This series consists of talks in the area of Mathematical Physics.
I will discuss how Costello's inductive renormalization
procedure for the construction of effective field theories can be
extended to manifolds with boundary.
Positive representations are infinite-dimensional bimodules for the quantum group and its modular dual where both act by positive essentially self-adjoint operators.
We study the effective BV quantization theory for chiral deformation of two dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. The generating functions are proven to be quasi-modular forms. As an application, we construct an exact solution of quantum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves.
We will describe a formulation of the Batalin-Vilkovisky formalism using derived symplectic geometry. In this setting, the classical master equation of the BV formalism describes a space of coisotropic structures. Using this approach, we resolve a conjecture of Felder-Kazhdan regarding BRST cohomology. Time permitting, we will also describe applications of these ideas to more general quantization problems.
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.
Quantum groups from character varieties