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Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces

Kang Lu, Evgeny Mukhin

TL;DR

This work develops a concrete bridge between Bethe ansatz solutions for Gaudin models tied to orthosymplectic Lie superalgebras and algebraic-geometry frameworks. By introducing a reproduction procedure, it generates populations of Bethe roots whose associated symmetric rational pseudodifferential operator $\mathcal{R}$ remains invariant; the superkernel $W$ becomes a self-dual superspace, and the population is canonically identified with the variety of isotropic full superflags in $W$ and with symmetric complete factorizations of $\mathcal{R}$. The authors carefully treat the two main osp families, $\mathfrak{osp}_{2m+1|2n}$ and $\mathfrak{osp}_{2m|2n}$, including type-B/C/D root structures, extended parity sequences, and fake reproduction procedures, to obtain a unified geometrical picture akin to the Langlands-dual flag variety in the purely bosonic setting. The main theorem asserts a bijection among the Bethe population, isotropic superflags in $W$, and symmetric factorizations of $\mathcal{R}$ under a superfertility condition, thus connecting integrable models with isotropic flag geometry and offering pathways to understanding eigenstructures of Bethe subalgebras in a super setting. These results generalize and unify known type-B, type-C, and even-type D cases and pave the way for further extension to quasi-periodic boundary conditions and XXX-type BAE for orthosymplectic algebras.

Abstract

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $\mathfrak{osp}_{2m+1|2n}$ and $\mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $\mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $\mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $\mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $\mathfrak{osp}_{2m|0}=\mathfrak{so}_{2m}$.

Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces

TL;DR

This work develops a concrete bridge between Bethe ansatz solutions for Gaudin models tied to orthosymplectic Lie superalgebras and algebraic-geometry frameworks. By introducing a reproduction procedure, it generates populations of Bethe roots whose associated symmetric rational pseudodifferential operator remains invariant; the superkernel becomes a self-dual superspace, and the population is canonically identified with the variety of isotropic full superflags in and with symmetric complete factorizations of . The authors carefully treat the two main osp families, and , including type-B/C/D root structures, extended parity sequences, and fake reproduction procedures, to obtain a unified geometrical picture akin to the Langlands-dual flag variety in the purely bosonic setting. The main theorem asserts a bijection among the Bethe population, isotropic superflags in , and symmetric factorizations of under a superfertility condition, thus connecting integrable models with isotropic flag geometry and offering pathways to understanding eigenstructures of Bethe subalgebras in a super setting. These results generalize and unify known type-B, type-C, and even-type D cases and pave the way for further extension to quasi-periodic boundary conditions and XXX-type BAE for orthosymplectic algebras.

Abstract

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras and . Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator . Under some technical assumptions, we show that the superkernel of is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in and with the set of symmetric complete factorizations of . In particular, our results apply to the case of even Lie algebras of type D corresponding to .

Paper Structure

This paper contains 32 sections, 32 theorems, 132 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on orthosymplectic Lie superalgebras
  3. General linear Lie superalgebras
  4. Orthosymplectic Lie superalgebras
  5. Root systems
  6. The case of $\mathfrak{osp}_{2m+1|2n}$
  7. The case of $\mathfrak{osp}_{2m|2n}$
  8. Borel subalgebras
  9. Representations
  10. Symmetric Rational pseudo-differential operators and isotropic superflags
  11. Rational pseudo-differential operators
  12. Factorizations and superflag varieties
  13. Symmetric differential operators and self-dual spaces
  14. Symmetric rational pseudo-differential operators and self-dual superspaces
  15. Bethe ansatz
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $\lambda$ be a dominant integral $\mathfrak{osp}_{2m+\iota|2n}$-weight, and ${\tilde{\bm s}} \in \widetilde{S}_{m|n}^\iota$ an extended parity sequence. For $1\leqslant i\leqslant r-1$, if $s_i\ne s_{i+1}$, then where $\eta=1$ if $\lambda_{[{\tilde{\bm s}}],i}^{{\tilde{\bm s}}}+\lambda_{[{\tilde{\bm s}}],i+1}^{{\tilde{\bm s}}}\ne 0$ and $\eta=0$ otherwise. If $\iota=0$ and $s_r=1$ (that is if

Theorems & Definitions (59)

  • Lemma 2.1
  • proof
  • Proposition 3.1: CDSK12
  • Proposition 3.2: HMVY19
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5: MV04
  • proof
  • Corollary 3.6
  • ...and 49 more