This series consists of talks in the area of 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.
In this talk first I will introduce and motivate the problem of finding finite energy Yang-Mills instantons on curved backgrounds.
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 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 grou
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.
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.