Unproved conjectures for the $Q$-functions of $\widehat{\mathfrak{sl}}_2$ Gaudin $\lambda$-opers: coincidence with the monster-potential $Q$-functions of the ODE/IM correspondence, classification of Bethe Ansatz solutions, and zeros

The ODE/IM correspondence connects the spectral theory of Schrödinger operators in the complex plane with the integrable structure of two-dimensional conformal field theory: the $Q$-functions of the quantum KdV model — the commuting integrals of motion built on the Virasoro algebra — arise as connection coefficients of differential operators. Masoero, Mukhin and Raimondo introduced the $Q$-functions $Q_\pm(\lambda)$ of the $\lambda$-opers, Schrödinger operators whose apparent singularities are parametrized by solutions of the Bethe ansatz equations of the affine $\widehat{\mathfrak{sl}}_2$ Gaudin model, and proved that these functions satisfy the same QQ-system (quantum Wronskian condition) as the quantum KdV $Q$-functions. They conjectured that the $Q$-functions of the $\lambda$-opers coincide, up to an explicit rescaling of the spectral parameter, with the $Q$-functions of the Bazhanov–Lukyanov–Zamolodchikov monster-potential operators of the classical ODE/IM correspondence, for all admissible values of the parameters. The coincidence is proved only in the elementary case without apparent singularities and, conditionally, for large momentum and $-2<k<-1$, assuming four unproved conjectures: the classification and counting of solutions of the affine Gaudin Bethe ansatz equations, the holomorphy and non-vanishing of $Q_\pm$ at the singular point $\lambda=0$, the asymptotic distribution, reality and simplicity of the zeros of $Q_+$, and the extension of the large-momentum uniqueness argument for the QQ-system to all momenta. The 2024–2026 literature shows genuine partial progress — Grigorev and Mukhin classified the relevant monodromy-free operators by populations of critical points of affine master functions, and Masoero and Ruzza proved the corresponding completeness statement on the monster-potential side at the free-fermion point $c=-2$ — but no work establishes or refutes the coincidence conjecture or its constituent analytic conjectures. The problem remains open.

Background

The ODE/IM (ordinary differential equation / integrable model) correspondence, discovered by Dorey and Tateo and developed by Bazhanov, Lukyanov and Zamolodchikov (BLZ), identifies connection coefficients of Schrödinger-type differential equations in the complex plane with the functions describing the spectrum of integrable models of two-dimensional conformal field theory; see the review of Dorey, Dunning and Tateo. On the integrable-model side stands the quantum KdV model — the quantization of the periodic Korteweg–de Vries equation, whose conserved quantities form a commutative family of operators acting on representations of the Virasoro algebra (the central extension of the algebra of vector fields on the circle, the symmetry algebra of two-dimensional conformal field theory; its central element, the central charge $c$, labels the theory). The relevant representation is the Verma module $M_{c,\Delta}$, the standard infinite-dimensional highest-weight representation of conformal dimension $\Delta$; writing $\Delta=\frac{c-1}{24}+l(l+1)$ introduces the momentum parameter $l$. The joint spectrum of the local integrals of motion (the commuting density-type conserved charges) together with the non-local integrals of motion (a larger commuting family containing them) on $M_{c,\Delta}$ is encoded in a pair of functions of a spectral parameter, the $Q$-functions $\overline Q_\pm(\bar\lambda)$, which satisfy the QQ-system, also called the quantum Wronskian condition,

$$\gamma_r\,\overline Q_+(q\bar\lambda)\,\overline Q_-(q^{-1}\bar\lambda)-\gamma_r^{-1}\,\overline Q_+(q^{-1}\bar\lambda)\,\overline Q_-(q\bar\lambda)=1,$$

a functional equation of Wronskian type whose constants $q$ and $\gamma_r$ depend on the model parameters and on an integer $r\in\mathbb{Z}$; the expansion of $\log\overline Q_\pm$ at infinity produces the eigenvalues of the integrals of motion.

On the differential-equation side, BLZ associated to the quantum KdV model the monster-potential Schrödinger operators

$$L^{BLZ}=\partial_x^2-\left(\frac{\bar l(\bar l+1)}{x^2}+\frac{1}{x}+\sum_{i=1}^{d}\left(\frac{2}{(x-\bar z_i)^2}+\frac{\bar k-2}{x(x-\bar z_i)}\right)-x^{\bar k-2}\,\bar\lambda\right),$$

with $\bar k\in(0,1)$ (related to the central charge by $c=13-6(\bar k+1/\bar k)$), momentum $\bar l$ with $\Re\bar l>-\frac12$, and poles $\bar z_i$ chosen so that each $x=\bar z_i$ is an apparent singularity: a singular point at which the monodromy — the linear transformation of a basis of solutions after analytic continuation around the point — is trivial. The $Q$-functions $\overline Q_\pm$ of $L^{BLZ}$ are defined as Wronskians (determinants pairing solutions with their derivatives) of the Frobenius solutions at the regular singular point $x=0$ (solutions with power-law behavior $x^{\bar l+1}$ and $x^{-\bar l}$) and the Sibuya solution at the irregular singular point $x=\infty$ (the unique, up to normalization, solution decaying super-exponentially in a prescribed sector at infinity); they are entire functions of the spectral parameter. Dorey and Tateo discovered this correspondence for the anharmonic oscillator (the case $d=0$), BLZ extended it to the monster potentials, which describe the excited states of quantum KdV, and Masoero and Raimondo constructed the opers for the higher states. Conti and Masoero proved that the solutions of the quantum KdV Bethe equations (the functional equations for the zeros of $\overline Q_+$) are unique at large momentum, counted the monster potentials (their number is bounded by the number of partitions of $d$), and established the asymptotics of their zeros. Frenkel and Hernandez related these spectra to affine opers through a Langlands-type duality.

Affine Gaudin models give a second, algebraically different construction of the same integrable structure. The generalized affine $\mathfrak{sl}(2)$ Gaudin model is an integrable system whose commuting Hamiltonians are built from the affine Kac–Moody algebra $\widehat{\mathfrak{sl}}_2$ (the central extension of the algebra of maps from the circle into $\mathfrak{sl}_2$) acting on a tensor product of its representations; Kotousov and Lukyanov formulated an ODE/IQFT (integrable quantum field theory) correspondence for it, and the non-local integrals of motion of quantum KdV are expected to coincide with its Hamiltonians. Its spectrum is described by the Bethe ansatz equations (BAE): an algebraic system for variables $s_1,\dots,s_{d_0}$ and $t_1,\dots,t_{d_1}$ (the Bethe roots), equivalent to the critical-point equations of a master function, a rational function of the roots whose stationary points are exactly the solutions (the explicit system is given in equations (2.21) and (3.7) of arXiv:2312.08842). Gaiotto, Lee, Vicedo and Wu, studying Kondo line defects (line impurities in two-dimensional integrable quantum field theory), associated to each BAE solution the $\lambda$-opers: one-parameter families of Schrödinger operators (the name reflects their interpretation as oper connections, a geometric reformulation of second-order differential equations)

$$L^G=\partial_x^2-\left(\frac{l(l+1)}{x^2}+\sum_{i=1}^{d_0}\frac{2}{(x-s_i)^2}+\sum_{i=1}^{d_0}\frac{k+s_i/(s_i-1)}{x(x-s_i)}+x^k(x-1)\,\lambda^2\right),$$

with $k>-2$, whose singularities $x=s_i$ are apparent for every value of the spectral parameter $\lambda\neq0$. Masoero, Mukhin and Raimondo defined the $Q$-functions $Q_\pm(\lambda)$ of $L^G$ by the same Wronskian construction — Frobenius solutions at $x=0$ paired with the Sibuya solution at $x=\infty$ — and proved (their Theorem 4.8) that $Q_\pm$ are holomorphic on the punctured plane $\mathbb{C}^*$, satisfy the QQ-system above with $q=e^{\frac{k+2}{k+3}\pi i}$ and $\gamma_r=e^{2\frac{l+\frac12-r(k+2)}{k+3}\pi i}$, and have zeros solving the Bethe equations of quantum KdV: an ODE/IM correspondence for the affine Gaudin opers.

Since both families of $Q$-functions satisfy the same QQ-system, and the asymptotics of the zeros of $Q_+$ and $\overline Q_+$ agree in the large-momentum limit $l\to\infty$ — a situation in which, by the results of Conti and Masoero, these properties determine the $Q$-function uniquely — the source conjectures that the two families coincide. Its main conjecture states that for every monster-potential operator and every integer $r$ there is a $\lambda$-oper with $d_0=d+r^2$ whose parameters are related by

$$\bar k=\frac{k+2}{k+3},\qquad \bar l+\frac12=\frac{l-(k+2)r+\frac12}{k+3},$$

and whose $Q$-functions coincide with those of the monster potential, the zeros being related by $\lambda_j=(k+3)^{\frac{2}{k+3}}\bar\lambda_j$; for the stratum labeled by a partition $\mu$ of $d_0-r^2$ (a partition is an unordered decomposition of an integer into positive summands), this is the identity (equation (7.5) of the source)

$$\overline Q_+(\bar\lambda,\underline z^\mu)=c\,Q_+\big((k+3)^{\frac{2}{k+3}}\bar\lambda,\underline s^\mu\big),\qquad c\neq0,$$

together with the analogous statement for $Q_-$ and the converse bijection: every $\lambda$-oper is matched by a monster potential with $d=d_0-r^2$, where $r=d_0-d_1$ is fixed by the Bethe-root counts.

The coincidence is elementary for the top opers with $d_0=0$, where it reduces to an explicit $\lambda$-dependent change of variables (and, for the other top opers, those with Bethe-root counts $(d_0,d_1)=(r^2,r^2-r)$, to that change of variables combined with a gauge transformation). It is proved conditionally for $l$ sufficiently large and $-2<k<-1$ (the source's Theorem 7.1, which combines the Conti–Masoero uniqueness theorem with the source's Conjectures 6.10 and 6.11). Four ingredients remain unproved, and they constitute the open core. (1) Classification: it is not known whether, for generic parameters, every $\lambda$-oper with distinct apparent singularities arises from a solution of the BAE, nor whether the BAE solutions are isolated, non-degenerate, and exactly $\mathrm{Part}(d_0-r^2)$ in number (where $\mathrm{Part}(n)$ is the number of partitions of $n$), nor what their large-$l$ asymptotics are; only the existence of one solution per non-degenerate partition is proven. (2) Analyticity at $\lambda=0$: the limit $\lambda\to0$ is singular for $L^G$, because the irregular singularity at $x=\infty$ degenerates to a Fuchsian (regular) singularity; that both $Q_\pm$ extend holomorphically and without zeros to $\lambda=0$, away from the resonance values $2l+1=(k+2)i+j$ with $i,j$ non-negative integers, is proven only for $Q_+$ in a half-plane of parameters, and for $Q_-$ only as an upper bound on the pole order. (3) Zero asymptotics for $d_0>0$: with apparent singularities present, the boundary-value problem for the zeros of $Q_+$ is not self-adjoint and the zeros need not be real and positive; the conjectured Bohr–Sommerfeld-type asymptotics (large-$\lambda$ and large-$l$ formulae with partition-dependent shifts) and the reality and simplicity of the zeros are supported by complex-WKB heuristics (the WKB method: asymptotic solution of differential equations by exponential ansatz, analytically continued in the complex plane) but not proved. (4) Extension to all $l$: the uniqueness of the solution of the QQ-system used in the comparison relies on the theory of logarithmic Bethe equations, available only at large momentum, and the authors state that its extension to all $l$ is not clear. On the monster-potential side the analogous statements are largely proven (counting and large-momentum uniqueness by Conti and Masoero; zero asymptotics by BLZ and Conti–Masoero), so the problem concentrates on the affine Gaudin side and on the assembly for all parameters. The source's further perspectives — the zero-asymptotics theory of $Q$-functions beyond $\widehat{\mathfrak{sl}}_2$, and the conceptual nature of the correspondence between the poles $s_i$ and $\bar z_i$, for which no algebraic map is expected and a description through dualities of quantum toroidal algebras (higher-dimensional analogues of quantum affine algebras) is anticipated — are out of scope for this task.

Problem Statement

Prove or refute the $Q$-function coincidence conjecture of Masoero–Mukhin–Raimondo, in the two-directional form of the source's main conjecture. First direction: for every BLZ monster-potential operator $L^{BLZ}$ with $d$ apparent singularities and every integer $r\in\mathbb{Z}$, there exists an affine $\widehat{\mathfrak{sl}}_2$ Gaudin $\lambda$-oper $L^G$ with $d_0=d+r^2$, with real $k>-2$ (equivalently $\bar k=(k+2)/(k+3)\in(0,1)$) and generic momentum $l$, constructed from a solution $(s_1,\dots,s_{d_0};\,t_1,\dots,t_{d_1})$ of the affine $\widehat{\mathfrak{sl}}_2$ Gaudin Bethe ansatz equations, whose parameters are related by

$$\bar k=\frac{k+2}{k+3},\qquad \bar l+\frac12=\frac{l-(k+2)r+\frac12}{k+3},$$

and whose $Q$-functions coincide with those of the monster potential, in the sense that, for a nonzero constant $c$, the zeros are related by $\lambda_j=(k+3)^{\frac{2}{k+3}}\bar\lambda_j$ and

$$\overline Q_+(\bar\lambda,\underline z^\mu)=c\,Q_+\big((k+3)^{\frac{2}{k+3}}\bar\lambda,\underline s^\mu\big),$$

together with the analogous statement for $Q_-$. Second direction (bijectivity of the correspondence): every admissible $\lambda$-oper $L^G$, constructed from a solution of the Bethe ansatz equations with real $k>-2$, generic $l$ and Bethe-root counts $d_0\ge r^2$, is matched by an integer $r$ and some monster-potential operator $L^{BLZ}$ with $d=d_0-r^2$ whose $Q$-functions coincide with those of $L^G$ as above.

A complete proof must be valid for all generic momenta $l$ (equivalently, for all $l$ with $2l+1\notin\{(k+2)i+j\,:\,i,j\in\mathbb{Z}_{\ge0}\}$) — in particular beyond the large-momentum regime and in regimes where the zeros of $Q_+$ are not all real and positive. By the source's own analysis, the currently missing ingredients are four constituent conjectures, which a solution may either establish or bypass by another route: (i) the classification of $\lambda$-opers with distinct apparent singularities by genuine Bethe ansatz solutions for generic parameters, together with the counting of BAE solutions by $\mathrm{Part}(d_0-r^2)$ and their large-$l$ asymptotics (Conjectures 2.4, 5.1 and 5.8 of the source); (ii) the holomorphy and non-vanishing of $Q_\pm$ at $\lambda=0$ for all BAE solutions, outside the resonance set (Conjecture 6.6); (iii) the asymptotic distribution, reality and simplicity of the zeros of $Q_+$ when apparent singularities are present, $d_0>0$ (Conjecture 6.10); (iv) the extension to arbitrary $l$ of the uniqueness theory for solutions of the QQ-system (Section 7.3 of the source).

Alternatively, refute the conjecture by exhibiting an admissible $\lambda$-oper, with certified Bethe ansatz data, whose $Q$-functions provably coincide with those of no monster-potential operator with the parameters above.

References

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