Rigorous foundations of the planar AdS$_5$/CFT$_4$ quantum spectral curve: pole-freeness for arbitrary operators, a first-principles derivation beyond the TBA, and exact state-universal equivalence with the Y-system/TBA

The quantum spectral curve (QSC) reformulates the exact planar spectrum of $\mathcal{N}=4$ super Yang–Mills theory (dual to type IIB strings on AdS$_5\times$S$^5$) as a finite closed system of Q-functions with prescribed analytic structure. Its foundational pillars, however, remain conjectural: (i) the regularity conjecture of Gromov–Kazakov–Leurent–Volin — that for every physical single-trace operator the fundamental QSC functions can be chosen free of poles on their Riemann surfaces, with singularities only at infinity and the prescribed Zhukovsky cuts — is proven only in restricted sectors, not for generic states; (ii) the QSC was extracted from the thermodynamic Bethe ansatz (TBA)/Y-system, and no first-principles derivation bypassing the TBA exists, in particular none valid at finite coupling; (iii) the equivalence between TBA and the QSC for arbitrary excited states rests on the still-unproven analytic-continuation trick for excited-state TBA. This record asks for the rigorous foundation of the QSC: a proof of pole-freeness for arbitrary operators, a TBA-free derivation, or a state-universal equivalence theorem.

Background

Planar $\mathcal{N}=4$ super Yang–Mills (SYM) theory, in the limit of a large number of colors, is conjecturally dual to free (tree-level) type IIB superstrings on the AdS$_5\times$S$^5$ background. Its spectral problem — computing the anomalous dimensions of all local single-trace operators (operators built as traces of products of fields and their covariant derivatives), or equivalently the string energy levels, at any value of the 't Hooft coupling $\lambda$ — is the central quantitative test of this gauge/string duality. Integrability, discovered in both descriptions, reduces this problem to solvable finite-difference equations. The asymptotic Bethe ansatz, valid when the operator is long enough (infinite volume), is corrected at finite length by wrapping effects, captured by the thermodynamic Bethe ansatz (TBA): the spectrum of the string is mapped to the ground and excited states of a mirror two-dimensional theory, whose infinite set of coupled integral equations (or the equivalent Y-system of functional relations) yields exact energies. Excited-state TBA equations are themselves constructed heuristically by analytically continuing the ground-state equations across the rapidity contour (the Dorey–Tateo trick), and explicit TBA/Y-system equations are known only for a limited set of states, making a state-by-state solution of the full spectrum impractical. The quantum spectral curve (QSC), proposed in a short form by Gromov–Kazakov–Leurent–Volin in 2014 and derived at length in their 2014–2015 paper 'Quantum spectral curve for arbitrary state/operator in AdS$_5$/CFT$_4$', reformulates the same problem as a finite Riemann–Hilbert-type system for Q-functions — the $2^8=256$ functions $Q_{A|I}(u)$ of a $\mathfrak{gl}(4|4)$ Q-system obeying bilinear QQ-relations (Plücker/Hirota relations), with $\mathbf{P}_a$ and $\mathbf{Q}_i$ (the one-index Q-functions) analytic except for a short or long Zhukovsky cut (a branch cut on $[-2g,2g]$ with $g=\sqrt{\lambda}/(4\pi)$, parametrized by the Zhukovsky variable $x(u)=(u+\sqrt{u^2-4g^2})/(2g)$), glued across the cuts by monodromy conditions (the $\mathbf{P}\mu$- and $\mathbf{Q}\omega$-systems), with large-$u$ asymptotics fixed by the operator's global charges. A key selection rule replaces the state-dependent TBA input: the regularity requirement of section 3.2.2, conjecturing that the fundamental QSC functions can be chosen with no poles anywhere on their Riemann surfaces (the only allowed singularities being at infinity and the prescribed cuts); via the Hasse diagram of QQ-relations this conjecture is directly linked to, and implies, the exact Bethe equations for arbitrary states (the converse is not established in the source, which also warns that the definition of exact Bethe roots as zeros of Q-functions is not universal). The QSC has become the workhorse of the field — solved numerically and perturbatively to high loop order and to very strong coupling, and generalized to ABJM/AdS$_4$, Cartan-twisted ($\gamma$-, $\beta$-) and $\eta$-deformed theories, cusped Wilson lines, and, recently, AdS$_3$/CFT$_2$ — yet its own foundations were left open by the founding paper: the regularity conjecture was supported by TBA examples but not proven for generic states; the derivation itself passes through the TBA-derived analytic Y-system, and the authors state they 'cannot fully rigorously demonstrate the equivalence with TBA for arbitrary state'; the Zamolodchikov-style finite-volume bootstrap that would produce the Q-system directly 'omitting the TBA procedure' was flagged as desirable but undeveloped; and an operatorial realization of the Q-functions (Q-operators, the building blocks familiar from spin-chain integrability, where Baxter Q-operators commute with transfer matrices and diagonalize them) at finite coupling was left as an open problem. Partial conceptual progress exists: the separation-of-variables program has reinterpreted the gluing conditions as a normalizability (finite-norm) condition on Q-functions, and in the bi-scalar fishnet limit of $\gamma$-deformed $\mathcal{N}=4$ SYM the QSC-type Baxter equations have been genuinely derived from Feynman graphs — but none of this lifts to the full theory at finite coupling.

Problem Statement

Establish the quantum spectral curve of planar AdS$_5$/CFT$_4$ as a rigorous, first-principles solution of the spectral problem, by resolving the linked questions left open in Gromov–Kazakov–Leurent–Volin (arXiv:1405.4857):

1. (Regularity) Prove, or refute by an explicit physical state, the conjecture of section 3.2.2: for every physical single-trace operator of planar $\mathcal{N}=4$ SYM at any value of the 't Hooft coupling, the fundamental QSC functions — equivalently the $\mathbf{P}_a$ and $\mu_{ab}$ of the $\mathbf{P}\mu$-system, or $T_{0,0}$ in the distinguished T-gauge — can be chosen with no poles anywhere on their Riemann surfaces, the only allowed singularities being at infinity (with asymptotics fixed by the operator's global charges) and the prescribed Zhukovsky cuts; and classify the possible state-dependent singularities. Establishing the regularity property would imply the exact Bethe equations along the Hasse diagram for arbitrary states; the converse is not established in the source, which also notes that the definition of exact Bethe roots as zeros of Q-functions is not universal, so the regularity property itself — not a Bethe-type reformulation — is the target of this facet. The classification of the possible state-dependent singularities is a related direction rather than a condition for completing this facet.

2. (First-principles derivation) Derive the analytic Q-system for planar $\mathcal{N}=4$ SYM directly, without passing through the TBA/Y-system, valid at finite coupling. The related question of proving that the closed QSC system admits only a discrete set of solutions (uniqueness/discreteness) is a separately recognized open problem rather than a condition for completing this facet.

3. (Equivalence) Turn the equivalence between the analytic Y-system/TBA and the QSC into a theorem for arbitrary excited states — in particular replacing the heuristic contour-deformation (analytic-continuation) step of excited-state TBA with a proof — including the treatment of exceptional state-dependent singularities. An explicit characterization of the auxiliary mirror-periodic functions $\Omega_{ab}$, defined in the source only through their discontinuities, is a source-mentioned optional direction rather than a condition of this facet.

These three facets serve one objective: removing the conjectural inputs on which the exact planar $\mathcal{N}=4$ SYM spectrum currently rests. Following the founding paper, they are posed as distinct open items that may be resolved separately rather than as a conjunctive package with a joint resolution clause; a genuine unavoidable-pole state would refute facet 1 and is an admissible answer for that facet.

References

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