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

Access information: https://us02web.zoom.us/j/84259232669?pwd=aXlkZTJ1dnFzT05Kb2ZkOXk1NEp6dz09

ATTENTION: The date of the talk has changed

Let $\mathcal{C}$ 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 $\Pi_Q$.  In each instance there is a Hall algebra structure on the Borel-Moore homology of the stack of objects in $\mathcal{C}$.  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 $\mathfrak{g}_\mathcal{C}^+$, a subspace of the Hall algebra that is closed under the commutator Lie bracket.

If $\mathcal{C}$ is the category of $\Pi_Q$-reps for $Q$ a finite type quiver, this Lie algebra is the Lie algebra associated to the Dynkin diagram underlying $Q$ and the Hall algebra is the positive half of the Yangian $\mathbb{Y}_Q$.  For all other choices of $\mathcal{C}$, the Lie algebra is more mysterious, but the Hall algebra should still be seen as the positive half of a Yangian $\mathbb{Y}_\mathcal{C}$ associated to $\mathfrak{g}_\mathcal{C}$.  I'll discuss conjectures and progress towards understanding imaginary simple roots of $\mathfrak{g}_\mathcal{C}^+$ using the "less" perverse filtration on the BM homology of the stack of objects in $\mathcal{C}$.

The slides are here

The video is here

Submitted by: Francesco Sala