Completeness of the Bethe ansatz for the dynamical elliptic (KZB) Bethe algebra and the global geometry of the elliptic Wronski map

The dynamical elliptic Bethe algebra of a tensor product of finite-dimensional irreducible $\mathfrak{sl}_2$-modules is the commutative algebra of differential operators generated by the coefficient $S_2(x)$ of the elliptic RST-operator. Thompson and Varchenko (arXiv:1810.09001, 2018) expressed $S_2(x)$ through the Felder–Wieczerkowski Knizhnik–Zamolodchikov–Bernard (KZB) operators, constructed eigenfunctions by the Bethe ansatz, and proved that Bethe eigenfunctions correspond to ordered pairs of theta-polynomials. However, all counting theorems — the $\binom{2m}{m}$ count of Bethe eigenfunctions and of the points of each labeled fiber of the elliptic Wronski map — were established only in an asymptotic regime with $|\operatorname{Im}\mu|$ large and generic marked points. The problem asks to remove these restrictions and to extend the theory beyond rank two: prove completeness of the Bethe ansatz for the dynamical elliptic Bethe algebra for all parameter values, determine the global geometry of the elliptic Wronski map (full fiber structure, singular fibers), and prove the Weyl-group invariance of the $\mathfrak{sl}_N$ RST-operator for $N\ge 3$, a direction the source explicitly promises and never delivers. No resolution of these questions is known in the literature to date; the only continuation of the program treats the rational, non-elliptic case and explicitly defers the elliptic one.

Background

Let $\mathfrak{sl}_2$ be the Lie algebra of traceless complex $2\times 2$ matrices, with one-dimensional diagonal (Cartan) subalgebra, and let $V=\otimes_{j=1}^n V_j$ be a tensor product of finite-dimensional irreducible $\mathfrak{sl}_2$-modules. The zero weight subspace $V[0]$ is the subspace on which the Cartan generator acts by zero; for $V=(\mathbb C^2)^{\otimes 2m}$ one has $\dim V[0]=\binom{2m}{m}$.

The Knizhnik–Zamolodchikov–Bernard (KZB) equations are the genus-one analog of the Knizhnik–Zamolodchikov equations of two-dimensional conformal field theory: they govern the dependence of torus conformal blocks (the chiral building blocks of correlation functions) on the moduli. Felder and Wieczerkowski (1996) introduced commuting differential operators $H_0(z,\tau),\dots,H_n(z,\tau)$, the KZB operators, acting on $V[0]$-valued functions of the coordinate $\lambda_{12}$ on the Cartan subalgebra, where $\operatorname{Im}\tau>0$ and $z_1,\dots,z_n$ are the positions of $n$ marked points on the elliptic curve $\mathbb C/(\mathbb Z+\tau\mathbb Z)$.

Rubtsov, Silantyev and Talalaev (2009) constructed, for tensor products of $\mathfrak{gl}_N$-modules, a commutative family of differential operators acting on zero-weight functions, presented through a universal differential operator $\mathcal D=\partial_x^N+\sum_{j=1}^N S_j(x)\,\partial_x^{N-j}$ (the RST-operator, after its authors). Thompson and Varchenko (2018) took up its $\mathfrak{sl}_2$ version, in which $\mathcal D=\partial_x^2+S_2(x)$ and the commutative algebra generated by the coefficient $S_2(x)$ is the dynamical elliptic Bethe algebra $\mathcal B^V(z,\tau)$; they proved that this operator is doubly periodic in $x$, invariant under the $\mathfrak{sl}_2$ Weyl group $W=\{\mathrm{id},s\}$ (the two-element reflection group acting on the Cartan subalgebra), and that its coefficient is an explicit combination of the KZB operators, $S_2(x,z,\tau)=-2\pi i H_0(z,\tau)-\sum_{j=1}^n\Big[H_j(z,\tau)\rho(x-z_j,\tau)+c_2^{(j)}\rho'(x-z_j,\tau)\Big],$ where $\rho=\theta'/\theta$ is the logarithmic derivative of the normalized first Jacobi theta function $\theta$ and $c_2^{(j)}$ is the quadratic Casimir element acting on the $j$-th tensor factor. Hence the dynamical elliptic Bethe algebra is generated by the KZB operators, so eigenfunctions of one are eigenfunctions of the other.

Eigenfunctions are constructed by the Bethe ansatz, the method that builds eigenvectors as products of elementary factors whose parameters are fixed by algebraic equations. Felder and Varchenko (1995) introduced an elliptic master function $\Phi(\mu,t,z,\tau)$ depending on a dynamical parameter $\mu$ and Bethe roots $t=(t_1,\dots,t_m)$, and showed that its critical points with respect to $t$ — the Bethe ansatz equations — produce $V[0]$-valued eigenfunctions $\Psi(\lambda_{12},\mu,t,z,\tau)$ of all KZB operators simultaneously. Whether the Bethe ansatz is complete (whether these eigenfunctions exhaust the joint spectrum, with no degeneracies beyond the Weyl symmetry) is the basic structural question of the subject. Its rational, non-dynamical analog for the $\mathfrak{gl}_N$ Gaudin model was settled by Mukhin, Tarasov and Varchenko through a correspondence between Gaudin eigenvectors and two-dimensional subspaces of $\mathbb C[x]$, that is, points of a Grassmannian connected to the classical Wronski map $(f,g)\mapsto\mathrm{Wr}(f,g)=fg'-f'g$, the tool behind their proof of the B. and M. Shapiro conjecture in real algebraic geometry.

The elliptic, dynamical analog of that geometry is built from theta-polynomials: functions $f(x)=e^{2\pi i\mu x}\prod_{j=1}^m\theta(x-t_j,\tau)$, the elliptic counterparts of exponential polynomials, quasi-periodic under the lattice $\mathbb Z+\tau\mathbb Z$. The Wronskian of two theta-polynomials of degree $m$ is a theta-polynomial of degree $2m$, which defines the elliptic Wronski map on ordered pairs $(f,g)$ whose first coordinate has unit first multiplier, that is, integer label $k$ in its exponential factor $e^{2\pi ikx}$; the target consists of theta-polynomials of degree $2m$ with non-unit first multiplier, and the labeled preimage $\mathrm{Wr}_k^{-1}(h)$ consists of the pairs mapping to $h$ whose first coordinate has label $k\in\mathbb Z$. Thompson and Varchenko proved, under genericity assumptions, a bijection between equivalence classes of solutions of the Bethe ansatz equations (with $\mu\notin\mathbb Z$) and pairs of theta-polynomials with generic first coordinate, showed that the transposition involution $(f,g)\mapsto(g,f)$ coincides with the Weyl involution $s$ on eigenfunctions, and showed that the Bethe algebra separates the Weyl orbits of eigenfunctions whose associated pairs have generic coordinates.

All counting statements, however, were proved only in an asymptotic window: for $z_1,\dots,z_{2m}$ distinct points of the open fundamental parallelogram and $|\operatorname{Im}\mu|$ sufficiently large, each labeled fiber of the elliptic Wronski map consists of exactly $\binom{2m}{m}$ nondegenerate points (Theorem 11.2 of the source), and the number of Bethe eigenfunctions equals $\dim V[0]=\binom{2m}{m}$ as $\mu\to\infty$. Nothing is known about the fibers, the singular locus, or the solution count for arbitrary $\mu$ and $z$; and the Weyl invariance of the RST-operator is proved only in rank two, the source explicitly deferring the $\mathfrak{sl}_N$ case to be 'discussed elsewhere'. (The solvability of the Wronskian equation $\mathrm{Wr}(f,g)=h$ for theta-polynomials, attributed to Borisov, Mukhin and Varchenko, carries a complete, self-contained proof in the source itself and lies outside the scope of the problem.)

Problem Statement

Extend the Thompson–Varchenko theory of the dynamical elliptic Bethe algebra beyond its asymptotic regime and beyond rank two. Concretely:

(1) Completeness and global geometry (rank two). For $V=(\mathbb C^2)^{\otimes 2m}$ with zero-weight subspace $V[0]$, $\operatorname{Im}\tau>0$, and arbitrary $\mu\in\mathbb C$ and $z=(z_1,\dots,z_{2m})$ (not necessarily distinct or inside the open fundamental parallelogram; for $\mu\in\mathbb Z$ the polynomial $h$ below lies in the excluded unit-multiplier locus and its labeled fibers are empty, and the degeneration towards such $\mu$ is part of the requested global description), determine the global geometry of the elliptic Wronski map on pairs of theta-polynomials of degree $m$. This splits into the following sub-claims, each of which is independently attackable and all of which are required for a complete answer:

• (1a) the number of points in each labeled fiber $\mathrm{Wr}_k^{-1}(h)$ for $h(x)=e^{-2\pi i\mu x}\prod_{a=1}^{2m}\theta(x-z_a,\tau)$;
• (1b) the discriminant locus of the elliptic Wronski map and the local structure of its singular (critical) fibers, in particular the cardinality and the multiplicity at each singular point;
• (1c) on the representation-theoretic side, whether the Bethe ansatz for the dynamical elliptic Bethe algebra $\mathcal B^V(z,\tau)$ is complete for all parameter values — that is, whether for every fixed $\mu\notin\mathbb Z$ and every $z$ the equivalence classes of solutions of the Bethe ansatz equations (modulo the natural lattice-shift equivalences) number exactly $\dim V[0]=\binom{2m}{m}$;
• (1d) that these solutions produce a complete set of joint eigenfunctions with no degeneracies of the spectrum beyond the Weyl involution;
• (1e) that the Bethe algebra separates the Weyl orbits of the Bethe eigenfunctions for all solutions, not only those whose associated pairs have generic coordinates (in the source this separation is proved for solutions whose associated pairs have generic coordinates, a condition guaranteed only inside the asymptotic window).

The bijection of the source between equivalence classes of Bethe-ansatz solutions and ordered pairs of theta-polynomials identifies the geometric counting sub-claim (1a) with the representation-theoretic sub-claim (1c); the sub-claims (1b), (1d) and (1e) have no counterpart under this bijection. A complete answer to (1) must settle all of (1a)–(1e).

(2) Higher-rank foundations. Prove that the RST-operator $\mathcal D=\partial_x^N+\sum_{j=1}^N S_j(x)\,\partial_x^{N-j}$ for $\mathfrak{sl}_N$, $N\ge 3$, acting on zero-weight functions for tensor products of $\mathfrak{sl}_N$-modules, is invariant under the $\mathfrak{sl}_N$ Weyl group. This is the direction the source explicitly defers; the identity of its Theorem 4.9 expressing the coefficient $S_2(x)$ through the KZB operators, $S_2(x,z,\tau)=-2\pi i H_0(z,\tau)-\sum_{j=1}^n[H_j(z,\tau)\rho(x-z_j,\tau)+c_2^{(j)}\rho'(x-z_j,\tau)]$, is already proved in the source for general $N$ as an identity of operators on zero-weight functions.

Components (1) and (2) are the parameter-range and rank axes of a single objective: replace the asymptotic, generic, rank-two correspondence between Bethe eigenfunctions and theta-polynomial geometry by an unrestricted theory of the dynamical elliptic Bethe algebra. The two components are independent mandates — each can be settled on its own — and the full problem is resolved only when both are.

References

1. Daniel Thompson, Alexander Varchenko, Dynamical elliptic Bethe algebra, KZB eigenfunctions, and theta-polynomials, arXiv:1810.09001 (2018), https://arxiv.org/abs/1810.09001
2. A. Slinkin, D. Thompson, A. Varchenko, Dynamical sl_2 Bethe algebra and functions on pairs of quasi-polynomials, arXiv:2007.09504 (2020), https://arxiv.org/abs/2007.09504
3. A. N. Varchenko, A. M. Slinkin, D. Thompson, Dynamical Bethe algebra and functions on pairs of quasi-polynomials, Russian Mathematical Surveys 76 (2021), no. 4, 653-684, doi:10.1070/rm10010, https://doi.org/10.1070/rm10010
4. G. Felder, C. Wieczerkowski, Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations, Communications in Mathematical Physics 176 (1996), 133-161, doi:10.1007/BF02099366, https://doi.org/10.1007/BF02099366
5. G. Felder, A. Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, International Mathematics Research Notices 1995 (1995), no. 5, 221-233, doi:10.1155/S1073792895000171, arXiv:hep-th/9502165, https://doi.org/10.1155/S1073792895000171
6. V. Rubtsov, A. Silantyev, D. Talalaev, Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model, SIGMA 5 (2009), 110, doi:10.3842/SIGMA.2009.110, arXiv:0908.4064, https://doi.org/10.3842/SIGMA.2009.110
7. E. Mukhin, V. Tarasov, A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Annals of Mathematics (2) 170 (2009), no. 2, 863-881, doi:10.4007/annals.2009.170.863, https://arxiv.org/abs/math/0512299