Proves Dolbeault geometric Langlands equivalence for GL_r and SL_r/PGL_r over the locus of spectral curves with at worst type A singularities, extending beyond the elliptic locus via Whittaker normalization.
A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves
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abstract
In our previous paper with Tudor P\u{a}durariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for $\mathrm{GL}_2$ over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.
fields
math.AG 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Constructs semiorthogonal decompositions for derived categories on quasi-smooth derived algebraic stacks indexed by component lattices, with examples for moduli stacks of G-bundles, G-Higgs bundles, and G-local systems.
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The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus
Proves Dolbeault geometric Langlands equivalence for GL_r and SL_r/PGL_r over the locus of spectral curves with at worst type A singularities, extending beyond the elliptic locus via Whittaker normalization.
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Semiorthogonal decompositions for stacks
Constructs semiorthogonal decompositions for derived categories on quasi-smooth derived algebraic stacks indexed by component lattices, with examples for moduli stacks of G-bundles, G-Higgs bundles, and G-local systems.