Communications in Number Theory and Physics

Volume 17 (2023)

Number 4

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds

Pages: 847 – 939

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n4.a2

Authors

Miroslav Rapčák (Center for Theoretical Physics, University of California, Berkeley, Calif., U.S.A.)

Yan Soibelman (Department of Mathematics, Kansas State University, Manhattan, Ks., U.S.A.)

Yaping Yang (School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, Australia)

Gufang Zhao (School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, Australia)

Abstract

We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi–Yau $3$-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via “raising operators”. Conjecturally the COHA action extends to an action of the Drinfeld double by adding the “lowering operators”.

In this paper, we show that the Drinfeld double is a generalization of the notion of the Cartan doubled Yangian defined earlier by Finkelberg and others. We extend this “$3d$ Calabi–Yau perspective” on the Lie theory furthermore by associating a root system to certain families of $X$. We formulate a conjecture that the above-mentioned action of the Drinfeld double factors through a shifted Yangian of the root system. The shift is explicitly determined by the moduli problem and the choice of stability conditions, and is expressed explicitly in terms of an intersection number in $X$. We check the conjectures in several examples, including a special case of an earlier conjecture of Costello.

Received 27 January 2021

Accepted 10 November 2023

Published 24 January 2024