This series consists of talks in the area of Mathematical Physics.
The affine Grassmannian is the analog of the Grassmannian for the loop group. They are very important objects in mathematical physics and the Geometric Langlands program. In this talk, I will explain my recent work on the central degeneration of semi-infinite orbits, Iwahori orbits and Mirkovic-Vilonen cycles in the affine Grassmannian. I will also use lots of convex polytopes to illustrate my results. In addition, I will explain the connections between my work and other parts of geometric representation theory and combinatorial algebraic geometry.
About a decade ago Eynard and Orantin proposed a powerful computation algorithm
known as topological recursion. Starting with a ``spectral curve" and some ``initial data"
(roughly, meromorphic differentials of order one and two) the topological recursion produces by induction
a collection of symmetric meromorphic differentials on the spectral curves parametrized by
pairs of non-negative integers (g,n) (g should be thought of as a genus and n as the number of punctures).
Integral values of zeta functions are important not only for what they say about other values of their respective functions, but also for what they say about transcendence degree questions for appropriate extensions of the rationals or other number fields. They also appear in some recent computations relevant to particle physics.
In this talk we will give a quick introduction to the theory of periods and motives, relate said theory to special values of zeta functions, and discuss a graphical definition of the associated category of motives.
We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras.
Quantum groups from character varieties
The Wigner-Eckart theorem is a well known result for tensor operators of SU(2) and, more generally, any compact Lie group. I will show how it can be generalised to arbitrary Lie groups, possibly non-compact. The result relies on the knowledge of recoupling theory between finite-dimensional and arbitrary admissible representations, which may be infinite-dimensional; the particular case of the Lorentz group will be studied in detail.
For a finite group G, a G-gerbe over a space B can be thought of as a fiber bundle over B with fibers the classifying orbifold BG. Hellerman-Henriques-Pantev-Sharpe studied conformal field theories on G-gerbes. Given a G-gerbe Y-> B, they constructed a disconnected space \widehat{Y} endowed with a locally constant U(1) 2-cocycle c. They conjectured that a CFT on Y is equivalent to a CFT on \widehat{Y} twisted by the "B-field" c.
I will talk about some connections among the GKZ (introduced by Gelfand-Kapranov-Zelevinsky) hypergeometric series, orbifold singularities of the system, and chain integrals in some geometry. The GKZ hypergeometric series appeared in some very interesting contexts including arithmetic geometry, enumerative geometry and mathematical physics in the last few decades. I will report some new geometric realizations and interpretations of them.
I'll describe a family of algebras called shifted Yangians, which arise as deformation quantizations of certain spaces related to loop groups. I'll also describe coproducts for these algebras, which are related to multiplication in the loop group. Physically, this fits into the story of Coulomb branches associated to 3d N=4 quiver gauge theories, where the above multiplication maps arises by taking products of scattering matrices.