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Superopers revisited

Anton M. Zeitlin

TL;DR

Addresses extending the geometric Langlands-type link between Gaudin spectra and opers to Lie superalgebras by introducing superopers on super Riemann surfaces and a $Z$-twisted Miura-Plücker framework. The paper develops $qp$- and $qq$-systems for Lie superalgebras, analyzes low-rank examples, and studies Weyl-group–type transformations and their extensions to odd roots. It shows, under nondegeneracy conditions, how these systems encode Bethe equations for Gaudin models, and discusses $q$-deformations and $(G,q)$-opers for supergroups. The work aims to unify Miura oper data across Dynkin diagrams via reproduction-type transformations, proposing a generalized $Z$-twisted superoper that captures Bethe data for a Langlands dual $^L\mathfrak{g}$ in the super setting.

Abstract

The relation between special connections on the projective line, called Miura opers, and the spectra of integrable models of Gaudin type provides an important example of the geometric Langlands correspondence. The possible generalization of that correspondence to simple Lie superalgebras is much less studied. Recently some progress has been made in understanding the spectra of Gaudin models and the corresponding Bethe ansatz equations for some simple Lie superalgebras. At the same time, the original example was reformulated in terms of an intermediate object: Miura-Plücker oper. It has a direct relation to the so-called $qq$-systems, the functional form of Bethe ansatz, which, in particular, allows $q$-deformation. In this note, we discuss the notion of superoper and relate it to the examples of $qq$-systems for Lie superalgebras, which were recently studied in the context of Bethe ansatz equations. We also briefly discuss the $q$-deformation of these constructions.

Superopers revisited

TL;DR

Addresses extending the geometric Langlands-type link between Gaudin spectra and opers to Lie superalgebras by introducing superopers on super Riemann surfaces and a -twisted Miura-Plücker framework. The paper develops - and -systems for Lie superalgebras, analyzes low-rank examples, and studies Weyl-group–type transformations and their extensions to odd roots. It shows, under nondegeneracy conditions, how these systems encode Bethe equations for Gaudin models, and discusses -deformations and -opers for supergroups. The work aims to unify Miura oper data across Dynkin diagrams via reproduction-type transformations, proposing a generalized -twisted superoper that captures Bethe data for a Langlands dual in the super setting.

Abstract

The relation between special connections on the projective line, called Miura opers, and the spectra of integrable models of Gaudin type provides an important example of the geometric Langlands correspondence. The possible generalization of that correspondence to simple Lie superalgebras is much less studied. Recently some progress has been made in understanding the spectra of Gaudin models and the corresponding Bethe ansatz equations for some simple Lie superalgebras. At the same time, the original example was reformulated in terms of an intermediate object: Miura-Plücker oper. It has a direct relation to the so-called -systems, the functional form of Bethe ansatz, which, in particular, allows -deformation. In this note, we discuss the notion of superoper and relate it to the examples of -systems for Lie superalgebras, which were recently studied in the context of Bethe ansatz equations. We also briefly discuss the -deformation of these constructions.
Paper Structure (22 sections, 9 theorems, 64 equations)

This paper contains 22 sections, 9 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Superopers on supercurves and their reduction
  3. A reminder on super Riemann surfaces and super holomorphic connections
  4. Miura Superopers
  5. Notations
  6. Definition of superopers
  7. Miura superopers and their pure even counterparts
  8. Miura-Plücker opers, $qp$-systems, and $qq$-systems
  9. $Z$-twisted Cartan connections
  10. Low rank cases
  11. Z-twisted Miura $SL(2)$-opers
  12. Z-twisted Miura $SL(1|1)$ opers and abelian connections
  13. Z-twisted Miura $OSP(1|2)$-opers
  14. Miura-Plücker opers
  15. $qp$-and $qq$-system for Lie superalgebra $\mathfrak{g}$.
  16. ...and 7 more sections

Key Result

Proposition 2.3

For any Miura $G$-oper on $SC^*$, there exists a trivialization of the underlying $G$-bundle $\mathcal{F}_G$ on an open dense subset of $SC^*$ for which the superoper connection has the form where $g_i(z,\theta), a_i(z)\in {S}(z)[\theta]$, so that $g_i(z,\theta)$ are all even and $a_i(z,\theta)$ are opposite in parity to $e_i$.

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Corollary 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 3.1
  • Proposition 3.2
  • ...and 11 more