Communications in Number Theory and Physics

Volume 5 (2011)

Number 2

Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants

Pages: 231 – 252



Maxim Kontsevich (Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France)

Yan Soibelman (Department of Mathematics, Kansas State University)


To a quiver with potential we assign an algebra in the category of exponential mixed Hodge structures (the latter is also introduced in the paper). We compute the algebra (which we call Cohomological Hall algebra) for quivers without potential and study factorization properties of its Poincaré–Hilbert series in general case. As an application we obtain an alternative approach to our theory of motivic Donaldson–Thomas invariants of 3-dimensional Calabi–Yau categories and prove their integrality properties. We discuss the relationship of Cohomological Hall algebra with other mathematical structures including cluster algebras and Chern– Simons theory.

Published 26 September 2011