Mathematical Physics

This is based on my joint work with Yaping Yang. In this talk, we use the equivariant elliptic cohomology theory to study the elliptic quantum groups.  We define a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. This definition is naturally obtained using the cohomological Hall algebra associated to the equivariant elliptic cohomology. After taking suitable rational sections, the sheafified elliptic quantum group becomes a quantum algebra consisting of the elliptic Drinfeld currents. The Drinfeld currents satisfy the relations of the elliptic quantum group studied by Felder and Gautam-Toledano Laredo. We show the elliptic quantum group acts on the equivariant elliptic cohomology of Nakajima quiver varieties.

In particular, the sheafified elliptic quantum group is an algebra object in a certain monoidal category of sheaves on the colored Hilbert scheme of an elliptic curve. This monoidal structure is related to Mirkovic’s refinement of the factorization structure on semi-infinite affine Grassmannian over an elliptic curve. If time permits, I will also talk about a work in progress, joint with Mirkovic and Yang, towards a construction of a double loop Grassmannian and vertex representations of the toroidal algebra, which in turn is related to representations of the elliptic quantum groups

We explain how a doubled version of the Beilinson-Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David Ben-Zvi and David Nadler. 

In my talk, I will briefly review the representation theoretical construction of conformal blocks attached to an affine Kac-Moody algebra and a smooth algebraic curve with marked points. I will focus on the case when the algebraic curve is an elliptic curve. The bundle of conformal blocks carries a canonical flat connection: the Knizhnik-Zamolodchikov-Bernard (KZB) equation. There are various generalizations of the KZB equation. I will talk about one generalization that constructed by myself and Toledano Laredo recently: the elliptic Casimir connection. It is a holonomic system of differential equations with regular singularities on elliptic curve with marked points, taking values in a deformation of the double current algebra g[u, v] defined by Guay. The monodromy of elliptic Casimir connection leads to interesting representations of the elliptic braid groups. 

By the non-abelian Hodge theory of Carlos Simpson, harmonic bundles
interpolate between  bundles with connections on a curve and
 Higgs bundes (precise formulations requires some additional data like parabolic structure and stability structure).

I will explain the framework for a generalization of the non-abelian Hodge theory
which unifies Simpson's story ("rational case") with those  for q-difference
equations ("trigonometric case") and elliptic difference equations
("elliptic case").

This unification leads to a class of examples of the new notion of ``twistor families of categories".
In the rational,trigonometric and elliptic cases twistor families of categories  involve partially wrapped Fukaya categories of certain complex symplectic surfaces, categories of
holonomic modules over quantizations of these surfaces and categories of coherent sheaves on the surfaces
with certain restrictions on the support.

In the trigonometric and elliptic cases doubly and triply periodic monopoles give an alternative description of harmonic objects, hence playing the same role
as harmonic bundles play in the case of Simpson theory.

Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the topic.
Natural targets for extended topological field theories are higher Morita categories: generalizations of the bicategory of algebras, bimodules, and homomorphisms.

After giving an introduction to topological field theories, I will explain how one can use geometric arguments to obtain results on dualizablity in a ``factorization version’’ of the Morita category and using this, examples of low-dimensional field theories “relative” to their observables. An example will be given by polynomial differential operators, i.e. the Weyl algebra, in positive characteristic and its center. This is joint work with Owen Gwilliam.

A great deal of progress has been made toward a classification of bosonic topological orders whose microscopic constituents are bosons. Much less is known about the classification of their fermionic counterparts. In this talk I will describe a systematic way of producing fermionic topological orders using the technique of fermion condensation. Roughly, this can be understood as binding a physical fermion to an emergent fermion and condensing the pair. I will discuss the `super pivotal categories' that describe universal properties of these phases and use them to construct exactly solvable string-net models. These string-net models feature conventional anyons and two flavours of vortices. I will show that one of the vortex types is similar to a vortex in a p+ip superconductor binding a Majorana zero mode, and will mention some possible applications.

In this talk first I will introduce and motivate the problem of finding finite energy Yang-Mills instantons on curved backgrounds.  

After the historical introduction, I will focus the example of the Euclidean Schwarzschild (ES) manifold from Quantum gravity. The ES manifold was first introduced by Hawking in the late 70's. It is an asymptotically locally flat solution of the Euclidean Einstein equation, but not hyper-Kahler, which makes instanton theory harder. Throughout the past four decades, related results have been sporadic: there have been only finitely many (at most 2) known solutions for each energy value.

In our work, making use of an old idea of Witten, and subsequently Taubes and Garcia-Prada, we produce infinitely many unit energy solutions, and also solutions of continuously varying energy, a feature (seemingly) unique to asymptotically locally flat spaces. These instantons describe vortex-like pseudoparticles; at the end of the talk I will present an intuitive picture in terms of planar (Ginzburg-Landau) vortices.

This is joint work with Gonçalo Oliveira (IMPA).

The construction of trial wave functions has proven itself to be very useful for understanding strongly interacting quantum many-body systems. Two famous examples of such trial wave functions are the resonating valence bond state proposed by Anderson and the Laughlin wave function, which have provided an (intuitive) understanding of respectively spin liquids and fractional Quantum Hall states. Tensor network states are another, more recent, class of such trial wave functions which are based on entanglement properties of local, gapped systems. In this talk I will discuss the use of tensor network states for topological phases, and what we can learn from this approach. I will consider one- and two-dimensional systems, consisting of both spins and fermions. The focus will be on the different connections that can be made using tensor networks, such as connecting theory to numerics, and physical properties to ground state entanglement.

The Hecke category is a certain monoidal category of constructible sheaves on a flag variety that categorifies the Hecke algebra and plays an important role in geometric representation theory. In this talk, I will discuss a monoidal Koszul duality relating the Hecke category of Langlands-dual (Kac–Moody) flag varieties, categorifying a certain involution of the Hecke algebra. In particular, I will try to explain why one needs to introduce a monoidal category of "free-monodromic tilting sheaves" to formulate this duality. (Joint with P.N. Achar, S. Riche, and G. Williamson.)

Talk is based on the joint work with Lev Rozansky. In my talk will outline a construction that provides  complex $C_b$ of coherent sheaves on the Hilbert scheme of $n$ points on the plane for every $n$-stranded braid $b$. The space of global sections of $C_b$ is a categorification of the HOMFLYPT polynomial of the closure $L(b)$ of the braid. I will also present a physical interpretation of our construction as a particular case of Kapustin-Saulina-Rozansky 3D topological field theory.

Making perturbative quantum field theory (QFT) mathematically rigorous is an important step towards understanding how the non-perturbative framework should look like. Recently, two approaches have been developed to address this issue: perturbative algebraic quantum field theory (pAQFT) and the factorization algebras approach developed by Costello and Gwilliam. The former works primarily in Lorentzian signature, while the later in Euclidean, but there are many formal parallels between them. In this talk I will show the recent results relating these two frameworks and discuss the consequences for future research (joint work with Gwilliam).