MIT Infinite Dimensional Algebra Seminar (Spring 2011)

Nonabelian multiplicative integration on curves is a classical theory, going back to Volterra in the 19-th century. In differential geometry this operation can be interpreted as the holonomy of a connection along a curve.

This talk is about the 2-dimensional case. A rudimentary nonabelian multiplicative surface integration was known since the 1920's (work of Schlesinger). I will present a much more sophisticated construction. My main result is a 3-dimensional nonabelian Stokes Theorem. This result is completely new; only a special case of it was predicted (without proof) in papers in mathematical physics.

The talk is fairly elementary, requiring only some knowledge of Lie groups and their Lie algebras. And there are many color pictures!

My motivation for this work has to do with a problem in twisted deformation quantization. I will say a few words about this at the end of the talk.

For full details see the lecture notes or the preprint on arXiv.

The nekrasov partition function is rigorous definition of a path integral in supersymmetric gauge theory, using equivariant integration on the moduli space of sheaves on P^2. Interestingly, the key objects for studying it are operators that appear in conformal field theory and representation theory. I'll explain about these connections, and how to obtain an exact formula in a special case, using an elementary formula for turning operator traces into a pfaffian of a "global" quadratic form.

We choose the termination for the sequence of the Laplace invariants for the linearization operator as a definition of Liouville (=Darboux) integrability. We generalize this approach to the multi-component case and prove that the open Toda lattices related to simple Lie algebras are Liouville integrable.

A vertex algebra A is called strongly finitely generated (SFG) if there exists a finite set of generators such that any element of A can be expressed as a normally ordered polynomial in the generators and their derivatives. Given an SFG vertex algebra A and a reductive group G of automorphisms of A, we are interested in the question of when the invariant subalgebra A^G is SFG. We prove that this holds for arbitrary G when A is the \beta\gamma system of rank n>=1, and when A is the universal affine vertex algebra V_k(sl_2) associated to sl_2, for generic level k. Our proof is essentially constructive, and combines tools from classical invariant theory with the representation theory of certain W-algebras.

We present relations between the operator algebraic approach to chiral superconformal field theory and the Connes noncommutative geometry. Based on an analogy between the conformal Hamiltonian of a chiral conformal field theory and the Laplacian of a closed Riemannian manifold, we give an analogue of the Dirac operator from the Ramond relations of the super Virasoro algebra. Various representation theoretic properties are also studied in this context.

I am going to talk about triangulated categories in algebra, geometry and physics and about differential-graded (DG) enhancements of triangulated categories. I will discuss such properties of DG enhancements as uniqueness and existing.

It can be proved that a uniqueness of DG enhancements exists for a large class of triangulated categories. This class includes all derived categories of quasi-coherent sheaves, bounded derived categories of coherent sheaves and category of perfect complexes on quasi-projective schemes, as well as on a noncommutative varieties.

This result shows that triangulated categories which have a geometric nature largely distinguished among all of triangulated categories; for which this property does not hold in general.

One consequence of these results is a theorem asserting that an existence of a fully faithful functor between such categories implies an existence of a fully faithful functor between them that has integral form, i.e. that is represented by an object on the product.

Moreover, for projective varieties there is a strong uniqueness for DG enhancements, which implies that any fully faithful functor from the category of perfect complexes on the projective variety to another category of this type has integral type, i.e. it is represented by an object on a product.

These results have also application to deformation theory of objects in derived categories and to homological mirror symmetry.

This is a joint paper with Valera Lunts.

We work over a field F of characteristic p. It is well known that the direct sum of the categories Rep(S_k) categorifies the basic representation of the Kac-Moody algebra \hat{sl}_p. Moreover, the symmetric groups and general linear groups are related via Schur-Weyl duality. Therefore it is natural to ask for an analogue of the direct sum of categories Rep(S_k) for the general linear groups. We will construct such a category as a limit category of polynomial representations of GL_n, and show that it categorifies the Fock space representation of \hat{sl}_p (in the sense of Rouquier). We then recover the crystal basis of Fock space from the simple objects in our category, and formulate Schur-Weyl duality as a categorification of the standard projection from Fock space to the basic representation. This is based on joint work with Jiuzu Hong.

Conformal Field Theory can be successfully studied by considering a family of von Neumann algebras attached to intervals of the circle or on the real line a "Local Conformal Net".

An algebraic approach to Boundary Conformal Field Theory has then been developed (K.-H. Rehren and R.L.). It turns out that Boundary Conformal Field Theory on the half-plane is described a (non necessarily local) conformal net on the real line.

In a recent work, starting with a local conformal net A on the real line, one considers a unitary semigroup E(A) associated with A. Each element V of this semigroup gives a new Boundary Quantum Field Theory on the half-plane (E. Witten and R.L.). The computation of second quantization elements of E(A) is obtained by an analog of the Beurling-Lax theorem theorem convening shift invariant subspaces of H^{\infty} of the disk. The corresponding subgroup of E(A) is isomorphic to the semigroup of symmetric, holomorphic inner functions on the disk.

A very recent description of Boundary QFT on the interior of the Lorentz hyperboloid is explained (K.-H. Rehren and R.L.). A different, but surprisingly isomorphic, semigroup plays a role here. The natural states are thermal states at Hawking temperature rather than ground states.

I will give several examples of categories which should be viewed as "F_1-linear", in that they are F_1 analogues of abelian categories. These include categories of Feynman graphs, incidence categories (built from partially ordered sets), quiver representations over F_1, and categories of coherent sheaves on F_1-schemes. In all these examples, one can make sense of exact sequences and define their Hall algebras, which are graded co-commutative Hopf algebras. In some cases, one can define their algebraic K-theory. For quiver representations and coherent sheaves on P^1, we obtain degenerate versions of theorems of Ringel-Green and Kapranov, relating the Hall algebras to certain quantum groups.

In the first talk I will explain my joint results with Ostrik on the classification of finite dimensional irreducible modules with integral central characters for W-algebras. We use different ingredients: the representation theory of W-algebras themselves, Harish-Chandra bimodules, Lusztig's theory of cells in Weyl groups, multi-fusion categories and their module categories, Springer representations -- I am going to discuss at least some of those.

In the second talk I will explain an application of our results to the classical problem of computing Goldie ranks for primitive ideals. The study of Goldie ranks was initiated by Joseph some 30 years ago.

All necessary information about W-algebras and Goldie ranks will be explained. Warning: speaker's opinion on what is necessary may be different from yours.