Lagrangian correspondences for moduli spaces of Higgs bundles and holomorphic connections

Abstract: On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast C$ (respectively, the twisted cotangent bundles of $C$). Central to these constructions are Higgs bundles (respectively, holomorphic connections) which are transversal to line subbundles of the underlying bundles: these naturally induce divisors on $C$ together with auxiliary parameters, namely lifts to divisors on spectral curves for Higgs bundles and residue parameters of apparent singularities for holomorphic connections. We discuss the evidence showing that the Dolbeault geometric Langlands correspondence is generically realized by these Lagrangian correspondences; we expect that the de Rham geometric Langlands correspondence can be realized by their quantization, following Drinfeld's construction of Hecke eigensheaves. We also discuss the relations of our constructions to various topics, including reductions of Kapustin-Witten equations, the conformal limit, separation of variables, and degenerate fields in conformal field theories.

• The paper constructs and analyzes explicit holomorphic Lagrangian correspondences between moduli spaces of Higgs bundles and holomorphic connections, linking them to Hilbert schemes.
• The paper demonstrates that the constructed subvarieties are generically Lagrangian or co-isotropic, ensuring precise dimension counts and connections with integrable systems.
• The paper advances the geometric Langlands program by relating these correspondences to Fourier-Mukai transforms and conjectural quantization frameworks.

Lagrangian Correspondences for Moduli Spaces of Higgs Bundles and Holomorphic Connections

Introduction and Motivation

This paper constructs and analyzes holomorphic Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (and holomorphic connections) on a compact connected Riemann surface $C$ (of genus $g \geq 2$) and Hilbert schemes of points on $T^*C$ (or its twisted variants). The key innovation is a systematic study of Higgs bundles and holomorphic connections transverse to line subbundles, which leads to explicit families of Lagrangians in moduli spaces, parameterized by effective divisors $D$ on $C$. These constructions not only provide geometric insights into the structure of Hitchin and de Rham moduli spaces, but also establish concrete links to the geometric Langlands program. The results further relate to reductions of Kapustin-Witten equations, separation of variables in integrable systems, as well as classical and quantum aspects of conformal field theory (CFT).

Figure 1: Illustration of Lagrangians $\mathbb{L}_H(D)$ (red) and $\mathbb{L}_{dR}(D)$ (blue) in rank $2$ moduli spaces as $\deg D$ varies, showing their stratification and projection structure.

Construction of Lagrangian Subvarieties

Triples and Induced Divisors

The central objects are triples $C$0 (for Higgs bundles) and $C$1 (for holomorphic connections), subject to generic transversality constraints. The choice of line subbundles $C$2 induces morphisms $C$3 and $C$4, whose zeros define divisors $C$5 and $C$6 on $C$7. These divisors encode spectral data: for Higgs bundles, lifts to spectral curves yield divisors on $C$8. For connections, apparent singularities correspond to divisors together with residue parameters, forming points in twisted cotangent bundles.

Lagrangian Subvarieties in Moduli Spaces

Fixing an effective divisor $C$9 and a suitable line bundle $g \geq 2$0, one forms subvarieties $g \geq 2$1 and $g \geq 2$2 inside the respective moduli spaces by requiring $g \geq 2$3 or $g \geq 2$4 for some embedding $g \geq 2$5. In the $g \geq 2$6 case, the compatibility condition $g \geq 2$7 appears naturally.

The paper proves:

• $g \geq 2$8 is non-empty with an open dense Lagrangian subset in $g \geq 2$9.
• $T^*C$0 is co-isotropic, and generically Lagrangian in $T^*C$1 for reduced divisors.

These results generalize earlier constructions for $T^*C$2 and for divisors of maximal or minimal degree, extending the Lagrangian property to arbitrary congruent degrees.

Lagrangian Correspondences with Hilbert Schemes

Higgs Bundles and Hilbert Schemes

Given a triple $T^*C$3 and a smooth spectral curve $T^*C$4, the spectral line bundle admits a canonical decomposition as $T^*C$5. Effective divisors $T^*C$6 on $T^*C$7 may be viewed as points in $T^*C$8.

Defining $T^*C$9 as the collection of pairs $D$0 matching spectral and divisor conditions, one obtains a Lagrangian correspondence:

$D$1

Holomorphic Connections and Twisted Hilbert Schemes

For connections, additional data — residue parameters at apparent singularities — yield points in the Hilbert scheme $D$2, where $D$3 is a $D$4-torsor. The paper constructs $D$5, a Lagrangian subscheme of $D$6, again providing a correspondence structure.

Relationship to Geometric Langlands and Related Frameworks

Dolbeault and de Rham Geometric Langlands

The constructed correspondences are shown to generically realize the Dolbeault geometric Langlands correspondence (GLC) via Fourier-Mukai transforms between Hitchin moduli and Hilbert schemes. The de Rham GLC is conjectured to arise via quantization of these correspondences, echoing Drinfeld's construction of Hecke eigensheaves. The paper provides evidence for these claims, both algebraic-geometric and from integrable systems.

Extended Kapustin-Witten Equations and Hodge Moduli

A reduction of Kapustin-Witten equations to extended Bogomolny equations relates solutions to the triple-based Lagrangian structures, particularly through conformal and Hodge limits. The interplay with $D$7-actions and stratifications in the Hodge moduli connects dynamical flows, Lagrangian strata, and the geometric representation theory of branes.

Separation of Variables and Conformal Field Theory

The technique of fixing divisors — corresponding to separated variables in integrable systems — finds classical and quantum avatars in Hitchin moduli and CFT partition functions. Apparent singularities and their residue parameters emerge naturally in BPZ equations with degenerate fields, placing the present constructions at the interface of Langlands, integrability, and quantum field theory.

Strong Results and Implications

• Dimension Counts: Lagrangian subvarieties have dimension exactly $D$8 of their ambient moduli spaces; corresponding correspondences are generically finite-to-one.
• Explicit Formulae: The morphisms $D$9, $C$0, and associated spectral divisors are described via explicit determinants and Wronskian constructions.
• Generic Lagrangian Property: The Lagrangian property is shown to hold for a broad class of divisors beyond previous results, both in the Hitchin and de Rham settings.
• Quantization Conjectures: The quantized Lagrangian correspondences are conjectured to realize functorial equivalence between categories of $C$1-modules, underpinning quantum GLC.

Applications, Examples, and Future Directions

The paper details the stratification and fiber structures of constructed Lagrangians, especially in rank $C$2 cases, linking them to Hodge moduli and wobbly bundles. The explicit descriptions facilitate computation of Darboux coordinates, analysis of brane-type subvarieties, and investigation of flows and stratifications in moduli. Practically, these results open new avenues for explicit parameterizations in geometric Langlands, for analysis of spectral data in integrable systems, and for understanding quantum phenomena in moduli spaces.

Theoretical implications include strengthened conjectures about quantization commutativity, the realization of Hecke eigensheaves and Whittaker models via Lagrangian microlocal supports, and new geometric perspectives on analytic and quantum Langlands correspondences.

Conclusion

By systematically exploiting transversality to line subbundles, this paper constructs explicit holomorphic Lagrangian correspondences between Hitchin/de Rham moduli spaces and Hilbert schemes (twisted and untwisted). These geometric structures generically realize classical geometric Langlands correspondences, with conjectural extensions to the quantum setting via functorial quantization. The results provide concrete bridges between algebraic geometry, integrable systems, and quantum field theory, offering new computational and conceptual tools for the study of moduli spaces and correspondences. Future work is likely to focus on higher rank generalizations, the explicit quantization framework, and deeper analytic aspects in Langlands and CFT.