| Résume | Coulomb branches are certain moduli spaces arising in supersymmetric field theory. They include as special cases many spaces of independent interest such as double affine Hecke algebras, certain open Richardson varieties, multiplicative Nakajima quiver varieties etc. In the four-dimensional case, one expects that their coordinate rings should be categorified (lifted) to tensor categories that carry a lot of interesting structure. The physics literature, in particular work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, provides various clues about these categories, but not a precise definition. However, recent advances have led to a satisfactory mathematical definition in an important special case (gauge theories of cotangent type), building on work of Braverman-Finkelberg-Nakajima.
We will illustrate with specific examples the ingredients which go into constructing and studying these categorified Coulomb branches based on recent work with Harold Williams. One of our main results is that these categories carry a natural t-structure consisting of what we call Koszul-perverse coherent sheaves. The classes of such simples sheaves provide canonical basis in a uniform way. We plan to discuss some basic geometry of the affine Grassmannian, the theory of perverse coherent sheaves, certain t-structures related to Koszul duality and perhaps other related topics such as the appearance of cluster algebras (depending on time and interest). |
