Mathematical Physics

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. As an application, the Wigner-Eckart theorem will be used to construct an analogue of the Jordan-Schwinger representation, previously known only for finite-dimensional representations of the Lorentz group, valid for infinite-dimensional ones.

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. In this talk, I plan to explain the constructions in this conjecture and the mathematical side of the story, in particular the viewpoints from noncommutative geometry and Gromov-Witten theory. This is based on joint work with Xiang Tang.

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.

I will report on progress understanding the 576-fold periodicity in TMF in terms of conformal field theoretic constructions.  Sporadic finite groups and their cohomology will play a role.

A Coxeter category is a braided tensor category which carries an action of a generalised braid group $B_W$ on its objects. The axiomatics of a Coxeter category and the data defining the action of $B_W$ are similar in flavor to the associativity and commutativity constraints in a monoidal category, but arerelated to the coherence of a family of fiber functors.

We will show how to construct two examples of such structure on the integrable category $\matcal{O}$ representations of a symmetrisable Kac–Moody algebra $\mathfrak{g}$, the first one arising from the quantum group $U_\hbar(\mathfrak{g})$, and the second one encoding the monodromy of the KZ and Casimir connections of $\mathfrak{g}$.

The rigidity of this structure, proved in the framework of $\mathsf{PROP}$ categories, implies in particular that the monodromy of the Casimir connection is given by the quantum Weyl group operators of $U_\hbar(\mathfrak{g})$.

This is a joint work with Valerio Toledano Laredo.

Topological factorization homology is an invariant of manifolds which enjoys a hybrid of the structures in topological field theory, and in singular homology.  These invariants are especially interesting when we restrict attention to the factorization homology of surfaces, with coefficients in braided tensor categories.  In this talk, I would like to explain a technique, related to Beck monadicity, which allows us to compute these abstractly defined categories, as modules for explicitly computable, and in many cases well-known, algebras.

The algebras produced in this way for the annulus and punctured torus are the so-called ``reflection equation algebra" and "quantum differential equation algebra", respectively.  When we close up punctures, a variation on the our formalism naturally reproduces the framework of quantum Hamiltonian reduction, and leads to simultaneous deformations of categories D(g/G) of character sheaves on g, on the one hand, and categories QC(Ch_G(T^2)), of quasi-coherent sheaves on the character variety of T^2, on the other.  We call these deformations "quantum character varieties", and they form the two-dimensional part of a four dimensional TFT related to Kapustin-Witten's geometric Langlands TFT, or "topologically twisted N=4 SYM".

We'll explain the slogan of the title: a cluster variety is a space associated to a quiver, and which is built out of algebraic tori.

They appear in a variety of contexts in geometry, representation theory, and physics. We reinterpret the definition as: from a quiver (and some additional choices) one builds an exact symplectic 4-manifold from which the cluster variety is recovered as a component in its moduli space of Lagrangian branes. In particular, structures from cluster algebra govern the classification of exact Lagrangian surfaces in Weinstein 4-manifolds.

This is joint work with Vivek Shende and David Treumann.

I will explain how to axiomatize the notion of a chiral WZW model using the formalism of VOAs (vertex operator algebras). This class of models is in almost bijective correspondence with pairs (G,k), where G is a connected (not necessarily simply connected) Lie group and k in H^4(BG,Z) is a degree four cohomology class subject to a certain positivity condition. To my surprise, I have found a couple extra models which satisfy all the defining properties of chiral WZW models, but which don't come from pairs (G,k) as above. The simplest such model is the simple current extension of the affine VOA E8xE8 at level (2,2) by the group Z/2. Thus from a quantum perspective, the Lie group E8 x E8 behaves as if it had a non-trivial center.

It is interesting to note that, unlike Chern–Simons theories whose gauge group can be disconnected, the gauge group of a chiral WZW model is necessarily connected. This raises the question of what is the chiral WZW model associated to a disconnected gauge group? Whatever the answer turns out to be, mathematicians will probably need to enlarge the class of objects that they agree to call ‘chiral conformal field theories’ in order to accommodate these yet to be defined models.

I will present some speculations on the matter.