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.

Project Report

Symmetries appearing in natural sciences (physics,chemistry,biology,etc.) are described (typically) in mathematics in terms of groups and algebras. Depending on the type of symmetry the required mathematical tools can be very abstract. The project deals with mathematical structures underlying certain symmetries in theoretical physics (more precisely, in string theory and gauge theory). The approach of the project leads to a clean rigorous definition of certain quantities known in physics as BPS invariants. Such a definition was known in certain cases,but not in general. The unifying power of mathematical point of view helps not only to define the known quantities, but also to compute them in many cases. The results of the project raise intriguing questions in algebra,geometry,mathematical physics. They have attracted attention of several research groups worldwide including those in China,France, Japan, Switzerland, UK,. They were subject of several international conferences,lecture courses,graduate courses, student seminars,etc. The project introduces a new mathematical structure which has already found interesting applications. More applications are expected. Those include mathematics (knot theory, representations of quivers, Donaldson-Thomas theory on Calabi-Yau threefolds,etc.) as well as physics (BPS algebras of gauge theories). The techniques proposed are original and non-trivial (e.g. exponential Hodge structures). On the other hand, the main definitions and structures can be explained to a good gradute student. Hence the subject will be (and actually is) attractive for young mathematicians and physicists looking for interesting "hot" topics for their research. This means that this area of research will be active for many years to come.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Program Officer
Tie Luo
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Kansas State University
United States
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