Mathematical Physics

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.

This is a joint work with A.Kuznetsov and L.Rybnikov.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety.

We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

We will discuss a (conjectural) explicit presentation for the equivariant cohomology of Nakajima quiver varieties of type ADE.  This presentation arises as a shadow of the expected symplectic duality between slices to Schubert varieties in the affine Grassmannian and Nakajima quiver varieties (a.k.a. the expected Coulomb and Higgs branches for a quiver gauge theory). 

Categorical symplectic geometry studies an invariant of symplectic manifolds called the "Fukaya (A-infinity) category", which consists of the Lagrangian submanifolds and a symplectically-robust intersection theory of these Lagrangians.  Over the last two decades the Fukaya category has emerged as a powerful tool: for instance, it has produced inroads to Arnol'd's Nearby Lagrangians Conjecture, and it allowed Kontsevich to formulate the the Homological Mirror Symmetry conjecture.

In this talk I will describe a project, joint with Satyan Devadoss, Stefan Forcey, and Katrin Wehrheim, which attempts to relate the Fukaya categories of different symplectic manifolds via a notion of functoriality.  After mentioning some analytic results about the singular quilts necessary for this construction, I will describe the combinatorial component: with Devadoss and Forcey we are constructing a family of polytopes that specialize to the associahedra in two different ways, and can be thought of as the 2-categorical version of associahedra.