Abstract: Both the Higgs bundle moduli space and the moduli space of flat connections have a natural stratification induced by a C* action. In both of these stratifications, each stratum is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and flat connections, this homeomorphism … Read More
Abstract: Wilson loops are important observables in gauge theory. In this talk, we study half-BPS Wilson loops of a large class of five dimensional supersymmetric quiver gauge theories with 8 supercharges, in a nontrivial instanton background. The Wilson loops are codimension 4 defects of the quiver gauge theory, and their interaction with self-dual instantons is captured by a 1d ADHM … Read More
Abstract: 2d mirror symmetry relates the algebraic and symplectic geometry of a pair of Kähler manifolds obtained by dualizing a Lagrangian torus fibration. We will discuss some known and expected results in the case when the spaces involved are actually hyperkähler, arising as moduli spaces of 4-dimensional supersymmetric gauge theories (for instance, by a K-theoretic version of the Braverman-Finkelberg-Nakajima construction), … Read More
The Affine Grassmannian is an ind-scheme associated to a reductive group G. It has a cell structure similar to the one in the usual Grassmannian. Transversal slices to these cells give an interesting family of Poisson varieties. Some of them admit a smooth symplectic resolution and have an interesting geometry related to the representation theory of the Langlands dual group. … Read More
Abstract: I will give an overview of some new invariants of 3- and 4-manifolds that arise naturally in the study of compactifications of M-theory. I will discuss homological invariants of 3-manifolds and their “decategorifications” that take the form of q-series with integer coefficients, and, if time permits, homotopy-theoretic invariants of 4-manifolds that generalize the more familiar Donaldson and Seiberg-Witten invariants.
G2 manifolds constitute a class of Einstein seven-manifolds and are of substantial interest both in Riemannian geometry and theoretical physics. At present a vast number of compact G2 manifolds is known to exist. In this talk I will discuss a gauge-theoretic approach to the construction of invariants of compact G2 manifolds. I will focus on an interplay between gauge theories … Read More
In this talk we discuss Koszul duality from a physics perspective, and emphasize its role in coupling quantum field theories to topological line defects. Using this physical translation of Koszul duality as inspiration, we propose a physical definition of Koszul duality for vertex algebras. The appearances of vertex algebras (physically: holomorphic conformal field theories) in physics are legion; one particularly … Read More
We will discuss the definition of Coulomb branches in N=4 SUSY field theories in 3 dimensions, adapting the Braverman-Finkelberg-Nakajima construction to the case of non-polarized representation. Their computation can be done by an Abelianization theorem. Time permitting, I will discuss the associated TQFT.
Abstract: It is well-known that the GIT quotient depends on a choice of an equivariant ample line bundle. Various different quotients are related by birational transformations, and their B-models (D^bCoh) are related by semi-orthogonal decompositions, or derived equivalences. If we apply mirror symmetry, it is natural to ask how the A-models of the mirror of various quotients are related. We … Read More
Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers the Yangian of a quiver Q as defined by Maulik-Okounkov. However, for general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem. One … Read More