Noncommutative Algebra Seminar

The standard Cartan calculus on polyvector fields and exterior forms can be naturally extended to the Hochschild cohomology and the Hochschild homology of an arbitrary associative algebra A. Recent results of M. Kontsevich and Y. Soibelman imply that this calculus on Hochschild (co)homology can be naturally upgraded to a homotopy calculus structure on the pair (C∙(A),C∙(A)) “Hochschild cochains + Hochschild chains” of an associative algebra A. In my talk I will consider the sheaf of homotopy calculi (C∙(OX),C∙(OX)) for a smooth algebraic variety X with OX being the structure sheaf. I will show that this sheaf (C∙(OX),C∙(OX)) of homotopy calculi is quasi-isomorphic to its cohomology. I will also talk about applications of this result. My talk will be based on joint paper arXiv:0807.5117 with D. Tamarkin and B. Tsygan. The main result of this paper was announced by D. Tamarkin in 2000 at the MoshŽ Flato memorial conference.

I'll discuss how to construct a "category O" attached to a symplectic resolution with an appropriate C^* action, using a quantization of the resolution. I'll show how in some cases, this can be related to equivariant perverse sheaves and the Fukaya category of the resolution. I'll give some examples of these, including the original category O for a semi-simple Lie-group, for a rational Cherednik algebra, a finite W-algebra, and the case of a hypertoric singularity, worked out by Braden, Licata, Proudfoot and myself. I'll discuss how in some of these cases, we've observed that the category in question is Koszul, and its Koszul dual is given by a "Higgs-Coulomb dual" resolution, which suggests connections to 3 dimensional conformal field theories in physics.

Given a smooth algebraic surface, the Hilbert scheme of points is a smooth variety parametrizing collections of points of the surface. The classical cohomology rings of these varieties have a rich structure developed by Nakajima, Grojnowski, and many others. This talk will be an introduction to studying the quantum cohomology of these spaces and the associated differential equation in the examples of C^2 (following Okounkov-Pandharipande) and ADE-resolutions (work with Oblomkov).

We will continue from last week, discussing the cases of C^2 and ADE, and giving the key ideas in the proofs. If time permits, we will discuss the relation to threefold geometries.

We continue from last week, discussing the ADE case in terms of affine algebra operators

I will review some important flat connections arising from representation theory - Knizhnik-Zamolodchikov, Dunkl-Cherednik, and Casimir connections, and discuss their monodromy and relations between them.

Periods of algebraic varieties, and more generally of mixed motives over Q, form an algebra, whose spectrum is conjecturally related to the motivic Galois group. However this relationship is difficult to see.

I will discuss a new way to present homotopy periods of an algebraic variety, which makes this relationship more transparent.

Namely, let X be a regular complex variety. We will introduce a Feynman integral related to X. Its correlators (defined via a perturbative series expansion) are complex numbers, which we call Hodge correlators. We show that they are homotopy periods, i.e. the real periods of the rational homotopy type of X; Moreover, they define a functorial real mixed Hodge structure of the latter.

Periods of algebraic varieties, and more generally of mixed motives over Q, form an algebra, whose spectrum is conjecturally related to the motivic Galois group. However this relationship is difficult to see.

I will discuss a new way to present homotopy periods of an algebraic variety, which makes this relationship more transparent.

Namely, let X be a regular complex variety. We will introduce a Feynman integral related to X. Its correlators (defined via a perturbative series expansion) are complex numbers, which we call Hodge correlators. We show that they are homotopy periods, i.e. the real periods of the rational homotopy type of X; Moreover, they define a functorial real mixed Hodge structure of the latter.

In the first part of the seminar, P. Etingof will give a short overview of Dunkl-Cherednik operators and flat connections arising from them.

In the second part (to start around 6.10pm), V. Toledano Laredo will describe the Casimir connection of a simple Lie algebra $\mathfrak{g}$ and explain how it is related to the Dunkl-Cherednik connection of the corresponding Weyl group and, when $\mathfrak{g}$ is $sl(n)$, to the Knizhnik-Zamolodchikov connection on $n$ points for a dual $sl(k)$. He will also explain how the monodromy of this connection is described by the quantum Weyl group operators of the quantum group $U_q(\mathfrak{g})$.

Part 1: "Introduction to BPS wall-crossing" - I will discuss the integer invariants that come from quantum field theories. These field theories have continuous moduli, often identified with the moduli of some geometric object, and one wants to know a "wall-crossing formula" describing how the integer invariants jump when one crosses special codimension-1 walls in the moduli space. I will describe two broad classes of example where this problem has been solved. One class was studied by Cecotti-Vafa in the early 1990's and mathematically has to do with singularity theory. The other class was studied by Kontsevich-Soibelman very recently and has to do with the geometry of Calabi-Yau threefolds (or categories).

Part 2: "Hyperkahler geometry and BPS wall-crossing" - Kontsevich-Soibelman's wall-crossing formula looks rather mysterious when first encountered. I will explain a way to understand this wall-crossing formula geometrically: it is a consistency condition for a new construction of hyperkahler spaces. This is work done in collaboration with Davide Gaiotto and Greg Moore.

Part 1: "Introduction to BPS wall-crossing" - I will discuss the integer invariants that come from quantum field theories. These field theories have continuous moduli, often identified with the moduli of some geometric object, and one wants to know a "wall-crossing formula" describing how the integer invariants jump when one crosses special codimension-1 walls in the moduli space. I will describe two broad classes of example where this problem has been solved. One class was studied by Cecotti-Vafa in the early 1990's and mathematically has to do with singularity theory. The other class was studied by Kontsevich-Soibelman very recently and has to do with the geometry of Calabi-Yau threefolds (or categories).

Part 2: "Hyperkahler geometry and BPS wall-crossing" - Kontsevich-Soibelman's wall-crossing formula looks rather mysterious when first encountered. I will explain a way to understand this wall-crossing formula geometrically: it is a consistency condition for a new construction of hyperkahler spaces. This is work done in collaboration with Davide Gaiotto and Greg Moore.

It is well-known that 2d topological sigma-models are related to geometrically interesting categories, such as the derived category of coherent sheaves and the Fukaya-Floer category. This relationship underlies the Homological Mirror Symmetry. In this talk I will discuss a 3d topological sigma-model constructed by Rozansky and Witten and its geometric interpretation. Going from 2d to 3d is roughly equivalent to categorifying the objects involved. I will show that the Rozansky-Witten model attaches a certain 2-category to every complex symplectic manifold. Its objects are complex Lagrangian submanifolds equipped with complex fibrations. This 2-category is a categorification of the derived category of coherent sheaves. It is also related to categories of matrix factorizations and deformation quantization of the derived category of coherent sheaves.