MIT Infinite Dimensional Algebra Seminar (Spring 2009)

Murnaghan-Kirillov theory seeks to associate to an irreducible representation π of a p-adic group G an Ad-invariant distribution δ_π supported on a finite union of coadjoint orbits such that the restriction of the character χ_π to a sufficiently small neighborhood U of e is equal to the F(δ_π)◦ln where F is the Fourier transform.

I'll discuss a construction of distributions δ_π for irreducible representation π of depth zero.

It is well-known that many objects that show up in conformal field theory have characters that exhibit good behavior with respect to an action of SL(2,Z), i.e., there is an action on some finite dimensional space of q-expansions. Examples include representations of affine Lie algebras, and more generally, rational vertex algebras and their modules. The characters arising from the monster vertex algebra obey an additional condition called genus zero, and its physical significance is still mysterious. However, this condition is a consequence of certain recursion relations that arise from equivariant Hecke operators, which made their first appearance in the context of power operations in elliptic cohomology.

I will summarize the main results of the theory of character sheaves on unipotent groups in positive characteristic that have been obtained so far.

First let k be an algebraically closed field of characteristic p>0, and let G be a unipotent group over k. I will define the notions of a character sheaf and of an L-packet of character sheaves on G, and I will describe the main properties of L-packets.

Next assume that k is an algebraic closure of a finite field F, and that G as above is defined over F. Let Fr denote the Frobenius endomorphism of G corresponding to the F-structure. One can ask about the relationship between irreducible characters of the finite group G(F) and the Fr-invariant character sheaves on G.

If G is connected, the trace-of-Frobenius functions corresponding to the Fr-invariant character sheaves on G form a basis of the space of conjugation-invariant functions on G(F), and (after suitable normalization) satisfy the natural orthogonality relations.

If G is disconnected, there are more Fr-invariant character sheaves than irreducible characters of G(F). In fact, an analogue of the statement formulated in the previous paragraph does hold in the disconnected case, but one has to take into account the groups of F-points of all inner forms of G over F. I will explain how this works in more detail during my talk.

If time permits, I will say a few words about the strategy of the proofs of the results described above.

The talk is based on joint work with Vladimir Drinfeld.

We present an operator algebraic approach to superconformal field theory and compare it with an approach based on vertex algebras. Ours is called algebraic quantum field theory and gives an unified treatment for various spacetimes with various symmetry groups. We emphasize representation theoretic aspects, and give classification results for small central charges, various Moonshine type results, and connections between the super Virasoro algebra and the Connes noncommutative geometry.

Let G be a compact Lie group. Then the complexification of G can be realized as the set of points of an algebraic group defined over the complex numbers. Better still, this algebraic group can be defined over the ring Z of integers. In the setting of derived algebraic geometry, one can ask whether this group is defined "over the sphere spectrum". In these talks, I'll explain the meaning of this question and describe how it can be addressed using recent joint work with Dennis Gaitsgory.

Let G be a compact Lie group. Then the complexification of G can be realized as the set of points of an algebraic group defined over the complex numbers. Better still, this algebraic group can be defined over the ring Z of integers. In the setting of derived algebraic geometry, one can ask whether this group is defined "over the sphere spectrum". In these talks, I'll explain the meaning of this question and describe how it can be addressed using recent joint work with Dennis Gaitsgory.

In the firt part of this talk, I will recall the classical definition of conformal nets. I will introduce a "coordinate free" description of that notion, and I will mention the subtleties that arise when one tries to introduce Z/2-graded versions of the above objects. This will all be made concrete by an important example: the free fermion conformal net. Then, I will introduce the one-morphisms (defects), the two-morphisms (sectors) and the three-morphisms of the 3-category CN3 of conformal nets. I will finish by giving a charcterisation of invertible nets: a net A is invertible in CN3 iff its mu-index mu(A) is equal to one. The latter is then the case iff the representation category of A is trivial.

In the second part of the talk, I will explain the notion of symmetric monoidal 3-category in which CN3 best fits. This consists of a rather long list of data, and of an even longer list of axioms. I will then explain how one can hope to verify that the above list of data and axioms is complete.

This is the first report of an ongoing project aimed at finding a geometric interpretation of the Witten genus in terms of vertex algebras and related algebraic structures. In their work on chiral differential operators, Gorbounov, Malikov and Schectman (GMS) have constructed sheaves of conformal vertex algebras on certain complex manifolds whose sheaf cohomology computes the Witten genus of these manifolds. In this talk, I will review the definition of the Witten genus, the GMS construction, and then describe some fine resolutions of the GMS sheaves, in the form of sheaves of dg conformal vertex algberas. The construction of these resolutions makes use of vertex algebroids and dg vertex algebroids. The resolutions should play the same role as the Dolbeault resolution does for the Todd genus, and may have a generalization that provides a geometric interpretation of the Witten genus in general.

The talk will discuss the general integrable structures behind Random Matrix Theory and also the role played by Virasoro algebras and how all this leads to PDE's and ODE's for the probabiliity distributions of interest in the theory, such as Painleve equations and their pde generalizations. Examples will be used to illustrate the theory.

Chiral equivariant cohomology is one that takes value in a vertex superalgebra, and contains and generalizes the classical equivariant cohomology of H. Cartan. This talk will include a brief overview of the chiral theory, and computations of this cohomology for some examples of G-spaces. Examples whose classical equivariant cohomology algebras are the same, but have different chiral equivariant cohomology algebras will also be discussed. The talk is based on joint work with B. Song and A. Linshaw.

We will furnish a vertex algebraic realization for the sporadic simple group of Held. This construction has many features. It is rather simple, and at the same time elucidates fascinating connections between the Held group, discrete groups of isometries of the hyperbolic plane, and affine Dynkin diagrams.

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Using Picard-Lefschetz periods, we construct a twisted representation of the lattice vertex algebra V associated to the root lattice of the corresponding finite-dimensional Lie algebra g. By the Frenkel-Kac construction, V is isomorphic to the basic representation of the corresponding affine Kac-Moody algebra, and in particular it admits an action of g by derivations. The kernel of this g-action is a subalgebra of V known as a W algebra. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W algebra. This is joint work with T. Milanov.