Schedule for: 18w5182 - Workshop on Geometric Quantization

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will describe a map from `D-cycles' for the twisted K-homology of a compact, connected, simply connected Lie group to the Verlinde ring. The induced map on K-homology is inverse to the Freed-Hopkins-Teleman isomorphism. An application is to show that two options for `quantizing' a Hamiltonian loop group space are compatible with each other. This talk is partly based on joint work with Eckhard Meinrenken and Yanli Song.

Complex line bundles are classified naturally up to isomorphism by degree two integer cohomology $H^2$, and it is of interest to find geometric objects which are similarly associated to higher degree cohomology. Gerbes (of which there are various versions, due respectively to Giraud, Brylinski, Hitchin and Chattergee, and Murray) provide a such theory associated to $H^3$. Various notions of"higher gerbes" have also been defined, though these tend to run into technicalities and complicted bookkeeping associated with higher categories. We propose a new geometric version of higher gerbes in the form of "multi simplicial line bundles", a pleasantly concrete theory which avoids many of the higher categorical difficulties, yet still captures key examples including the string (aka loop spin) obstruction associated to $\frac{1}{2}\ p_1$ in $H^4$. In fact, every integral cohomology class is represented by one of these objects in the guise of a line bundle on the iterated free loop space equipped with a "fusion product" (as defined by Stolz and Teichner and further developed by Waldorf) for each loop factor. This is joint work in progress with Richard Melrose.

In the classical geometric quantization procedure with the "half-form correction", one cannot quantize a complex projective space of even complex dimension (there is no "half form bundle"), and one cannot equivariantly quantize any symplectic toric manifold (there is no "equivariant half form bundle"). I will describe a geometric quantization procedure that uses metaplectic-c structures to incorporate the "half form correction" into the prequantization stage and that does apply to these examples. This follows work of Harald Hess from the late 1970s, with recent contributions of Jennifer Vaughan.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This is an expository talk about C*-algebra K-theory for reductive groups. I’ll try to explain what it is, what it actually says about representation theory, and what else it suggests about representation theory, at least to a willing mind. The story begins with Harish-Chandra’s parametrization of the discrete series representations, and the realization of discrete series representations using the Dirac operator. I’ll discuss these things, and then touch on other parts of Harish-Chandra’s theory of tempered representations that are prominent from the K-theoretic point of view.

In 1980s, Connes and Moscovici studied index theory of G-invariant elliptic pseudo-differential operators acting on non-compact homogeneous spaces. They proved a $L^2$ -index formula using the heat kernel method, which is related to the discrete series representation of Lie groups. In this talk, I will discuss the orbital integral of heat kernel and its relation with Plancherel formula. This is a generalization of the analytic index studied by Connes-Moscovici to the limit of discrete series case. In a recent work by Hochs-Wang, they obained a fixed point theorem for the topogical side of the index. This is a joint work with Xiang Tang.

K-theory of reduced group $C^*$-algebras and their trace maps can be used to study tempered representations of a semisimple Lie group from the point of view of index theory. For a semisimple Lie group, every K-theory generator can be viewed as the equivariant index of some Dirac operator, but also interpreted as a (family of) representation(s) parametrised by A in the Levi component of a cuspidal parabolic subgroup. In particular, if the group has discrete series representations, the corresponding K-theory classes can be realised as equivariant geometric quantisations of the associated coadjoint orbits. Applying orbital traces to the K-theory group, we obtain a fixed point formula which, when applied to this realisation of discrete series, recovers Harish-Chandra's character formula for the discrete series on the representation theory side. This is a noncompact analogue of Atiyah-Segal-Singer fixed point theorem in relation to the Weyl character formula. This is joint work with Peter Hochs.

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank 1. The purpose of this talk is to establish holomorphic Morse inequalities, analogous to Demailly's one, for the invariant part of the Dolbeault cohomology of tensor powers of $L$, twisted by $E$. To do so, we define a moment map $\mu$ by the Kostant formula and then the reduction of $M$ under a natural hypothesis on $\mu^{-1}(0)$. Our inequalities are given in term of the curvature of the bundle induced by $L$ on this reduction, in the spirit of "quantization commutes with reduction"

Let $G$ be a real semisimple Lie group, and $K < G$ a maximal compact subgroup. A tempered representation $\pi$ of G is an irreducible representation that occurs in the Plancherel decomposition of $L^2(G)$. The restriction $\pi|_K$ of $\pi$ to $K$ contains a substantial amount of information about $\pi$. (This is roughly analogous to the fact that an irreducible representation of $K$ is determined by its restriction to a maximal torus.) By realising this restriction as the geometric quantisation of a suitable space, which is a coadjoint orbit under a regularity assumption on $\pi$, we can apply a suitable version of the quantisation commutes with reduction principle to obtain geometric expressions for the multiplicities of the irreducible representations of $K$ in $\pi|_K$ (the $K$-types of $\pi$). This was done for the discrete series by Paradan in 2003. In recent joint work with Song and Yu, we extended this to arbitrary tempered representation. The resulting multiplicity formula was obtained in a different way for tempered representations with regular parameters by Duflo and Vergne in 2011. In independent work in progress with Higson and Song, we give a new proof of Blattner's formula for multiplicities of $K$-types of discrete series representations using geometric quantisation. This formula was first proved by Hecht and Schmid in 1975, and later by Duflo, Heckman and Vergne in 1984.

For manifolds including metric-contact manifolds with non-resonant Reeb flow, we prove a Gutzwiller type trace formula for the associated magnetic Dirac operator involving contributions from Reeb orbits on the base. As an application, we prove a semiclassical limit formula for the eta invariant.

For a compact Lie group action on a smooth manifold, we will introduce a complex of basic relative forms on the inertia space, which was originally constructed by Brylinski. We will explain how basic relative forms can be used to study the Hochschild homology of the convolution algebra. This is work in progress with Markus Pflaum and Hessel Posthuma.

The hypoelliptic Laplacian gives a natural interpolation between the Laplacian and the geodesic flow. This interpolation preserves important spectral quantities. I will explain its construction in the context of compact Lie groups: in this case, the hypoelliptic Laplacian is the analytic counterpart to localization in equivariant cohomology on the coadjoint orbits of loop groups. The construction for noncompact reductive groups ultimately produces a geometric formula for the semisimple orbital integrals, which are the key ingredient in Selberg trace formula. In both cases, the construction of the hypoelliptic Laplacian involves the Dirac operator of Kostant.

Abstract: Given a finite dimensional irreducible complex representation of $G=SO_o(d,1)$, one can associate a canonical flat vector bundle $E$ together with a canonical bundle metric $h$ to any finite volume hyperbolic manifold $X$ of dimension $d$. For $d$ odd and provided $X$ satisfies some mild hypotheses, we will explain how, by looking at a family of compact manifolds degenerating to $X$ in a suitable sense, one can obtain a formula relating the analytic torsion of $(X,E,h)$ with the Reidemeister torsion of an associated manifold with boundary. As an application, we will indicate how, in the arithmetic setting, this formula can be used to derive exponential growth of torsion in cohomology for various sequences of congruence subgroups. This is a joint work with Werner Mueller.

The first part of my talk will be an introduction to Berezin-Toeplitz quantization on Kähler manifolds. Then I will consider a particular class of Berezin-Toeplitz operators whose symbols are characteristic functions. I will discuss their spectral distribution and present a two-terms Weyl law. As an application, I will explain the area law for the entanglement entropy in Quantum Hall effect. Joint work with B. Estienne.

Abstract: In the early eighties, Connes developed his Noncommutative Geometry program, mostly to extend index theory to situations where usual tools of differential topology are not available. A typical situation is foliations whose holonomy does not necessarily preserve any transverse measure, or equivalently the orbit space of the action of the full group of diffeomorphisms of a manifold. In the end of the nineties, Connes and Moscovici worked out an equivariant index problem in these contexts, and left a conjecture about the calculation of this index in terms of characteristic classes. The aim of this talk will be to survey the history of this problem, and explain partly our recent solution to Connes-Moscovici's conjecture, focusing on the part concerning `quantization'. No prior knowledge of Noncommutative Geometry will be assumed, and part of this is joint work with Denis Perrot.

We study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian on a symplectic manifold, corresponding to eigenvalues localized near the origin. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product. This is joint work with L. Ioos, W. Lu and X. Ma.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.