arXiv subjects

Robert Hanson

Publications and source records attributed to Robert Hanson.

5 recordsLinked to original sources

arXiv2025

Geometric Eisenstein series in non-abelian Hodge theory and hyperholomorphic branes from supersymmetry

Using geometric Eisenstein series, foundational work of Arinkin and Gaitsgory constructs cuspidal-Eisenstein decompositions for ind-coherent nilpotent sheaves on the de Rham moduli of local systems. This article extends these constructions to coherent (not ind-coherent) nilpotent sheaves on the Dolbeault, Hodge and twistor moduli from non-abelian Hodge theory. We thus account for Higgs bundles, Hodge filtrations and hyperk\"ahler rotations of local systems. In particular, our constructions are shown to decompose a hyperholomorphic sheaf theory of so-called BBB-branes into cuspidal and Eisenstein components. Our work is motivated, on the one hand, by the `classical limit' or `Dolbeault geometric Langlands conjecture' of Donagi and Pantev, and on the other, by attempts to interpret Kapustin and Witten's physical duality between BBB-branes and BAA-branes in 4D supersymmetric Yang--Mills theories as a mathematical statement within the geometric Langlands program.

math.AG↗

arXiv2025

The CompGIT package: a computational tool for Geometric Invariant Theory quotients

We describe CompGIT, a SageMath package to describe Geometric Invariant Theory (GIT) quotients of projective space by simple groups. The implementation is based on algorithms described by Gallardo--Martinez-Garcia--Moon--Swinarski. In principle the package is sufficient to describe any GIT quotient of a projective variety by a simple group -- in practice it requires that the user can construct an equivariant embedding of the polarised variety into projective space. The package describes the non-stable and unstable loci up to conjugation by the group, as well as describing the strictly polystable loci. We discuss potential applications of the outputs of CompGIT to algebraic geometry problems, a well as suggesting directions for future developments.

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arXiv2024

Fourier-Mukai transforms and normalisation of nodal curves

We study Arinkin's Poincar\'e sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $\Sigma \to C$. We prove that the Poincar\'e sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincar\'e sheaf $\mathcal{P}_{\Sigma}$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_{\Sigma}$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems.

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arXiv2023

The Dirac-Higgs complex and categorification of (BBB)-branes

Let $\mathcal{M}_{\mathrm{Dol}}(X,G)$ denote the hyperk\"ahler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang-Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of $\mathcal{M}_{\mathrm{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperk\"ahler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne-Hitchin twistor space $\mathrm{Tw}(\mathcal{M}_{\mathrm{Dol}}(X,G))$. Following Gaiotto's suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\mathrm{Tw}(\mathcal{M}_{\mathrm{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann-Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

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arXiv2022

Fourier-Mukai transform for fine compactified Prym varieties

Consider a finite covering $\beta : C \to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q'$, one can construct the corresponding compactified Prym varieties $\overline{\mathrm{P}}_\beta(q)$ and $\overline{\mathrm{P}}_\beta(q')$. Consider $\Gamma$ to be the group of line bundles whose torsion coincides with the order of $\beta$. In this article we construct a Fourier-Mukai transform between the derived categories of $\overline{\mathrm{P}}_\beta(q)$ and the $\Gamma$-equivariant derived category of $\overline{\mathrm{P}}_\beta(q')$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves.

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