Mathematical Physics

The notion of Positive Representations is a new research program devoted to the representation theory of split real quantum groups, initiated in a joint work with Igor Frenkel. It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finite-dimensional representation theory of compact quantum groups, but at the same time also serves some new properties that are not available in the compact case. In this talk, I will survey the recent developments and describe some of its relations to other areas of mathematics. 

I will discuss some results on double loop groups that point to geometric phenomena about double affine flag varieties and double affine Grassmannian. One result of this study is a definition of double affine Kazhdan-Lusztig polynomials.

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 second half of the talk will concern applications to affine Macdonald theory.  I will present refinements of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr.  I will then explain how to prove the first non-trivial cases of these conjectures by combining the methods of the first half and well-chosen applications of the elliptic beta integral.   The second half of this talk is joint work with E. Rains and A. Varchenko.

The physics proof of the Atiyah-Singer index theorem relates the Hamiltonian and Lagrangian approaches to quantization of N=1 supersymmetric mechanics. Similar ideas applied to the N=(0,1) supersymmetric sigma model construct two versions of elliptic cohomology: elliptic cohomology at the Tate curve over the integers and the universal elliptic cohomology theory over the complex numbers. Quantization procedures give analytic constructions of wrong-way maps in these cohomology theories. Relating these to the Ando-Hopkins-Strickland-Rezk string orientation of topological modular points to intricate torsion invariants associated with these sigma models. 

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-dimensional moduli (perfect of characteristic p), but they are more "stable" than ordinary local systems in the GIT sense.  When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way.  On S^1 x R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it.

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. Finally, we apply our results to the two "Grassmannians of finite mutation type," proving in these cases versions of conjectures made by Fomin and Pylyavskyy describing cluster variables as SL_k web invariants. 

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 Ruoran Zhang.