Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for 22-dimensional Calabi-Yau categories (with an appendix by Ben Davison)

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).Comment: The revised version contains the Appendix written by Ben Davison about the relationship of Kontsevich-Soibelman product with the one of Schiffmann-Vassero