Lagrangian correspondences for moduli spaces of Higgs bundles and holomorphic connections

TL;DR

This paper constructs Lagrangian correspondences linking moduli spaces of Higgs bundles and holomorphic connections to Hilbert schemes and twisted cotangent bundles on Riemann surfaces, with implications for geometric Langlands and related topics.

Contribution

It introduces new Lagrangian correspondences between moduli spaces and Hilbert schemes, providing evidence for geometric Langlands correspondence realization.

Findings

Lagrangian correspondences relate moduli spaces to Hilbert schemes.

Evidence suggests these correspondences realize the Dolbeault geometric Langlands.

Discussion includes potential realization of the de Rham Langlands via quantization.

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.