Speaker
Description
In arXiv:2105.13334, Gyenge, Koppensteiner and Logvinenko constructed a 2-categorification of the Heisenberg algebra of a smooth and proper DG category, and decategorified it via Grothendieck group. In this talk, I will explain the ongoing effort to make this work with the Hochschild homology $HH_*$, instead. Effectively, this means extending it from a lattice in $HH_0$ to the whole Hochschild homology.
This first raises a question of what is the Heisenberg algebra of a graded vector space. Then, one has to construct the crucial map from the Heisenberg algebra of $HH_*$ of a DG category to the $HH_*$ of the Heisenberg 2-category. The payoff is a direct generalisation of Nakajima’s original result on the Heisenberg algebra acting on the cohomology of Hilbert schemes of points on a surface.
