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GRT at home Seminar

Ben Davison, University of Edinburgh, UK

Cuspidal cohomology for 2CY categories

Organizer's time: 2021-04-27 17:00 Europe/Rome

Duration: 1 hour 15 minutes

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ATTENTION: The date of the talk has changed

Let be a 2CY category, e.g. the category of Higgs sheaves on a smooth projective curve, or coherent sheaves on a surface with trivial canonical bundle, or representations of a preprojective algebra .  In each instance there is a Hall algebra structure on the Borel-Moore homology of the stack of objects in .  Via dimensional reduction from the shifted cotangent stack and the perverse filtration on the vanishing cycles of this stack, one may define the BPS cohomology , a subspace of the Hall algebra that is closed under the commutator Lie bracket.

If is the category of -reps for a finite type quiver, this Lie algebra is the Lie algebra associated to the Dynkin diagram underlying and the Hall algebra is the positive half of the Yangian .  For all other choices of , the Lie algebra is more mysterious, but the Hall algebra should still be seen as the positive half of a Yangian associated to .  I'll discuss conjectures and progress towards understanding imaginary simple roots of using the "less" perverse filtration on the BM homology of the stack of objects in .

The slides are here

The video is here

Submitted by: Francesco Sala