D-module and nilpotent limit categories for genus-zero vector bundles

From papers

Let C=P1C=\mathbb{P}^1 and let BunGLr(0)\operatorname{Bun}_{\operatorname{GL}_r}(0) be the degree-zero component of the moduli stack of rank-rr vector bundles. Let IndLN(HiggsGLr(0))0\operatorname{IndL}_{\mathcal{N}}(\mathrm{Higgs}_{\operatorname{GL}_r}(0))_0 denote the neutral component of the nilpotent limit category. Genus-zero limit-category conjecture. There is an equivalence

D-mod(BunGLr(0))IndLN(HiggsGLr(0))0.\mathrm{D\text{-}mod}\left(\operatorname{Bun}_{\operatorname{GL}_r}(0)\right)\simeq\operatorname{IndL}_{\mathcal{N}}\left(\mathrm{Higgs}_{\operatorname{GL}_r}(0)\right)_0.

This is proposed as the automorphic counterpart of the trivial genus-zero degeneration on the spectral side. No resolution is supplied in the statement.

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Sources & referencesView supporting material

Primary source

Tudor Pădurariu and Yukinobu Toda, “The Dolbeault geometric Langlands conjecture via limit categories”, arXiv:2508.19624 (2026).

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