Completing the dynamical elliptic Bethe algebra correspondence: elliptic Wronski map geometry beyond the asymptotic regime, Bethe ansatz completeness for arbitrary $\mu$, and the source-deferred extension beyond $\mathfrak{sl}_2$

For the $\mathfrak{sl}_2$ dynamical elliptic Bethe algebra of Rubtsov–Silantyev–Talalaev acting on the zero weight subspace of a tensor product of finite-dimensional $\mathfrak{sl}_2$-modules, Thompson and Varchenko established a correspondence between Bethe ansatz eigenfunctions, solutions of the Bethe ansatz equations, and pairs of theta-polynomials linked by the elliptic Wronski map, and proved a fiber-counting theorem ($\binom{2m}{m}$ points in each labeled fiber) together with completeness-type counting of Bethe eigenfunctions — but only in the asymptotic regime where $|\operatorname{Im}\mu|$ is large, with $z$ of distinct coordinates. The open problem, formulated in the later literature as developing "the elliptic version of this correspondence", is to complete this theory within the domain in which the dynamical elliptic Bethe algebra is defined: (i) describe the fibers of the elliptic Wronski map for arbitrary $\mu\in\mathbb{C}$ and arbitrary $\tau$ with $\operatorname{Im}\tau>0$, with $z=(z_1,\dots,z_{2m})$ of distinct coordinates in the interior of a fundamental parallelogram, thereby extending the fiber-counting theorem beyond the asymptotic regime; and (ii) determine whether the Bethe ansatz yields a complete set of joint eigenfunctions of the dynamical elliptic Bethe algebra for all such parameter values, with no degeneracies beyond the Weyl involution, or else exhibit parameter values where completeness fails. A separate open item, explicitly deferred by the authors of the source paper ("will be discussed elsewhere"), is the Weyl-group invariance of the RST-operator for $\mathfrak{sl}_N$ with $N\geq 3$. Directions (i)–(ii) constitute the elliptic dynamical analogue of the Mukhin–Tarasov–Varchenko correspondence between Bethe algebras, Wronski maps, and geometry.

Background

A quantum Gaudin model is a family of commuting operators (the Bethe algebra, generated by the coefficients of an auxiliary 'transfer' differential operator) acting on a tensor product of irreducible modules over a Lie algebra such as $\mathfrak{gl}_2$ or $\mathfrak{sl}_2$. The Bethe ansatz is a method for constructing joint eigenvectors of such a commuting family from solutions of an algebraic system, the Bethe ansatz equations, whose unknowns are the Bethe roots $t_1,\dots,t_m$. The algebra acts on the zero weight subspace $V[0]$, the subspace of the tensor product on which the Cartan element acts by zero. For the rational $\mathfrak{gl}_2$ Gaudin model, Mukhin, Tarasov and Varchenko proved a complete geometrization: joint eigenvectors of the Bethe algebra correspond to two-dimensional subspaces of the polynomial ring $\mathbb{C}[x]$ — equivalently, to ordered pairs $(f,g)$ of polynomials up to change of basis — and this correspondence is governed by the classical Wronski map $(f,g)\mapsto\operatorname{Wr}(f,g)=fg'-f'g$; the Bethe ansatz is complete (it produces $\dim V[0]$ eigenvectors for generic parameters), and the whole picture is equivalent to Schubert calculus on Grassmannians.

The source paper, Thompson–Varchenko, 'Dynamical elliptic Bethe algebra, KZB eigenfunctions, and theta-polynomials' (arXiv:1810.09001), pushes this program to the elliptic, dynamical setting. Here 'elliptic' refers to theta-functional structures with two periods $1$ and $\tau$ ($\operatorname{Im}\tau>0$), and 'dynamical' to dependence on the Cartan variable $\lambda$ in addition to the spectral variable. The KZB (Knizhnik–Zamolodchikov–Bernard) equations are the elliptic analogue of the Knizhnik–Zamolodchikov equations for conformal blocks on elliptic curves; their coefficient operators $H_a(z,\tau)$, the KZB operators, were introduced in the form used here by Felder and Wieczerkowski, and are invariant under the Weyl group of $\mathfrak{sl}_2$ (the two-element group $\{\mathrm{id},s\}$ acting on the Cartan subalgebra). Rubtsov, Silantyev and Talalaev constructed, via Manin matrices (matrices with noncommuting entries satisfying a right-commutativity condition under which they behave like commutative matrices in determinant and inverse formulas), a commutative algebra of differential operators in the Cartan variable acting on functions with values in $V[0]$; the fundamental object of that construction is the RST-operator (named for Rubtsov–Silantyev–Talalaev), the column determinant of an $N\times N$ matrix of first-order operators, which expands as a scalar operator $D=\partial_x^N+S_1(x)\partial_x^{N-1}+\cdots+S_N(x)$ whose commuting coefficients $S_j(x)$ generate the dynamical elliptic Bethe algebra. Thompson–Varchenko proved for $V=(\mathbb{C}^2)^{\otimes 2m}$ that the generating coefficient $S_2(x)$ of the fundamental operator $D=(d/dx)^2+S_2(x)$ is expressible through the KZB operators, that the whole Bethe algebra is Weyl-group invariant, and that Bethe eigenfunctions are governed by theta-polynomials: an entire function $f$ is a theta-polynomial of degree $m$ if $f(x+1)=Af(x)$ and $f(x+\tau)=Bf(x)$ for fixed multipliers $(A,B)$, in which case $f$ factors as a product of Jacobi theta functions $\theta(x-t_j,\tau)$ times an exponential; when the first multiplier is $A=1$ this exponential is $e^{2\pi ikx}$ with an integer $k$, the label $l(f)=k$ of $f$. The elliptic Wronski map sends an ordered pair of degree-$m$ theta-polynomials to their Wronskian, a degree-$2m$ theta-polynomial. Their central geometric result (Theorem 11.2) states: if $z=(z_1,\dots,z_{2m})$ has distinct coordinates in the interior of a fundamental parallelogram $\Lambda$ of the lattice $\mathbb{Z}+\tau\mathbb{Z}$ and $|\operatorname{Im}\mu|$ is sufficiently large, then for each label $k\in\mathbb{Z}$ the labeled fiber $\operatorname{Wr}^{-1}_k(h)=\{(f,g):\operatorname{Wr}(f,g)=h,\ l(f)=k\}$ over $h(x)=e^{-2\pi i\mu x}\prod_{a=1}^{2m}\theta(x-z_a,\tau)$ consists of exactly $\binom{2m}{m}$ explicitly described points, each with nonvanishing Jacobian, while the full fiber $\operatorname{Wr}^{-1}(h)$ is the disjoint union of the labeled fibers $\operatorname{Wr}^{-1}_k(h)$ over $k\in\mathbb{Z}$. From this they deduce (Theorems 12.1, 12.2, in the regime $|\mu|$ large) that the number of relevant Bethe eigenfunctions equals $\dim(\mathbb{C}^2)^{\otimes 2m}[0]=\binom{2m}{m}$, and that the Weyl involution on eigenfunctions coincides with the analytic involution $(f,g)\mapsto(g,f)$ acting on pairs of theta-polynomials (Theorem 10.3).

What the paper explicitly leaves open: the Weyl-group invariance of the RST-operator is proved only for $\mathfrak{sl}_2$ (Lemma 4.11), with the $\mathfrak{sl}_N$ case deferred ('will be discussed elsewhere'); the identification of the Bethe algebra with KZB-type generators is likewise complete only at rank two — the coefficients $S_1(x)=0$ (Lemma 4.8) and $S_2(x)$ (Theorem 4.9) are identified through the KZB operators for all $N$, but the generation of the Bethe algebra by the KZB operators (Corollary 4.10) is established only for $N=2$, and the paper nowhere frames the coefficients $S_3(x),\dots,S_N(x)$ at higher rank as an open question; and both the fiber description of Theorem 11.2 and the counting of eigenfunctions in Theorems 12.1–12.2 are proved only in the controlled regime of large $\mu$ ($|\operatorname{Im}\mu|$ sufficiently large in Theorem 11.2, $|\mu|$ sufficiently large in Theorems 12.1–12.2) and $z$ with distinct coordinates in the interior $\Lambda'$ of a fundamental parallelogram, so nothing is known about the fibers of the elliptic Wronski map or about completeness of the Bethe ansatz at finite $\mu$, even within the genericity domain in which the dynamical elliptic Bethe algebra itself is defined (Theorem 3.1: no pairwise difference of the $z_i$ belongs to $\mathbb{Z}+\tau\mathbb{Z}$). The follow-up work of Slinkin–Thompson–Varchenko developed the trigonometric/quasi-polynomial (non-elliptic) analogue and explicitly formulates the elliptic version as the open direction, recording the obstruction that the elliptic KZB operator $H_0$ depends on $z$ and has no apparent analogues of the finite-dimensional invariant subspaces used in the degenerate cases.

Problem Statement

Develop the elliptic version of the correspondence between the dynamical elliptic Bethe algebra, its Bethe ansatz eigenfunctions, and the geometry of theta-polynomials begun in Thompson–Varchenko (arXiv:1810.09001) and formulated as an open direction in Slinkin–Thompson–Varchenko (arXiv:2007.09504): extend the geometrization from the asymptotic regime of Theorems 11.2 and 12.1–12.2 to arbitrary values of the dynamical parameter, within the domain in which the source defines its objects.

Throughout, $V=(\mathbb{C}^2)^{\otimes 2m}$, $\tau\in\mathbb{C}$ with $\operatorname{Im}\tau>0$ is arbitrary, $\mu\in\mathbb{C}$ is arbitrary (with $\mu\notin\mathbb{Z}$ where the bijection of Theorem 9.7 is invoked), and $z=(z_1,\dots,z_{2m})$ has distinct coordinates in the interior $\Lambda'$ of a fundamental parallelogram $\Lambda$ of the lattice $\mathbb{Z}+\tau\mathbb{Z}$; this genericity implies that no pairwise difference of the $z_i$ lies in $\mathbb{Z}+\tau\mathbb{Z}$, the hypothesis under which the dynamical elliptic Bethe algebra and the RST-operator are defined (Theorem 3.1 of the source).

1. Fibers of the elliptic Wronski map beyond the asymptotic regime. Describe the labeled fibers $\operatorname{Wr}^{-1}_k(h)=\{(f,g):\operatorname{Wr}(f,g)=h,\ l(f)=k\}$, $k\in\mathbb{Z}$, of the map $(f,g)\mapsto\operatorname{Wr}(f,g)$ on pairs of degree-$m$ theta-polynomials over $h(x)=e^{-2\pi i\mu x}\prod_{a=1}^{2m}\theta(x-z_a,\tau)$ for arbitrary $\mu\in\mathbb{C}$ and $z$ as above (the full fiber $\operatorname{Wr}^{-1}(h)$ is the disjoint union of these labeled fibers over $k\in\mathbb{Z}$) — not only in the regime of sufficiently large $|\operatorname{Im}\mu|$ covered by Theorem 11.2.

2. Completeness of the Bethe ansatz beyond the asymptotic regime. Determine whether the Bethe ansatz for the $\mathfrak{sl}_2$ dynamical elliptic Bethe algebra on $(\mathbb{C}^2)^{\otimes 2m}[0]$ produces a complete set of joint eigenfunctions for all such parameter values $(\mu,z,\tau)$ — that is, whether the appropriately counted equivalence classes of solutions of the Bethe ansatz equations always number $\dim(\mathbb{C}^2)^{\otimes 2m}[0]=\binom{2m}{m}$, with no degeneracies beyond those forced by the Weyl involution, as Theorems 11.2 and 12.2 establish only for $|\mu|$ sufficiently large — or else exhibit explicit parameter values in this domain where completeness fails.

The common scientific objective is the geometrization of the dynamical elliptic Bethe algebra by spaces of pairs of theta-polynomials, valid for arbitrary $\mu$ rather than only asymptotically. Item 1 supplies the geometric side (fibers of the elliptic Wronski map) and item 2 the spectral side (the joint spectrum and completeness of the Bethe ansatz); in the domain above and for $\mu\notin\mathbb{Z}$ the two are locked together by the bijection of Theorem 9.7 between equivalence classes of Bethe ansatz solutions and pairs of theta-polynomials, so that a fiber count and a completeness count are two readings of one statement.

Separate source-deferred item (not a conjunctive completion requirement of the above): prove (or refute) the Weyl-group invariance of the RST-operator for $\mathfrak{sl}_N$ with $N\geq 3$ — the statement the authors explicitly deferred ('will be discussed elsewhere', Section 4.5 of arXiv:1810.09001).

Related context, not part of the stated completion goal: at higher rank the coefficients $S_1(x)=0$ (Lemma 4.8) and $S_2(x)$ (Theorem 4.9) are identified through the KZB operators for all $N$, while the generation of the dynamical elliptic Bethe algebra by the KZB operators (Corollary 4.10) is proved only for $N=2$; expressions of the coefficients $S_3(x),\dots,S_N(x)$ of the $\mathfrak{sl}_N$ RST-operator $D=\partial_x^N+S_1(x)\partial_x^{N-1}+\cdots+S_N(x)$ through the $\mathfrak{sl}_N$ KZB operators, and a higher-rank generation statement, are natural extrapolations of this rank-two picture, but the sources do not state them as open questions.

References

1. D. Thompson, A. Varchenko, Dynamical elliptic Bethe algebra, KZB eigenfunctions, and theta-polynomials, arXiv:1810.09001 (2018), doi:10.48550/arXiv.1810.09001, https://arxiv.org/abs/1810.09001 (journal version: CJAM 1 (2020), no. 1, 78–125)
2. A. Slinkin, D. Thompson, A. Varchenko, Dynamical $\mathfrak{sl}_2$ Bethe algebra and functions on pairs of quasi-polynomials, arXiv:2007.09504 (2020), doi:10.48550/arXiv.2007.09504, https://arxiv.org/abs/2007.09504; journal version, with author order as published: A. N. Varchenko, A. M. Slinkin, D. Thompson, Dynamical $\mathfrak{sl}_2$ 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
3. 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, https://arxiv.org/abs/0908.4064
4. G. Felder, C. Wieczerkowski, Conformal blocks on elliptic curves and the Knizhnik–Zamolodchikov–Bernard equations, Communications in Mathematical Physics 176 (1996), no. 1, 133–161, doi:10.1007/BF02099366
5. E. Mukhin, A. Varchenko, Quasi-polynomials and the Bethe Ansatz, Geometry & Topology Monographs 13 (2008), 385–420, doi:10.2140/gtm.2008.13.385
6. E. Mukhin, V. Tarasov, A. Varchenko, Schubert calculus and representations of the general linear group, Journal of the American Mathematical Society 22 (2009), no. 4, 909–940, doi:10.1090/S0894-0347-09-00640-7