Gromov, Stefański jr. and Thull (arXiv:2601.21992) solved the QSC numerically from weak to strong coupling for short strings, finding the expected flat-space Regge trajectories (linear energy–spin relations of highly excited strings) with Kaluza–Klein splitting between states carrying different compact S3S^3 angular momenta. Meanwhile the mirror TBA side produced independent finite-coupling data, notably tensionless-limit spectra (Brollo, le Plat, Sfondrini and Suzuki, arXiv:2303.02120 and arXiv:2308.11576) dominated by massless excitations. What has still not been achieved, as the later papers themselves state, is any first-principles derivation of the QSC assumptions, any resolution of the massless–massless phase mismatch, any derivation of the TBA from the QSC, and any QSC computation of the finite-coupling spectrum of states containing massless excitations or its direct comparison with the TBA.

Problem Statement

Establish, from first principles, the exactness and completeness of the conjectured Quantum Spectral Curve for planar type IIB string theory on AdS3×S3×T4AdS_3\times S^3\times T^4 with pure R–R flux — equivalently, of the planar spectral problem of the dual CFT — with the massless worldsheet sector included at finite coupling. Concretely, a complete resolution should: (i) prove or derive the analyticity inputs that the conjecture only assumes, namely the monodromy (mirror ii-periodicity and effectively quadratic branch cuts) of the μ\mu-functions in the asymptotic limit and the constant off-diagonal form of the gluing matrices GG, or replace them with derived equivalents; (ii) resolve the status of the massless–massless dressing phase, reconciling the QSC discontinuity constraints with the Frolov–Sfondrini dressing factors (or showing which must be modified); (iii) derive the exact mirror TBA equations of Frolov and Sfondrini — or an equivalent set of exact finite-coupling equations — from the QSC (or vice versa), closing the derivation loop that exists in the AdS5/CFT4AdS_5/CFT_4 case; and (iv) compute and validate, at finite coupling, the QSC spectrum of states containing massless excitations against the independent TBA results. These four sub-claims are logically independent and may be attained separately, whereas a complete resolution requires all four.

References

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