Kavli Affiliate: Yukinobu Toda
| First 5 Authors: Tudor Pădurariu, Tudor Pădurariu, , ,
| Summary:
We introduce limit categories for cotangent stacks of smooth stacks as an
effective version of classical limits of categories of D-modules on them. We
develop their general theory and pursue their relation with categories of
D-modules. In particular, we establish the functorial properties of limit
categories such as the smooth pull-back and projective push-forward. Using the
notion of limit categories, we propose a precise formulation of the Dolbeault
geometric Langlands conjecture, proposed by Donagi-Pantev as the classical
limit of the de Rham geometric Langlands equivalence. It states an equivalence
between the derived categories of moduli stacks of semistable Higgs bundles and
limit categories of moduli stacks of all Higgs bundles. We prove the existence
of a semiorthogonal decomposition of the limit category into quasi-BPS
categories, which are categorical versions of BPS invariants on a non-compact
Calabi-Yau 3-fold. This semiorthogonal decomposition is interpreted as a
Langlands dual to the semiorthogonal decomposition constructed in our previous
work on the category of coherent sheaves on the moduli stack of semistable
Higgs bundles. We also construct Hecke operators on limit categories for Higgs
bundles. They are expected to be compatible with Wilson operators under our
formulation of Dolbeault geometric Langlands conjecture. The conjectured
equivalence implies an equivalence between BPS categories for semistable Higgs
bundles, which we expect to be a categorical version of the topological mirror
symmetry conjecture for Higgs bundles by Hausel-Thaddeus.
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