This series consists of talks in the area of Mathematical Physics.
This talk concerns a family of special functions common to the study of quantum conformal blocks and hypergeometric solutions to q-KZB type equations. In the first half, I will explain two methods for their construction -- as traces of intertwining operators between representations of quantum affine algebras and as certain theta hypergeometric integrals we term Felder-Varchenko functions. I will then explain our proof by bosonization the first case of Etingof-Varchenko's conjecture that these constructions are related by a simple renormalization.
The physics proof of the Atiyah-Singer index theorem relates the Hamiltonian and Lagrangian approaches to quantization of N=1 supersymmetric mechanics.
An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. You can study local systems of vector spaces over this local system of fields. On a 3-manifold, they are are rigid, and the rank one local systems are counted by the Alexander polynomial. On a surface, they come in positive
We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k,n), provided k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k,n). Our results can be translated to statements about clusters in Fock-Goncharov configuration spaces of affine flags, provided the number of flags is even.
The Feynman diagram expansion for a Wilson loop observable in Chern-Simons gauge theory generates an infinite series of topological invariants for framed knots. In this talk, I will describe a new perturbative formalism which conjecturally generates the same invariants for Legendrian knots in the standard contact R^3. The formalism includes a `perturbative' localization principle which drastically simplifies the structure of calculations. As time permits, I will provide some examples and applications. This talk is based upon joint work with Brendan McLellan and Ruo
Representations of the fundamental groups of surfaces appear so often in geometry that it's tempting to see them primarily as geometric structures. In recent years, however, researchers have uncovered beautiful new features of these representations by thinking of them instead as dynamical systems. As an invitation to the dynamical point of view, I'll describe how geometric tools from the study of billiards can be used to build invariants of surface group representations.
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.