MIT Infinite Dimensional Algebra Seminar (Spring 2012)

Classical descent is about gluing a global geometric object out of local information. Or conversely, it is about classifying global geometric objects using open coverings and cocycles.

I will begin the lecture with a rather thorough discussion of how descent theory let's us classify twisted forms of a sheaf on a topological space (this is 1st nonabelian cohomology).

Next I will recast this geometric construction in terms of cosimplicial groups, in this way getting something that is of purely combinatorial nature.

Higher descent refers to the classification of twisted forms of a stack on a topological space (a sort of 2nd nonabelian cohomology). But in this talk I will adhere to the combinatorial point of view, so stacks will only appear as motivations, and in one or two examples. Thus, in the talk, "higher descent" will be mostly a study of cosimplicial crossed groupoids and their descent data. (These concepts will be defined.)

I will present a recent result, the ``Equivalence Theorem'', and mention its role in my work on twisted deformation quantization. Finally I will briefly discuss how model structures enter the picture, and a very new proof of this theorem by Prezma.

There will be a few pictures.

- Lecture notes are at http://www.math.bgu.ac.il/~amyekut/lectures/higher-descent/

notes.pdf

- The paper is at http://arxiv.org/abs/1109.1919

This is a report on the joint project with M. Mueger, D. Nikshych and V. Ostrik. It turns out that Drinfeld centers of fusion categories can be characterized as non-degenerate braided fusion categories containing a Lagrangian algebra. One can look at the quotient of the monoid of non-degenerate braided fusion categories modulo the submonoid of the Drinfeld centers. Formal properties of this quotient are similar to those of the classical Witt group. Although the precise structure of this group (explicit generators and relations) is out of reach at the moment, certain questions about the torsion part can be answered. Namely the torsion is a 2-group of infinite rank with 32 being the maximal order.

We will show how to use the recent result on the contractibility of the space of rational maps to the group G to prove that higher conformal blocks for WZW (a.k.a. chiral homology of the integrable quotient of the Kac-Moody vertex algebra) is given by (the dual of) the cohomology of the corresponding line bundle on the moduli stack Bun(G).

The theory of associators was developed by Drinfeld in relation with certain problems of quantum groups theory. One of its main ingredients is a connection on the configuration spaces of points in the complex plane, called the universal KZ connection. It gives rise to explicit relations between multiple zeta values, and explains the "torsor" structure of these relations ; it also gives rise to a Lie algebra, which is the target of a morphism from a motivic Lie algebra. An elliptic analogue of the KZ connection, living over the moduli space of elliptic curves with marked points, was later constructed (D. Calaque, P. Etingof and the speaker). It allows for an analytic proof of the computation of the prounipotent completion of elliptic pure braid groups (Bezrukavnikov). A theory parallel to the theory of associators can then be developed. The analogues of multiple zeta values are certain functions of the elliptic parameter, which are shown to satisfy certain relations. The theory gives rise to a Lie algebra, which seems to be related to motivic theory, and is shown to possess a semidirect product structure. A new feature with respect to genus zero it the appearance of iterated integrals of Eisenstein series: they are shown to satisfy some algebraic relations with multiple zeta values. Another byproduct is the computation of the Zariski closure of the genus one mapping class group in the automorphism groups of the prounipotent completions of pure braid groups in genus one.

(Partially based on joint work with D. Calaque and P. Etingof.)

Given a complex, reductive algebraic group G and a G-module V, the m-th jet scheme G_m acts on the m-th jet scheme V_m, for all m>0.

In the first hour, we will discuss the ring of G_m invariant functions on V_m, and its relationship to the ring of functions on (V//G)_m, where V//G is the categorical quotient. These invariant rings were previously studied in some special cases by D. Eck, E. Frenkel, and D. Eisenbud.

In the second hour, we will discuss an application of these results to vertex algebras. Given a vertex algebra V and a subalgebra A, a basic problem is to describe the commutant Com(A,V) by giving generators, operator product expansions, and normally ordered polynomial relations among the generators. We solve this problem for a family of examples where V is a "free field" algebra and A is an affine vertex algebra. This is a joint work with Bailin Song (University of Science and Technology of China).

I will explain the construction of a categorification of the affine Lie algebra of gl_n on the category O of cyclotomic rational Cherednik algebras and explain how to use it to prove some conjectures of P. Etingof on the filtration of the category O by the support of modules (joint work with E. Vasserot). I will also discuss some relations between the category O of cyclotomic rational Cherednik algebras and parabolic category O of affine Lie algebras at negative level.

Irreducible finite-dimensional representations of rational Cherednik algebras of type A are classified by a single rational number m/n. It turns out that such a representation is tightly related to the different invariants of the plane curve singularity x^m=y^n. I will describe some of these relations and, in particular, explain the surprising "rank-level duality" exchanging m and n.

In the mid 1990s Zhu proved that characters of certain modules over certain rational vertex algebras are (vector-valued) modular forms on $SL(2, Z)$. In this talk I describe some generalizations of Zhu's theorem. First I remove an assumption that certain numbers, called the conformal weights of the vertex algebra, are integers, I also pass to the superalgebra case. Modular invariance is then recovered by enlarging the scope to include twisted modules, and by including certain 'queer' traces alongside the usual characters. There are several natural classes of examples, affine Kac-Moody algebras at admissible level being one. Next I remove the assumption of rationality of the vertex algebra and prove (under some technical conditions) that certain 'logarithmic characters' are invariant under $SL(2, Z)$. These generalized modular forms are not expressible as q-series, but rather as series in $q$ and $log(q)$.

The lattice vertex operator algebra plays a central role in the quantization of the sigma model with target a flat torus. When this torus is twisted by a non-trivial gerbe or, in its T-dual side, if one studies non-trivial torus fibrations over tori, dilogarithmic singularities appear in the operator product expansions of fields. We will show how the classical identities of the dilogarithm function appear as factorizations of the corresponding correlation functions.

I will review some of the Kac-Peterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, K-theory, elliptic cohomology and cobordism.

This is work in progress with Nora Ganter.

We'll discuss combinatorial structures that appear in the study of totally positivity on the Grassmannian. The totally nonnegative part of the Grassmannian is a CW complex, whose cells (called positroid cells) can be parametrized in terms of certain planar graphs with vertices colored in two colors. Combinatorics of these graphs is an extension of combinatorics of reduced decompositions in the symmetric group; and positroid cells generalize (type A) double Bruhat cells. This extension reveals new symmetries, which were hidden on the level of reduced decompositions and double Bruhat cells.

Interestingly, the same combinatorial structures have recently appeared in several different (and unrelated to each other) areas of mathematics and physics, for example, in super-Yang-Mills theory, KP-equations and solitons, statistical models, Teichmuller theory, and many other areas. This might be an evidence of new links between these areas.

We'll talk about the Grassmannian formula for (N=4 SYM) scattering amplitudes by Nima Arkani-Hamed et al., where our graphs play a role similar to Feynman diagrams.

Simple singularities are classified by Dynkin diagrams of type $ADE$.

Let $\mathfrak{g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak{g}$-invariants in the basic representation of the affine Kac--Moody algebra $\hat{\mathfrak{g}}$ is known as a $\mathcal{W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal{F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal{F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal{F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the $\mathcal{W}$-algebra. (Joint work with T. Milanov.)