Mathematical Physics

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.

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 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).

In this talk I will define the quantum K-theory of Nakajima quiver varieties and show its connection to representation theory of quantum groups and quantum integrable systems on the examples of the Grassmannian and the flag variety. In particular, the Baxter operator will be identified with operators of quantum multiplication by quantum tautological classes via Bethe equations. Quantum tautological classes will also be constructed and, time permitting, an explicit universal combinatorial formula for them will be shown.
Based on joint works with P.Koroteev, A.Smirnov and A.Zeitlin

It is easy to prove that d-dimensional complex Hilbert space can contain at most d^2 equiangular lines.  But despite considerable evidence and effort, sets of this size have only been proved to exist for finitely many d.  Such sets are relevant in quantum information theory, where they define optimal quantum measurements known as SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures).  They also correspond to complex projective 2-designs of the minimum possible cardinality.   Numerical evidence points to their existence for all d as orbits of finite Heisenberg groups, the current record being d=844 [Scott-Grassl '17].  However, to date, they are only proven to exist for finitely many d (the current record being d=323 [SG17]) via computer-assisted calculations in number fields of degree increasing with d.  In this talk, I will discuss the structure of these number fields, which turn out to be specific abelian extensions of specific real quadratic number fields [Appleby, Flammia, McConnell, Y. 1604.06098].  Such fields are known to exist by general theorems of class field theory, but until now, had never been found 'explicitly' in Nature.  This contrasts the classical situation for abelian extensions of CM fields, which are generated by the torsion points of abelian varieties with complex multiplication.   All known Heisenberg-covariant SIC-POVMs have unitary symmetries under the associated Weil representation that are intimately related to the structure of the underlying number fields.  A proper understanding of this relationship may ultimately lead to a general proof of their existence in all dimensions, rather than the finite number of examples currently proved to exist. 

I will talk about a recent proof, joint with M. Gröchenig and D. Wyss, of a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

Geometric invariant theory (GIT) is an essential tool for constructing moduli spaces in algebraic geometry. Its advantage, that the construction is very concrete and direct, is also in some sense a draw-back, because semistability in the sense of GIT is often more complicated to describe than related intrinsic notions of semistability in moduli problems. Recently a theory has emerged which treats the results and structures of geometric invariant theory in a broader context. The theory of Theta-stability applies directly to moduli problems without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme. I will discuss some new progress in this program: joint with Jarod Alper and Jochen Heinloth, we give a simple necessary and sufficient criterion for an algebraic stack to have a good moduli space. This leads to the construction of good moduli spaces in many new examples, such as the moduli of Bridgeland semistable objects in derived categories. Time permitting, I will also discuss applications to enumerative geometry and wall crossing formulas.

Moore and Tachikawa conjecture that there exists a functor from the category of 2-bordisms to a certain category whose objects are algebraic groups and morphisms between $G$ and $H$ are given by affine symplectic varieties with an action of $G\times H$.  I will explain a proof of this conjecture due to Ginsburg and Kazhdan, and its relation to Coulomb branches of certain quiver gauge theories which allows to make interesting calculations.

I'll do my best to explain my approach to the BFN construction of (quantum) Coulomb branches.  This approach is based on viewing the BFN algebra as an endomorphism algebra in a larger category that's easier to present (and which we can draw some pretty pictures for).  In particular, this approach is helpful in understanding the representation theory of this algebra, and in constructing and analyzing tilting generators on Coulomb branches.