This series consists of talks in the area of Mathematical Physics.
The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra in the category of corepresentations of a coquasitriangular Hopf algebra gives a new larger coquasitriangular Hopf algebra, for example taking c_q[SL_2] to c_q[SL_3] for these quantum groups reduced at certain odd roots of unity.
We use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes.
Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation. An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure.
I will present some results on three-dimensional gauge theory from the point of view of extended topological field theory. In this setting a theory is specified by describing its collection of boundary conditions - in our case, a collection of categories (standing in for 2d TFTs) with a prescribed symmetry group G.
The notion of singular support for coherent sheaves was introduced by Arinkin and Gaitsgory in order to carefully state the geometric Langlands conjecture. This is a conjectural equivalence of categories of sheaves on certain moduli spaces: in order to make the conjecture reasonable one needs to restrict to sheaves which satisfy a certain "singular support condition". In this talk I'll explain how to think about this singular support condition from the point of view of boundary conditions in twisted N=4 gauge theory. Specifically, Arinkin and Gaitsgory's singular su
In this talk, we will discuss an open Gromov-Witten invariant on hyperKahler surfaces, including K3 surfaces and certain Hitchin moduli spaces. The invariant is defined via the Lagrangian Floer theory and satisfy the Kontsevich-Soibelman wall-crossing formula and are expect to recover the generalized Donaldson-Thomas invariants studied in the work of Gaiotto-Moore-Neitzke.
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.
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 physics proof of the Atiyah-Singer index theorem relates the Hamiltonian and Lagrangian approaches to quantization of N=1 supersymmetric mechanics.