Mathematical Physics

One can associate to any finite graph Q the skew-symmetic Kac-Moody Lie algebra g_Q. While this algebra is always infinite, unless Q is a Dynkin diagram of type ADE, g_Q shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to Q gives a geometric representations of g_Q. Encouraged by this story, one could hope to define the Yangian of g_Q, for general Q, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties. In fact there are two approaches to doing this: firstly via the stable envelope construction of Maulik and Okounkov, secondly via the preprojective Hall algebra of Schiffmann and Vasserot. Via another Hall algebra construction, due to Kontsevich and Soibelman, and work of myself and Meinhardt on BPS Lie algebras, these approaches turn out to be more or less the same. In showing this we show that the correct Lie algebra of endomorphisms associated to Q is cohomologically graded, with zeroeth piece the old Kac-Moody Lie algebra - the best way to build a (Borcherds) Lie algebra out of Q is directly from geometric representation theory.

Work of Bezrukavnikov on local geometric Langlands correspondence and works of Gorsky, Neguţ, Rasmussen and Oblomkov, Rozansky on knot homology and matrix factorizations suggest that there should be a categorical version of a certain natural homomorphism from the affine Hecke algebra to the finite Hecke algebra in type A, sending basis lattice elements on the affine side to Jucys-Murphy elements on the finite side. I will try to explain some of the structures involved and will talk about recent progress towards a construction of such a categorification in the setting of Hecke categories.

I will sketch why self-dual versions of the moduli of G-Higgs bundles are expected to arise physically from the study of 4d theories of class S. I will then describe an extension of the Langlands duality results of Hausel-Thaddeus (G=SL(n)) and Donagi-Pantev (arbitrary reductive G) that yields self-dual moduli spaces as a corollary.

The Alday-Gaiotto-Tachikawa correspondence connects gauge theory on a fourfold with conformal field theory. We are interested in a certain algebro-geometric incarnation of this framework, where the fourfold is an algebraic surface and instantons/differential geometry are replaced with sheaves/algebraic geometry. In this talk, we will present a certain approach to AGT that yields partial results for quite general surfaces, and ask questions about what still needs to be done to state and prove the full correspondence in the language of algebraic geometry.

The localization theorem, which has played a central role in representation theory since its discovery in the 1980s, identifies a regular block of Category O for a semisimple Lie algebra with certain D-modules on its flag variety. In this talk we will explain work in progress which produces a similar picture for the Virasoro algebra and more generally for affine W-algebras. Some new purely algebraic input is (i) a version of the Feigin-Fuchs duality between Verma modules for Vir at central charges c and 26 - c, which applies to all smooth representations and other affine W-algebras, and (ii) a linkage principle for representations in category O of a W-algebra. As geometric input, we will explain how to (i) adapt the Beilinson-Drinfeld construction of vertex algebras via factorization spaces to also produce representations and in particular (ii) develop a factorizable version of affine Borel--Weil--Bott.  

Quantized lattices, or q-lattices, appear naturally through categorification constructions (for example from zigzag-algebras), but they haven't been studied from a lattice theory point of view. After establishing the necessary background, we'll explain the q-versions of various lattice theory concepts illustrated with examples from combinatorics and graph theory, and we'll list a number of open problems.

In this talk we will discuss three (intimately related) examples of 4-manifold invariants based on higher structures:
   - VOA[M4] from transgression of EFTs;
   - SW and Donaldson invariants as chiral algebra correlators;
   - Massey triple products from trisections.
These topics are based, respectively, on work with A.Gadde, P.Putrov (and work to appear with B.Feigin); recent paper with M.Dedushenko and P.Putrov; and a solo paper of the speaker.

Skew Howe duality is based on a very simple observation: the set of n by m matrices has commuting action of SL_n and SL_m.  We can use this observation to study morphisms of GL_m representations using GL_n.  This perspective has proven very useful in recent years for studying quantum knot invariants and their categorifications.  I will survey work in this direction from the last 10 years, including more recent developments concerning annular skew Howe duality and annular knot invariants.

In this talk, intended for a broader audience, I promise to use techniques only at the level of university calculus. While staying at this level, our goal will be to learn conceptual lessons for categorification of quantum group invariants of knots and 3-manifolds, also known as the Witten-Reshetikhin-Turaev (or WRT) invariants. In particular, we will introduce new q-series invariants of 3-manifolds that have integer powers and integer coefficients and, if time permits, discuss their various constructions and properties. The talk is based on several papers with D. Pei and/or P. Putrov, C. Vafa, as well as ongoing work with C. Manolescu.

The cobordism hypothesis gives a functorial bijection between oriented 

n-dimensional fully local topological field theories, valued in some 

higher category C, and the fully dualizable object of C equipped with 

the structure of SO(n)-fixed point.  In this talk I'll explain recent 

works of Haugseng, Johnson-Freyd and Scheimbauer which construct a 

Morita 4-category of braided tensor categories, and I'll report on joint 

work with Brochier and Snyder which identifies two natural subcategories 

therein which are 3- and 4-dualizable.  These are the rigid braided 

tensor categories with enough compact projectives, and the braided 

fusion categories, respectively.  I'll also explain work in progress by 

us to construct SO(3)- and SO(4)-fixed point structures in each case, 

starting from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain 3- and 4-dimensional fully 

local TFT's, which extend the 2-dimensional TFT's we constructed with 

Ben-Zvi and Brochier, and which conjecturally relate to a number of 

constructions in the literature, including: skein modules, quantum 

A-polynomials, Crane-Kauffmann-Yetter invariants; hence our construction 

puts these on firm foundational grounds as fully local TFT's.  A key 

feature of our construction in dimension 3 is that we require the input 

braided tensor category neither to be finite, nor semi-simple, so this 

opens up new examples -- such as non-modularized quantum groups at roots 

of unity -- which were not obtainable by earlier methods.