Séminaire Laurent Schwartz — EDP et applications

In this talk I will focus on BPS states in supersymmetric field theories with N=2. In this theories one can consider a certain class of supersymmetric line operators. Such operators support a new class of BPS states, called framed BPS states. I will discuss a formalism based on quivers to understand these objects and their properties. Time permitting I will discuss a relation with the theory of cluster algebras.

The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier:

affine spheres as level surfaces
metrization of the interior of K by the Hessian metric F”
primal-dual symmetry
interpretation as a minimal Lagrangian submanifold in a certain para-Kähler space form
complex-analytic structure on 3-dimensional cones.

Kitaev’s quantum double models provide a rich class of examples of two-dimensional lattice systems with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the Hamiltonians. (joint work with Matthew Cha and Pieter Naaijkens)

We discuss some quite general properties of infinite discrete groups G acting on compact spaces. The spaces we mainly have in mind are horoboundaries of metric spaces which admit an isometric action of G. As an application, we show that the mapping class group of a surface of finite type admits a proper action on some $L^p$-space.

The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group $pi_1$(X) of X into PGL(2,R). It is a contractible space. For closed orientable surfaces, « higher analogues » of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of $pi_1$(X) that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yields new examples of higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the derivations' arguments. Let us explore the implications of this principle: in effect, we develop the calculus of variations on the infinite jet spaces for maps from sheaves of free associative algebras over commutative manifolds to the generators and then, quotients of free associative algebras over the linear relation of equivalence under cyclic shifts. In the frames of such variational noncommutative symplectic (super)geometry we prove the main properties of the Batalin-Vilkovisky Laplacian and variational Schouten bracket.

 

We show as by-product that the structures which arise in the variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory. I will explain how string theory unifies these phenomena.

 

This is based on my work with Kevin Costello [arXiv:1810.01970].

I will describe a remarkable symmetric monoidal category associated to a reductive group G, which acts centrally on any G-category. This construction quantizes the universal centralizer group scheme, together with its action on Hamiltonian G-spaces used by Ngo in his proof of the Fundamental Lemma. The category and its central action appear most naturally in a Langlands dual incarnation, which is phrased in terms of convolution on the affine Grassmannian, via work of Bezrukavnikov, Finkelberg and Mirkovic. This is joint work with David Ben-Zvi.

The purpose of the talk is to explain a result in collaboration with B. Pozzetti and A. Wienhard expressing the Hausdorff dimension of certain attractors as a critical exponent. This class of attractors consists of limit sets of Anosov representations in PGLd (hence of non-conformal nature) that verify an extra open condition. If time permits, we will discuss implications of the formula to the geometry of the Hitchin component.

Mapping class groups associated to surfaces whose fundamental group is not finitely generated are called « big », and their study is linked to classical problems in dynamics. In this talk, we discuss some of the basic properties of big mapping class groups, their simplicial actions, and how these can be used to prove that big mapping class groups « detect » surfaces or (if time allows) that the space of non-trivial quasimorphisms of a big mapping class group is infinite dimensional.