Mathematical Physics

The underlying holomorphic structure of a generalized Kahler manifold has been recently understood to be a square in the double category of holomorphic symplectic groupoids (or (1,1)-shifted symplectic stacks). I will explain what this means and how it allows us to describe the generalized Kahler metric in terms of a single real scalar function, resolving a conjecture made by physicists Gates, Hull, and Rocek in 1984. This is based on joint work with Yucong Jiang and Daniel Alvarez available at https://arxiv.org/abs/2407.00831.

I'll discuss the representation theory of line operators in 3d holomorphic-topological theories, following recent work with Wenjun Niu and Victor Py. Examples of the line operators we have in mind include half-BPS lines in 3d N=2 supersymmetric theories (reinterpreted in a holomorphic twist). We compute the OPE of line operators, which endows the category with a meromorphic tensor product, and establish a perturbative nonrenormalization theorem for the OPE. Then, applying Koszul-duality methods of Costello and Costello-Paquette, we represent the category of lines as modules for a new sort of mathematical object, which we call a dg-shifted Yangian. This is an A-infinity algebra, with a chiral coproduct whose data includes a Maurer-Cartan element that behaves like an infinitesimal r-matrix. The structure is a cohomologically shifted version of the ordinary Yangians that represent lines in 4d holomorphic-topological theories.

Higgs branches in theories with 8 supercharges change as one tunes the gauge coupling to critical values.  This talk will focus on six dimensional (0,1) supersymmetric theories in studying the different phenomena associated with such a change. Based on a Type IIA brane system, involving NS5 branes, D6 branes and D8 branes, one can derive a "magnetic quiver” which enables the construction of the Higgs branch using a “magnetic construction” or as a more commonly known object “3d N=4 Coulomb branch”.  Interestingly enough, the magnetic construction opens a window to a new set of Higgs branches which were not available using the well studied method of hyperkähler quotient.  It turns out that exceptional global symmetries are fairly common in the magnetic construction, and few examples will be shown. In all such cases there are strongly coupled theories where Lagrangian description fails, and the magnetic construction is helpful in finding properties of the theory.  Each Higgs branch can be characterized by a phase diagram which describes the different sets of massless fields around vacua. We will use such diagrams to study how Higgs branches change.  If time permits we will show an interesting exceptional sequence consisting of SU(3) — G2 — SO(7).

We will discuss some aspects of my recent preprint, joint with Victor Ginzburg, on Kostant-Whittaker reduction, a (deformation) quantization of restriction to a Kostant slice. We will explain how this functor can be used to prove conjectures of Ben-Zvi and Gunningham on parabolic induction, as well as a convolution exactness conjecture of Braverman and Kazhdan in the D-module setting. While this talk will occasionally reference facts from a talk I gave at Perimeter on other aspects of this preprint, the overlap and references will be minimal.

Skein theory forms a once-categorified 3d TQFT and assigns skein algebras to surfaces and skein modules to 3-manifolds. Motivated by physics, these modules are expected to satisfy a certain holonomicity property, generalizing Witten's finiteness conjecture of skein modules. In this talk, we will recall the basic notions of skein theory as a deformation quantization theory, and then state and discuss the generalized Witten's finiteness conjecture.

Chern-Simons theory is a topological quantum field theory which leads to link invariants, such as the Jones polynomial, through the expectation values of Wilson loops. The same link invariants also appear in a mathematical construction of Reshetikhin and Turaev which uses a trace on Yang-Baxter operators. Several algebraic structures are involved in these frameworks for computing link invariants, including the braid group, quantum algebras and centralizer algebras (such as the Temperley-Lieb algebra). In this talk, I will explain how the Askey-Wilson algebra, originally introduced in the context of orthogonal polynomials, can also be understood within the Chern-Simons theory and the Reshetikhin-Turaev link invariant construction.

The operator algebraic approach to quantum field theory is called algebraic quantum field theory. In this setting, we consider a family of operator algebras generated by observables in spacetime regions. This has been particularly successful in 2-dimensional conformal field theory. We present roles of tensor categories, modular invariance, classification theory, induction machinery and connections to vertex operator algebras.

I will review some recent progress in derived differential geometry, in particular pertaining to moduli stacks of solutions of elliptic partial differential equations on manifolds (with boundaries, and also with `logarithmic' boundaries, which include, for instance, manifolds with asymptotically cylindrical ends). In particular, this framework allows one to work efficiently with the compactified moduli spaces of symplectic topology and gauge theory. In another direction, I will explain some work in progress on the derived geometry of jet spaces, which can be used to endow moduli stacks of solutions of EOMs of a classical field theory with shifted symplectic structures.