New Connections of Representation Theory to Algebraic Geometry and Physics
Representation theory is formally a branch of algebra which studies algebraic structure of symmetries. Its role in contemporary mathematics is due to the fact – realized through major discoveries of the 19th and 20th centuries mathematics – that the structure of symmetries holds key to understanding basic objects of interest in algebra, number theory and quantum physics. Thus the whole history of the subject suggests that important insights are to be derived from further expanding the rich web of connections between representation theory and neighboring fields.
Some conceptually new connections of this kind have been discovered lately. Notable examples include: the interpretation of geometric Langlands duality in terms of gauge theory by Kapustin, Witten et. al.; new homotopy-theoretic foundations of algebraic geometry (by J. Lurie, B. Toen et. al.) and their applications to representation theory including geometric Langlands duality; wall-crossing formalism in quantum field theory (by M. Douglas, G. Moore et. al.) and its algebro-geometric interpretation (T. Bridgeland, M. Kontsevich – Y.Soibelman) which opens a way to use quantum cohomology and other deep algebraic geometry as a tool for study of numerical invariants of representations.
The goal of the Special Year was to explore and advance these and other new directions. The Special Year Activity can be classified into three categories:
1. Several lectures and mini-course were given to introduce experts in representation theory to fundamentals of other subjects. Two seminars presenting the basics of quantum field theory to mathematicians (led by S. Gukov, with participation of E. Diaconescu, M. Douglas, A. Neitzke and others) ran for the whole of the fall semester. The seminars have accomplished the difficult task of introducing mathematicians to the background material necessary to understand the recent results in the physics literature which were discussed in other seminars later in the year. Another fall seminar provided an introduction to geometric Langlands correspondence theory. In the second semester a mini-course on categorical knot invariants was given by one of the creators of the theory Mikhail Khovanov. This part of low-dimensional topology has many intersections with representation theory, some of these are subject of research by IAS members C. Stroppel and S. Gukov reported in other seminars. Also, a series of lectures delivered by member C. Barwick and visitors J. Lurie, B. Toen forms a mini-course on new homotopy-theoretic foundations of algebraic geometry, the so-called derived algebraic geometry. These have already found major applications to representations of real groups in the work of members D. Ben-Zvi and D. Nadler. Many more applications are expected to appear in the near future.
2. Seminar talks by members, visitors and invited speakers informed participants of recent and current research results. Only a sample can be described here. A seminar on instanton counting was organized by H. Nakajima. This is one of the mathematical incarnations of Seiberg-Witten theory providing powerful identities between Gromov-Witten invariants. The approach of A. Braverman (member in the 2nd semester) links this to deep representation theory. Members A. Braverman and M.Finkelberg gave talks about “classical” and geometric versions of representation theory of affine Kac-Moody groups over a local field, mostly due to them. This is one of the approaches to building the 2-dimensional version of geometric Langlands duality. The analogues of basic facts of the 1-dimensional theory present rather difficult and beautiful problems resolved mostly by these authors. Another series of talks was devoted to wall-crossing phenomena and the related formalism of stability conditions on derived categories by T. Bridgeland. Most of the talks were given by V. Toledano Laredo who is collaborating with Bridgeland on the exciting connection of this formalism to Stokes phenomenon for ordinary differential equations. This theory is expected to provide the necessary tool for extending the established classical results of representation theory, such as Kazhdan-Lusztig theory, to a much wider context. Ramification and developments of this topic were discussed in the conference talks by T. Bridgeland, A. Okounkov and Y. Soibelman. Last but not least, the discussion of results from physical literature culminated in a series of talks by E. Witten explaining the ideas of his ongoing project (joint with Gaiotto). There the spaces familiar in representation theory arise as spaces of vacua of certain quantum field theory and Langlands duality appears as a duality of boundary conditions in Yang-Mills theory.
3. Collaborations between participants and individual work has led to important new results.