This series consists of talks in the area of Mathematical Physics.
In this talk I will present a construction of relative Bridgeland stability conditions that appeared in my work on stability of topological Fukaya categories of surfaces. This construction gives a local-to-global tool for constructing stability conditions. I will explain what technical features of such Fukaya categories render this construction useful in their context, and time allowing discuss the challenges involved in applying these ideas to other contexts such as Bridgeland-Smith type stability conditions and topological Fukaya categories with coefficients.
A recent construction of HOMFLY-PT knot homology by Oblomkov-Rozansky has its physical origin in “B-twisted” 3D N=4 gauge theory, with adjoint and fundamental matter. Mathematically, the construction uses certain categories of matrix factorization. We apply 3D Mirror Symmetry to identify an A-twisted mirror of this construction. In the case of algebraic knots, we find that knot homology on the A side gets expressed as cohomology of affine Springer fibers (related but not identical to work if Gorsky-Oblomkov-Rasmussen-Shende).
There is a close relationship between derived loop spaces, a geometric object, and Hochschild homology, a categorical invariant, made possible by derived algebraic geometry, thus allowing for both intuitive insights and new computational tools. In the case of a quotient stack, we discuss a "Jordan decomposition" of loops which is made precise by an equivariant localization result. We also discuss an Atiyah-Segal completion theorem which relates completed periodic cyclic homology to Betti cohomology.
I discuss an application of a recent construction of 2d integrable field theories from 4d Chern-Simons theory by Costello and Yamazaki. After a review of the construction, I consider integrable line defects in purely chiral models, such as the chiral WZW model and products/cosets thereof. Here, Wilson lines of the 4d Chern-Simons theory descend to line defects in 2d theories which break the conformal symmetry but preserve integrability, and the role of the spectral parameter is played by the complexified RG scale of line defects.
I will talk about compact hyperkahler manifolds, which generalize the famous K3 surface to the higher dimensions. Given a compact simple hyperkahler manifold $M$, I will describe how the structure of cohomology algebra H*(M) is related with the so(b_2+2) Lie algebra action and the second cohomology group. I will explain how this is applied to the generalization of Kuga-Satake construction which allows us to assign for K3-type Hodge structure a Hodge structure of weight one (i.e. complex torus).
The category of coherent sheaves on an interesting variety X has an extremely annoying property: does not have enough projectives, so it cannot be equivalent to the category of modules over an algebra. However, if you pass to the derived category, this defect can be fixed in many interesting cases, by finding a tilting generator: that is, a vector bundle T such that any coherent sheaf can be resolved by a complex consisting of sums of copies of T, and Ext^i(T,T)=0 for all i>0.
I will discuss joint work with Roman Bezrukavnikov on a categorical version of Hikita duality, which relates coherent sheaves on a symplectic resolution to constructible sheaves on the loop space of the dual resolution. I will focus on a basic case, where this can be made very explicit, and finish with some wild speculation on further generalisations.
I will discuss a theorem, joint work in progress with Constantin Teleman, in which we characterize which topological 3-dimensional Chern-Simons theories admit nonzero boundary theories. Accepting some physical heuristics, it tells which gapped systems in 2+1 dimensions admit only gapless boundary systems.
I will discuss recent developments in describing the chiral algebras associated to 4d N=2 theories introduced by Beem et al. in terms of Omega backgrounds, and give a description of the class S chiral algebras following this perspective, in terms of boundary conditions, interfaces, and junctions in 4d N=4 SYM.
In the 90s Turaev, Viro, Barrett and Westbury constructed a (2+1)D state sum TQFT associated to any fusion category. The associated phases of matter were popularized by Kiteav, Levin and Wen and are now central examples in condensed matter physics and quantum information theory.
Despite the importance of these phases, many of the computational techniques for working with fusion categories have not percolated into condensed matter physics. Many of these techniques are "folk theorems"
and have not appeared in the literature in a digestible form.