Cohomological Hall algebra and motivic Donaldson-Thomas invariants

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The project is an example of a successful interaction of ideas originated in physics with those in mathematics. Mathematically, the project is devoted to an approach to motivic Donaldson-Thomas invariants based on the new mathematical object, called Cohomological Hall algebra. It was introduced in the joint work of PI and Maxim Kontsevich. Proposed work simplifies some old results in the area (e.g. it gives a transparent explanation of the so-called wall-crossing formulas which show how Donaldson-Thomas invariants depend on a stability condition). It also opens new directions of work. Some of them have already attracted attention of mathematical community (e.g. the conjecture about the structure of Cohomological Hall algebra for symmetric quiver). The approach is developed in the framework of quivers with potential. The latter give rise to 3-dimensional Calabi-Yau categories generated by critical points of the potential. Donaldson-Thomas invariants are defined in terms of the sheaf of vanishing cycles of the potential. Cohomological Hall algebra encodes the structure of the cohomology of Milnor fiber of the potential near the critical locus. This makes a link with the earlier work of the PI and Kontsevich on the approach to motivic Donaldson-Thomas invariants which utilizes motivic integration. The new approach is more direct and easier. The interplay between quivers and categories mentioned above is used in both directions, e.g. in the course of study of the behavior of Donaldson-Thomas invariants with respect to mutations. This leads to an interesting application to cluster transformations. Furthermore, the analogy with Chern-Simons theory suggests a new application to topological invariants of 3-dimensional manifolds. From another perspective, Cohomological Hall algebra is a mathematical incarnation of the algebra of BPS states envisioned by string theorists in the middle of 90's. Motivic Donaldson-Thomas invariants correspond to refined BPS states in some supersymmetric theories. In a sense the project gives first mathematically rigorous definition of the notion of BPS state (and refined BPS state) which is ``model independent". Wall-crossing formulas for BPS states, which play a role e.g. in the conjectures about the entropy of black holes, can be written in a new non-trivial way and proven mathematically. Maybe this is one of the reasons for the attention of different groups of physicists to the results of the project. Growing interest to the new approach has already generated a flow of papers written by senior and young researchers in both physics and mathematics communities. Several conferences and workshops recently organized in the US, Europe and Japan were influenced by the developments related to the project.