Mathematical Physics

Consider a polynomial differential operator in one variable, depending on a small parameter (Planck constant). Under appropriate conditions, the low-energy spectrum admits an asymptotic expansion in hbar.
I will present a way to calculate such a series via a purely "commutative problem", a mixture of variations of Hodge structures and of the Stirling formula. This result came from discussions with A.Soibelman. It seems that we obtain an explanation of an old observation by J.Zinn-Justin of the 

 universal appearance of Bernoulli numbers.

In recent years, there has been a great deal of progress on ideas related to twisted supergravity, building on the definition given by Costello and Li. Much of what is explicitly known about these theories comes from the topological B-model, whose string field theory conjecturally produces the holomorphic twist of type IIB supergravity. Progress on eleven-dimensional supergravity has been hindered, in part, by the lack of such a worldsheet approach. I will discuss a rigorous computation of the twist of the free eleven-dimensional supergravity multiplet, as well as an interacting BV theory with this field content that passes a large  number of consistency checks. Surprisingly, the resulting holomorphic theory on flat space is closely related to the infinite-dimensional exceptional simple Lie superalgebra E(5,10). This is joint work with Surya Raghavendran and Brian Williams.

Zoom Link: https://pitp.zoom.us/j/99622967785?pwd=YmlQWW1sNW1qS1FhQkV4NXFlY0Nsdz09

I will connect approaches to classical integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations. In particular, I will consider holomorphic Chern-Simons theory on twistor space, defined using a range of meromorphic (3,0)-forms. On shell these are, in most cases, found to agree with actions for anti-self-dual Yang-Mills theory on space-time. Under symmetry reduction, these space-time actions yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces the holomorphic Chern-Simons action to 4d Chern-Simons theory.

Zoom Link: https://pitp.zoom.us/j/99193672959?pwd=RUJ3N3h2V3RFK3ZNVVVCK1E3bXJ2Zz09

Factorization homology is a local-to-global invariant which "integrates" disk algebras in symmetric monoidal higher categories over manifolds. In this talk I will discuss how to compute categorical factorization homology on oriented surfaces with principal D-bundles, for D a finite group, in terms of categories of modules over algebras defined in purely combinatorial terms. This is an extension of the work of Ben-Zvi, Brochier and Jordan to D-decorated surfaces. The main example for us comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups associated to a reductive group G. We will see that in this case factorization homology gives rise to a quantization of character varieties which are twisted by the group of outer automorphisms of G.

This talk is based on joint work with L. Müller.

Zoom Link: https://pitp.zoom.us/j/93950433494?pwd=WXI2VE9IdnRweEh5RmZsZ21BV1BQQT09

I will describe a concrete, computable example of holographic geometry emerging purely from the combinatorics of large matrices. Correlation functions of determinant operators in a matrix-valued free chiral algebra will be matched to holomorphic curves in SL(2,C).

Zoom Link: https://pitp.zoom.us/j/99341570607?pwd=TVVmSkF3d1haa0hBOWlPeDhGNi9aQT09

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a
genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Finally, we shall look into the appearance of branes within the moduli space of generalized hyperpolygons as well as of Higgs bundles, and consider mirror symmetry for such branes. Time permitting, we will mention some other recent results in various areas of science. 

Zoom Link: https://pitp.zoom.us/j/91592778202?pwd=WnM2VS9pS2c0QVIxVWFOdGhFMTdEdz09

The program of Ben-Zvi--Sakellaridis--Venkatesh connects the construction of L-functions in number theory with S-duality of boundary conditions in 4d. In particular this predicts certain equivalences of categories between equivariant D-modules on the formal loop space of a smooth variety X and equivariant quasi-coherent sheaves on a Hamiltonian manifold. I discuss an extension of this conjecture to certain singular varieties X and the possibility of quantizing the equivalence. I will explain joint work with Yiannis Sakellaridis on computing a certain factorization algebra which plays a role in the story. 

Zoom Link: https://pitp.zoom.us/j/95543248994?pwd=bmZIRnEyLzZnNmlEWW5oNTEwaEhNUT09

Human contact patterns are highly heterogeneous in terms of the both the number and nature of interactions. To incorporate these heterogeneities into infectious disease models one naturally represents a population as a weighted network. While there is a large literature on the spread of diseases on networks, most techniques are highly computational in nature. In this talk I will talk about an analytical framework for modeling infectious diseases as a percolation process on weighted networks based on probability generating functions. 

In the context of vaccination, human contact heterogeneities become a resource: Vaccinating individuals with greater total exposure leads to a greater reduction in disease spread. We have proposed that exposure notification apps, such as COVID Alert, can be leveraged to improve vaccine uptake among high exposure individuals, thereby optimizing our limited COVID-19 vaccine supply. We demonstrate the efficiency of this proposal using our weighted percolation theory framework.

We introduce the notions of (G,q)-opers and Miura (G,q)-opers, where G is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q)-opers of a certain kind and the set of nondegenerate solutions of a system of XXZ Bethe Ansatz equations. This can be viewed as a generalization of the so-called quantum/classical duality which I studied with D. Gaiotto several years ago. q-Opers generalize classical side, while on the quantum side we have more general XXZ Bethe Ansatz equations. The generalization goes beyond the scope of physics of N=2 supersymmetric gauge theories.

The subject of equivariant elliptic cohomology finds itself at the interface of topology, string theory, affine representation theory, singularity theory and integrable systems. These connections were already known to the founders of the discipline, Grojnowski, Segal, Hopkins, Devoto, but have come into sharper focus in recent years with a number of remarkable developments happening simultaneously: First, there are the geometric constructions, due to Kitchloo, Rezk and Spong, Berwick-Evans and Tripathy, building on the older program of Stolz-Teichner, and of course Segal. These are now all known to be equivalent to Grojnowski's original definition, thanks to work of Spong. Second, there are new applications, to integrable systems (Aganagic-Okounkov, Felder-Rimanyi-Varchenko-Tarasov) and to representation theory (Yang-Zhao-Zhong, G-Ram). Finally, there is the work of Weber-Rimanyi-Kumar, who build on the techniques of Borisov and Libgober to define Schubert classes in equivariant elliptic cohomology (with an added dynamical parametre), recovering the stable envelopes in type A. This talk will give a gentle introduction to the topic and attempt an overview over these recent developments.

I will present recent work (to appear soon) on the homological mirror symmetry about the universal centralizers $J_G$, for any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category of $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center).