Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for $2$-dimensional Calabi-Yau categories (with an appendix by Ben Davison)

TL;DR

This paper explores the relationship between cohomological Hall algebras, semicanonical bases, and Donaldson-Thomas invariants within 2-dimensional Calabi-Yau categories, extending concepts from 3-dimensional categories and proposing new conjectures.

Contribution

It introduces a framework connecting semicanonical bases and motivic Donaldson-Thomas invariants in 2D Calabi-Yau categories, including a new conjecture on defining Kac polynomials.

Findings

Proposes a conjecture for defining Kac polynomials in 2D Calabi-Yau categories.

Discusses the dimensional reduction from 3D to 2D Calabi-Yau categories.

Relates semicanonical bases to Cohomological Hall algebras in this setting.

Abstract

We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the "dimensional reduction" from 3-dimensional Calabi-Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson-Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi-Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi-Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).