Cohomological Hall algebra for matrix integrals

Maxim Kontsevich

IHES

Mon, May. 25th 2009, 11:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

This is a recent joint work with Y.Soibelman.
With a quiver with polynomial potential we associate an associative algebra (called cohomological Hall algebra) graded by a discrete Heisenberg group. This algebra is an analog of the algebra of BPS states proposed by Harvey and Moore. Graded components of our algebra are equivariant cohomology groups for contours in the matrix model, for all finite sizes of matrices. It is the algebra of ($m-1$) free fermions for the potential Trace($X^m$), and is quite non-trivial for a general quiver with vanishing potential. The generating series for dimensions of graded components is a generalization of the quantum dilogarithm, and is closely related to BPS counting and Donaldson-Thomas invariants for local Calabi-Yau, or for N=4 gauge theories. Also one can identify the wall-crossing with Poincare-Birkhoff-Witt decomposition.

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