arXiv subjects

Duong Dinh

Publications and source records attributed to Duong Dinh.

4 recordsLinked to original sources

arXiv2026

Lagrangian correspondences for moduli spaces of Higgs bundles and holomorphic connections

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.

math.AG↗

arXiv2025

Rank-2 wobbly bundles from special divisors on spectral curves

We study rank-2 wobbly bundles on a Riemann surface $C$ of genus $g\geq 2$, i.e. semi-stable bundles admitting nonzero nilpotent Higgs fields, in terms of direct images of line bundles on smooth spectral curves $\tilde{C} \oversetπ{\rightarrow} C$. We give a sufficient condition for a semi-stable bundle $E$ to be wobbly: $E$ is a twist of $π_\ast \left(\mathcal{O}_{\tilde{C}}(\tilde{D}) \right)$ where the norm of $\tilde{D}$ is a summand of the divisor of a quadratic differential on $C$. We sketch the proof of the necessary condition statement, namely all rank-2 wobbly bundles can be characterised as such, and discuss how certain singularities of the wobbly locus arise from the Brill-Noether loci of spectral curves.

math.AG↗

arXiv2025

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

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.

math.DG↗

arXiv2024

From $λ$-connections to $PSL_2(\mathbb{C})$-opers with apparent singularities

On a Riemann surface of genus $> 1$, we discuss how to construct opers with apparent singularities from $SL_2(\mathbb{C})$ $λ$-connections $(E, \nabla_λ)$ and sub-line bundles $L$ of $E$. This construction defines a rational map from a space which captures important data of triples $(E, L, \nabla_λ)$ to a space which parametrises the positions and residue parameters of the induced apparent singularities. We show that this is a Poisson map with respect to natural Poisson structures. The relations to wobbly bundles and Lagrangians in the moduli spaces of Higgs bundles and $λ$-connections are discussed.

math.DG↗