Description
I will discuss a precise formulation of the Dolbeault geometric
Langlands conjecture, introduced by Donagi–Pantev as the classical
limit of the de Rham geometric Langlands correspondence. On the
automorphic side, the formulation involves limit categories, which may
be viewed as classical limits of categories of D-modules on moduli
stacks of bundles over curves. It predicts an equivalence between the
derived categories of moduli stacks of semistable Higgs bundles and
the limit categories associated with moduli stacks of all Higgs
bundles. The definition of limit categories, as well as this
formulation of the Dolbeault geometric Langlands conjecture, is
motivated by categorical Donaldson–Thomas theory for Calabi–Yau
3-folds. In particular, these categories carry semiorthogonal
decompositions into quasi-BPS categories, categorifying BPS invariants
in Donaldson–Thomas theory. This is joint work with Tudor Pădurariu,
arXiv:2508.19624.
I will then explain work in progress on the proof of the Dolbeault
geometric Langlands equivalence for GL_r and SL_r/PGL_r over an open
locus of the Hitchin base which strictly contains the elliptic locus.
This gives a nontrivial case in which the relevant moduli stacks are
not quasi-compact, and the use of limit categories is essential both
for the formulation and for the proof.