MIT Infinite Dimensional Algebra Seminar (Spring 2017)

Abstract: Let A be a commutative ring, and let \a := \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and the derived \a-adic completion functors to be nicely behaved is the weak proregularity of the ideal \a. In particular, the MGM Equivalence holds under this condition. Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. Consider a torsion class T in the category M(A) of left modules over a ring A. We introduce a new condition on T: weak stability. Our first main theorem is that when A is commutative, an ideal \a in A is weakly proregular if and only if the corresponding torsion class T in M(A) is weakly stable. It turns out that when the ring A is noncommutative, one must impose two more conditions on the torsion class T: quasi-compactness and finite dimensionality (these are new names for old conditions). We prove that for a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived T-torsion functor is isomorphic to a left derived tensor functor. This result involves derived categories of bimodules. Some examples will be given. The third main theorem is the Noncommutative MGM Equivalence, under the same assumptions on T. Finally, there is a theorem about derived left-sided and right-sided torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and it corrects an error in a paper of Yekutieli-Zhang from 2003. We expect that the approach outlined in this talk will open up the way to a useful theory of rigid dualizing complexes in the arithmetic noncommutative setting (namely without a base field). The work above is joint with Rishi Vyas.

We prove the conjecture on the GKO construction of the minimal series (principal) W-algebras in ADE types, which goes back to the work of Kac and Wakimoto in 1990. This is a joint work with Thomas Creutzig and Andrew Linshaw.

We discuss some interesting relations among 4d Argyles-Douglas (AD) theories, vertex operator algebras (VOA) and wild Hitchin system. We use the Coulomb branch index of AD theories to study geometric and topological data of moduli spaces of wild Hitchin system. These data show an one to one map between fixed points on the moduli space and irreducible modules of the VOA. Moreover, a limit of the Coulomb branch index of AD theories can be identified with matrix elements of the modular transform ST^kS in certain two-dimensional VOAs. The appearance of VOAs, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising. This is based on our recent paper

We define pentagram maps on polygons in any dimension, which extend R.Schwartz's definition of the 2D pentagram map. Many of those maps turn out to be discrete integrable dynamical systems, while the corresponding continuous limits of such maps coincide with equations of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We discuss their geometry, Lax forms, and interrelations between recent pentagram generalizations. This is a joint work with Fedor Soloviev.

We describe a method for constructing the generators, and their commutation relations, for the finite W-algebras of type A. We also see how the analogue result in the classical affine case can be used to construct integrable Hamiltonian hierarchies of Lax type

Geometric Quantization is a program of assigning to Classical mechanical systems (Symplectic manifolds and the associated Poisson algebras of $C^\infty$ functions) their quantizations --- algebras of operators on Hilbert spaces. Geometric Quantization has had many applications in Mathematics and Physics. Nevertheless the main proposition at the heart of the theory, invariance of polarization, though verified in many examples, is still not proved in any generality. This causes numerous conceptual difficulties: For example, it makes it very difficult to understand the functoriality of theory. Nevertheless, during the past 20 years, powerful topological and geometric techniques have clarified at least some of the features of the program. In 1995 Kontsevich showed that formal deformation quantization can be extended to Poisson manifolds. This naturally raises the question as to what one can say about Geometric Quantization in this context. In recent work with Victor Guillemin and Eva Miranda, we explored this question in the context of Poisson manifolds which are "not too far" from being symplectic---the so called b-symplectic or b-Poisson manifolds---in the presence of an Abelian symmetry group. In this talk we review Geometric Quantization in various contexts, and discuss these developments, which end with a surprise.

The category of representations over a quantum group $U_q(g)$ form a braided tensor category that produces an action of the (pure) braid groups on tensor products. Respectively, the category of crystals (which is a limit for q tends to zero) form a coboundary category together with an action of (pure) cacti group on tensor products. The little discs operad is an operad whose space of $n$-ary operations is the Eilenberg-MacLane space of the pure braid groups with $n$ braids. Correspondingly, the real locus of the moduli space of stable rational curves with marked points assemble an operad of the Eilenberg-MacLane spaces of pure cacti groups. I will present the detailed description of the latter operad as well as its deformation theory and relationships with the little discs operad. Among different applications I will prove rational K(\pi,1) property of the latter moduli spaces as well as other interesting properties of pure cacti groups that were conjectured by P.Etingof, A.Henriques,J.Kamnitzer, E.Rains in the seminal paper

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. The cohomology class Poincar´e dual to the locus of any given singularity admits a universal expression in terms of certain basic characteristic classes. Our expressions hold for any family of curve-to-curve maps satisfying the above conditions. They extend universal expressions for isolated singularities, which exist due to R. Thom’s principle, to the nonisolated case. We also obtain universal expressions for residual polynomials describing classes Poincar´e dual to multisingularities. The aim of the project is to develop a tool for computing cohomology classes in spaces of mappings of curves required for computing Gromov-Witten invariants of curves. This is a joint work in progress with M. Kazarian and D. Zvonkine.

I will discuss a question of Beilinson and Drinfeld regarding the vanishing of the higher chiral homology of V (the integrable quotient of the affine Kac-Moody vertex algebra at positive integral level). In particular I will describe an approach (joint with J. Van Ekeren) that relates this question in the particular case of Elliptic curves to classical homological constructions on the Zhu algebra of V.

In the early 90's it was proposed by several people (Feigin-Frenkel, Finkelberg, Lusztig) that there should exist a category of sheaves/D-modules on the "semi-infinite flag manifold", though of as the quotient of the loop group G((t)) by its subgroup N((t))T[[t]]. The problem is that as an algebro-geometric object, this quotient is "very infinite-dimensional", and it is not clear how to approximate it by finite-dimensional objects in order to have a well-behaved category of sheaves. As a test, this category should have a distinguished object--the Intersection Cohomology sheaf of the closure of G[[t]]/B[[t[[ inside G((t))/N((t))T[[t]]. This category should be related (in fact, equivalent) to (the regular block of) the category of modules over the small quantum group, and also to modules over the Kac-Moody algebra at the critical level. In this talk, we will survey several approaches to this problem, and discuss a particular one, recently proposed by the speaker.

A key feature of integrability for systems of evolution PDEs u_t=F(u), u=(u_1,...,u_k) is to be part of an infinite hierarchy of commuting generalized symmetries. In all known examples, these generalized symmetries are constructed by means of Lenard-Magri sequences involving a pair of matrix differential operators (A,B). We show that in the scalar case k=1 a necessary condition for a pair of differential operators (A,B) to generate a Lenard-Magri sequence is that the ratio L=AB^{-1} lies in a class of operators which we call integrable and contains all ratios of compatible Poisson (or Hamiltonian) structures. We give a sufficient condition on an integrable pair of matrix differential operators (A,B) to generate an infinite Lenard-Magri sequence when the rational matrix differential operator L=AB^{-1} is weakly non-local. This is applied to many systems of evolution PDEs to prove their integrability.

In logarithmic conformal field theory the operator product expansion (OPE) of quantum fields involves logarithms. I will introduce a notion of a logarithmic vertex algebra, which provides a rigorous algebraic formalism for studying such OPEs. I will derive a Borcherds identity for logarithmic vertex algebras and present the examples of free bosons and symplectic fermions.

The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others. In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories, and we will also give some concrete examples to have a better understanding of their structures. We will emphasize some of the interesting properties and structures that modular categories carry with them. Then we will give an overview about the current situation of the classification program for modular categories. We will explain some of the techniques that we found useful to push further the classification. We will also present some of the constructions that give rise to modular categories. If time allows, we will mention some open questions and some results for the pre-modular case.