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Classical limit of the geometric Langlands correspondence for $SL(2, \mathbb{C})$

Duong Dinh, Joerg Teschner

TL;DR

The paper develops a concrete, coordinate-based description of the integrable structure on Hitchin moduli spaces for $SL_2(\mathbb{C})$, by building explicit parameterisations of strata and deploying Separation of Variables (SoV) to relate Higgs data to BA-divisors on spectral curves. It introduces and analyzes the BA-divisor framework, the SoV map, and its inverse, proving that SoV is generically étale with a $2^{2g}$-fold fiber and that, near generic points, SoV is a local symplectomorphism between the Hitchin-side cotangent data and a symmetric product of the spectral curve. The work connects these explicit geometric constructions to the geometric Langlands program via Drinfeld’s approach, and outlines paths toward the analytic Langlands program through quantisation and the KZB/BPZ framework, with potential applications to isomonodromic deformations and $\lambda$-connections. Overall, it provides a detailed, constructive bridge between Hitchin integrable systems, BA-divisors, and Langlands-type dualities, offering tools for explicit computation and for future quantisation of Hitchin systems. The results illuminate how BA-data encode the Prym-fibration structure and enable local symplectomorphisms that link Hitchin moduli to torus fibrations, thereby clarifying the classical limit of Langlands-type correspondences in this setting.

Abstract

The goal of this paper is to give an explicit description of the integrable structure of the Hitchin moduli spaces. This is done by introducing explicit parameterisations for the different strata of the Hitchin moduli spaces, and by adapting the Separation of Variables method from the theory of integrable models to the Hitchin moduli spaces. The resulting description exhibits a clear analogy with Drinfeld's first construction of the geometric Langlands correspondence. It can be seen as a classical limit of a version of Drinfeld's construction which is adapted to the complex number field.

Classical limit of the geometric Langlands correspondence for $SL(2, \mathbb{C})$

TL;DR

The paper develops a concrete, coordinate-based description of the integrable structure on Hitchin moduli spaces for , by building explicit parameterisations of strata and deploying Separation of Variables (SoV) to relate Higgs data to BA-divisors on spectral curves. It introduces and analyzes the BA-divisor framework, the SoV map, and its inverse, proving that SoV is generically étale with a -fold fiber and that, near generic points, SoV is a local symplectomorphism between the Hitchin-side cotangent data and a symmetric product of the spectral curve. The work connects these explicit geometric constructions to the geometric Langlands program via Drinfeld’s approach, and outlines paths toward the analytic Langlands program through quantisation and the KZB/BPZ framework, with potential applications to isomonodromic deformations and -connections. Overall, it provides a detailed, constructive bridge between Hitchin integrable systems, BA-divisors, and Langlands-type dualities, offering tools for explicit computation and for future quantisation of Hitchin systems. The results illuminate how BA-data encode the Prym-fibration structure and enable local symplectomorphisms that link Hitchin moduli to torus fibrations, thereby clarifying the classical limit of Langlands-type correspondences in this setting.

Abstract

The goal of this paper is to give an explicit description of the integrable structure of the Hitchin moduli spaces. This is done by introducing explicit parameterisations for the different strata of the Hitchin moduli spaces, and by adapting the Separation of Variables method from the theory of integrable models to the Hitchin moduli spaces. The resulting description exhibits a clear analogy with Drinfeld's first construction of the geometric Langlands correspondence. It can be seen as a classical limit of a version of Drinfeld's construction which is adapted to the complex number field.
Paper Structure (65 sections, 25 theorems, 141 equations)

This paper contains 65 sections, 25 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Integrable structure of Hitchin's moduli spaces
  3. Cotangent spaces of moduli of pairs (subbundles, bundles)
  4. The Separation of Variables maps
  5. Relation to geometric and analytic Langlands correspondence
  6. Further generalisations
  7. Acknowledgements.
  8. Moduli spaces
  9. Moduli spaces of stable bundles
  10. Segre stratification.
  11. Secant varieties.
  12. "Overcounting" bundles.
  13. Explicit construction of $\pmb{\mathbb{P}_L}$.
  14. Moduli spaces of pairs of bundles and subbundles
  15. Local coordinates.
  16. ...and 50 more sections

Key Result

Theorem 1.1

Theorems & Definitions (56)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Example 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 46 more