Mathematical Physics

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

Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogs of classical objects provide insights into their stable behavior patterns as n goes to infinity.
I will talk about some of my results on Harish-Chandra bimodules in Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.
Another interesting phenomenon arising in complex rank is that there are two ways to define Harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however, in Deligne categories that is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of Deligne categories.

Zoom Link: https://pitp.zoom.us/j/93951304913?pwd=WVk1Uk54ODkyT3ZIT2ljdkwxc202Zz09

There is an extremely rich history of interaction between physics and moduli spaces, for instance cohomological Hall algebras/algebras of BPS states, or vertex/chiral algebras. In this talk, I will explain a link between Joyce vertex algebras and one dimensional CoHAs, based on my paper 2110.14356, where quantum vertex algebras play a central role. The main technical tool is a ``bivariant" Euler class which makes torus localisation work in this context. I will also talk about ongoing work, in this area as well as in closely related projects.

Zoom Link: https://pitp.zoom.us/j/93045858347?pwd=N25FNVVjNEFFN09sdmFUV2M2YlZ5QT09

In this talk, I will discuss a new duality that was recently discovered in joint work with David Ayala and Nick Rozenblyum, which we refer to as reflection.

In essence, reflection amounts to two dual methods for reconstructing objects, based on a stratification of the category that they live in. As a classical example, an abelian group can be reconstructed on the one hand in terms of its p-completions and its rationalization, or on the other (reflected) hand in terms of its p-torsion components and its corationalization; and these both come from a certain "closed-open decomposition" of the category of abelian groups.

Examples and applications of reflection are abundant, because stratifications are abundant. In algebra, reflection recovers the derived equivalences of quivers coming from BGP reflection functors (hence the terminology "reflection"). In topology, reflection is closely related to Verdier duality, a generalization of Poincaré duality that applies to singular spaces. Moreover, an explicit description of reflection leads to a categorification of the classical Möbius inversion formula, a Fourier inversion theorem for functions on posets.

Zoom Link: https://pitp.zoom.us/j/92439401154?pwd=cGZsUGJGcmNXdFZuclREL3gyYnhQdz09

 I will report on my progress, joint with David Reutter, to construct and analyze the algebraic closure of nVec – in other words, the universal n-category of framed nD TQFTs. The invertibles are Pontryagin dual to the stable homotopy groups of spheres. The Galois group is almost, but not quite, the stable PL group. An invertible TQFT can be condensed from the vacuum if and only if it trivializes on (possibly-exceptional) spheres.