Mathematical Physics

The coordinate functions on a Poisson variety are log-canonical if the Poisson bracket of two coordinate functions equals a constant times the product of these functions. We consider the symplectic groupoid of unipotent upper-triangular matrices equipped with canonical Poisson bracket. We described a system of log-canonical coordinates and the corresponding cluster structure. As a bonus, we discovered a system of log-canonical coordinates on Teichmueller space of closed genus 2 surfaces. This is joint work with L. Chekhov.

Zoom link:  https://pitp.zoom.us/j/94716952708?pwd=R2RiQWRpcHFMYlJLMlB0UjlPVGZkQT09

A factorization algebra is a cosheaf-like local-to-global object which is meant to model the structure present in the observables of classical and quantum field theories. In the Batalin-Vilkovisky (BV) formalism, one finds that a factorization algebra of classical observables possesses, in addition to its factorization-algebraic structure, a compatible Poisson bracket of cohomological degree +1. Given a ``sufficiently nice'' such factorization algebra on a manifold $N$, one may associate to it a factorization algebra on $N\times \mathbb{R}_{\geq 0}$. The aim of the talk is to explain the sense in which the latter factorization algebra ``knows all the classical data'' of the former. This is the bulk-boundary correspondence of the title. Time permitting, we will describe how such a correspondence appears in the deformation quantization of Poisson manifolds.

Zoom Link: https://pitp.zoom.us/j/96701822187?pwd=NDh5ZFpGZ2JCNmVNOVVIYzVPV2wvdz09

In 2008 jointly with Maxim Kontsevich we introduced the notion of stability data on graded Lie algebras. In the case of the Lie algebra of vector fields on a symplectic torus it underlies the wall-crossing formulas for Donaldson-Thomas invariants of 3-dimensional Calabi-Yau categories. In 2013 we introduced the notion of wall-crossing structure, which is a locally-constant sheaf of stability data. Wall-crossing structures naturally appear in complex integrable systems, Homological Mirror Symmetry and many other topics not necessarily related to Donaldson-Thomas theory. Recently, in 2020 we introduced a sublass of analytic wall-crossing structures. We formulated a general conjecture that analytic wall-crossing structure gives rise to resurgent (i.e. Borel resummable) series.

Many wall-crossing structures have geometric origin, and moreover they naturally appear in our Holomorphic Floer Theory program. Aim of my talk is to discuss wall-crossing structures associated with a pair of holomorphic Lagrangian submanifolds of a complex symplectic manifold (in most cases it will be the cotangent bundle). These wall-crossing structures underly Cecotti-Vafa wall-crossing formulas, and as such they appear naturally in the study of exponential integrals in finite and infinite dimensions. I am going to explain our conjectural approach to Chern-Simons theory  which is based on the idea of wall-crossing structure. In some aspects this approach is related to the work of Witten on analytic continuation of Chern-Simons theory.

Zoom link:  https://pitp.zoom.us/j/99446428842?pwd=aDRzbFJoNytDNURDUVFMNGQzNjBFQT09

We describe a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the so-called Madelung transform between the Schrödinger-type equations on wave functions and Newton's equations on densities turns out to be a Kähler map between the corresponding phase spaces, equipped with the Fubini-Study and Fisher Rao information metrics. This is a joint work with G.Misiolek and K.Modin.

Zoom Link: https://pitp.zoom.us/j/92283820673?pwd=dXQzYWppaUdBVnhtVG1TZTFQQmx4UT09

In recent years, the classification of fermionic symmetry protected topological phases has led to renewed interest in classical constructions of invariants in homotopy theory. In this talk, we focus on the description of Steenrod squares for triangulated spaces at the cochain level, introducing new formulas for the cup-i products and discussing their universality through an axiomatic approach. We also examine the interaction between Steenrod squares and the algebra structure in cohomology, providing a cochain level proof of the Cartan relation as requested by Kapustin. Time permitting, we will also study the Adem relation from this perspective.

Zoom link:  https://pitp.zoom.us/j/98288876236?pwd=cHJVM3M1K3FsUmdtbENZenhKMnBkdz09

Factorization algebras are local-to-global objects, much like sheaves, and it is natural to ask what kind of topology, geometry, and physics they are sensitive to. We will examine this question with a focus on less-perturbative phenomena, touching on topics like moduli of vacua for 4-dimensional gauge theories and Dijkgraaf-Witten-type TFTs. Apologies hereby issued in advance to the (hopefully) friendly audience (and to my collaborators!) for speaking before achieving complete clarity.

Zoom link:  https://pitp.zoom.us/j/94417858154?pwd=ak54UFpPb3hFbnBwcUlnMnhCdG1odz09

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.