Mathematical Physics

The Fukaya category of a symplectic manifold is an excellent toy model for exploring the general structure of boundary conditions in two-dimensional topological field theories. In this talk I will explain an analogue for three dimensional theories. The natural playground for boundary conditions in d=3 is the 3d A-model into a hyperKahler target space. Here the fundamental BPS equation is the Fueter equation. I will explain how studying certain boundary value problems for the Fueter equation, one is lead to something resembling a higher categorical structure. I will then argue how a similar structure also seems to appear after performing a massive deformation by a hyperKahler moment map. 

Zoom link:  https://pitp.zoom.us/j/96574302292?pwd=dkpIVXN0UmFiRkdkenFzc2o2Nko0dz09

The Deligne-Mumford moduli space of genus 0 curves plays many roles in representation theory.  For example, the fundamental group of its real locus is the cactus group which acts on tensor products of crystals.

I will discuss a variant on this space which parametrizes "cactus flower curves".  The fundamental group of the real locus of this space is the virtual cactus group.  This moduli space of cactus flower curves is also the parameter space for inhomogeneous Gaudin algebras.

Zoom link:  https://pitp.zoom.us/j/96658223425?pwd=NUxRN2FsdWJ1SWtHMlRDcTdHMGNPQT09

I'll explain work in progress, joint with Miroslav Rapcak, on geometric constructions of vertex algebras associated to divisors in toric Calabi-Yau threefolds, in terms of moduli stacks of objects in certain exotic abelian subcategories of complexes of coherent sheaves on the underlying threefold. These vertex algebras were originally proposed by Gaiotto-Rapcak, and constructed mathematically in the example of affine space by Rapcak-Soibelman-Yang-Zhao, building on Schiffmann-Vasserot's proof of the AGT conjecture. We give a geometric explanation and generalization of the quivers with potential that feature in the latter results, and outline the analogous construction of vertex algebras in this setting.

Zoom link:  https://pitp.zoom.us/j/93749710253?pwd=Y3NUTHBXZ3FPQUdPU0E0d0ttVzFFdz09

The Morita 2-category has as objects associative algebras, 1-morphisms are bimodules and 2-morphisms are given by bimodule homomorphisms. Equivalent objects in this category are exactly Morita equivalent algebras. A vast generalisation of this as a higher category is the so-called higher Morita category, denoted Alg_n. It has two constructions, one due to Haugseng, and one due to Scheimbauer which uses (constructible) factorization algebras. In the latter, Gwilliam-Scheimbauer has proven that every object of Alg_n is n-dualizable. Hence, by the Cobordism Hypothesis, every object gives rise to an n-dimensional (fully extended framed) topological field theory. A natural question to ask is “Which objects of Alg_n are also (n+1)-dualizable?”. This talk is on work in progress (for n=2) to prove a conjecture due to Lurie answering this question. 

Zoom link: https://pitp.zoom.us/j/98595913913?pwd=Vlo1aWtXVlZBVjYxNFlTM2VZY2s3Zz09

The Lagrangian Floer homology of a pair of holomorphic Lagrangian submanifolds of a hyperkahler manifold is expected to simplify, by work of Solomon-Verbitsky and others. This occurs in part because, in this setting, the symplectic action functional, the gradient flow of which computes Lagrangian Floer homology, is the real part of a holomorphic function. As noted by Haydys, thinking of this holomorphic function as a superpotential on an infinite-dimensional symplectic manifold gives rise to a quaternionic analog of Floer's equation for holomorphic strips: the Fueter equation. I will explain how this line of thought gives rise to a `complexification' of Floer's theorem identifying Fueter maps in cotangent bundles to Kahler manifolds with holomorphic planes in the base. This complexification has a conjectural categorical interpretation, giving a model for Fukaya-Seidel categories of Lefshetz fibrations, which should have algebraic implications for the study of Fukaya categories. This is a report on upcoming joint work with Aleksander Doan.

I will discuss a method to associate new types of quantum groups to certain holomorphic Lagrangian fibrations. This algebraic structure controls the enumerative geometry of the variety admitting the Lagrangian fibration in a way analogous to how a quantum group controls the enumerative geometry of a symplectic singularity. Various questions, including the monodromy of the quantum differential equation, can be answered using local-to-global techniques.

Zoom Link: https://pitp.zoom.us/j/99571261442?pwd=NDZTL1dIbXJPNHZwWVcvSEM4M2crZz09

 I will discuss a brane construction of rational gl(m|n) spin chains with spins valued in Verma modules and show that string dualities map the spin chain spectra to the vacua of certain 2d N=(2,2) gauge theories, thus realizing the 2d Bethe/Gauge correspondence for superspin chains. The talk is based on arXiv:2110.15112.

Zoom Link: https://pitp.zoom.us/j/98827189404?pwd=blNjdy9oYnc2cGUrQXdwV2xoOVpOdz09

In this talk I will review the recent progress in constructing factorisation algebras that represent QFT models on Lorentzian manifolds. I will also discuss their relation to local nets of algebras in AQFT. Renormalization in these models is done using the Epstein-Glaser approach. This is joint work with O. Gwilliam, E Hawkins and B. Visser.

Zoom Link: https://pitp.zoom.us/j/93242972298?pwd=Z0taa3hBSzJ5VWtpQWlaTVRkbTBaUT09

Let K be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to K, and in this way obtain a link invariant. In other words the number of minimal surfaces doesn’t depend on the isotopy class of the link. These counts of minimal surfaces can be organised into a two-variable polynomial which is perhaps a known polynomial invariant of the link, such as HOMFLYPT. “Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how these minimal surface invariants can be seen as Gromov-Witten invariants for the twistor space of hyperbolic 4-space. Whilst Gromov-Witten theory suggests the overall strategy for defining the minimal surface link-invariant, there are significant differences in how to actually implement it. This is because the geometry of both hyperbolic space and its twistor space become singular at infinity. As a consequence, the PDEs involved (both the minimal surface equation and J-holomorphic curve equation) are degenerate rather than elliptic at the boundary. I will try and explain how to overcome these complications. 

Zoom link:  https://pitp.zoom.us/j/95847819111?pwd=MVg5dWFsNUpiZWVPL1l4Uk9PV2tZZz09

In this talk, I will discuss: (1) the new BGG-type resolutions of finite dimensional representations of simple Lie algebras that lead to BGG-relations expressing finite-dimensional transfer matrices via infinite-dimensional ones, (2) the factorization of infinite-dimensional ones into the product of two Q-operators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R.Frassek, I. Karpov, and V.Pestun.

Nakajima’s quiver varieties play important roles in mathematical physics and representation theory. They are defined as symplectic reduction of the space of representations of the doubled quivers, and they are equipped with natural scheme structures. It is not known in general whether this scheme is reduced or not, and the reducedness issue does show up in certain scenario, for example the integration formula of the K-theoretic Nekrasov’s partition function. In this talk I will show that the quiver variety is reduced when the moment map is flat, and I will also give some applications of this result. This talk is based on my work arXiv: 2201.09838.

Zoom Link: https://pitp.zoom.us/j/97405405211?pwd=dEtVeHhQVjNrdGN4Vkh0ZlRrbEpVQT09