### Course listing

Graduate Seminar on Differential Geometry (S4D1)

Hauptseminar Geometrie (S2D1)

University of Bonn, Winter semester 2015/16

Instructors: Christian Blohmann, Saskia Voß

Time/venue: Wednesday 14:15-16:00, Max Planck Institute for Mathematics, seminar room

#### Seminar description

In mathematics, quantization is a broad term for a variety of methods that associate an algebraic to a geometric structure following more-or-less the concept of quantization in physics, where the geometry describes a classical mechanical system and the algebra its quantum mechanical counterpart. In physics, quantization ought to associate to a symplectic or Poisson manifold $M$ an algebra $A$ of operators on a Hilbert space and a linear map $Q: C^\infty(M) \to A$ that maps the constant function 1 to the identity operator and Poisson brackets to commutators, such that the image of $Q$ acts irreducibly on the Hilbert space. Satisfying this wishlist from Physics on the nose turns out to be impossible. The search for modifications and rigorous implementations of the concept of quantization is a difficult yet fruitful mathematical problem which has led to many new developments in pure mathematics, such as noncommutative geometry, quantum groups, deformation theory, Floer homology, symplectic field theory, topological quantum field theory, factorization algebras, just to name a few.

In this seminar we will study one particular method of quantization called geometric quantization. It consists of three steps: 1.) Prequantization, which associates to a symplectic manifold $M$ the Hilbert space of square integrable sections of a complex line bundle over $M$ and maps smooth functions on $M$ to differential operators. 2.) Polarization, which uses foliations of $M$ to find an irreducible subspace of the Hilbert space. 3.) The metaplectic correction, which replaces square integrable sections of the line bundle by half-densities. This is necessary because the subspace of polarized sections turns out to be too small or even empty already in the most basic cases such as cotangent bundles.

This procedure, which relates the geometry of the symplectic manifold closely with the quantized algebraic structures, involves many interesting concepts such as connections, curvature, characteristic classes, holonomy, foliations, lagrangian submanifolds, extensions of Lie groups, projective and induced representations, operator algebra, which we will learn or review as needed for the seminar.

#### Prerequisites

A background in differential geometry on the level of Geometry I is required. Knowledge of some material from Geometry II (basic symplectic geometry, curvature forms, principal bundles) is helpful but can be reviewed in the seminar or on an individual basis as needed. On the algebraic side, basic undergraduate algebra and functional analysis will be sufficient. This is a math seminar, so no prior knowlegde of physics is required.

#### Seminar organization

The seminar consists of short talks (ca. 30-45 minutes) by the participants on well-defined parts of the seminar material (e.g. a presentation of an important result with proof or an introduction of a new concept) which are framed and connected by short introductory talks by the seminar organizers.

**Registration**

If you are interested in participating you can send an email to blohmann@mpim-bonn.mpg.de and/or sign up at the first meeting on **October 21**. In addition, you will have to register officially via Basis. If you have a preference for one or several of the talks, please let me know by email.

#### Course credit

In order to obtain credit you will have to participate actively. This means that you will have to a) come to all seminar meetings and b) give one of the short talks listed below. The seminar is officially registered for both, Bachelor studies (S2D1) and Master studies (S4D1).

#### Seminar plan

**Introduction to (geometric) quantization**(21 Oct 2015)

The mathematical apparatus of quantum mechanics; the classical-quantum dictionary; Dirac's wishlist for quantization; no-go theorems; canonical quantization of a symplectic vector space; overview of geometric quantization**Review of basic geometric notions**(28 Oct 2015)

Symplectic manifolds, connections, curvature, holonomy, the Ambrose-Singer theorem**Some Chern-Weil theory**(4 Nov 2015), Chair: Saskia

Hermitian line bundles, the Chern-Weil homomorphism, constructing a complex line bundle for a given Chern class**Prequantization**(11 Nov 2015)

The prequantization on smooth sections of the quantum line bundle; the prequantum Hilbert space; an aside on unbounded operators, Hellinger-Töplitz theorem; prequantization of the cotangent bundle; the energy spectrum of the harmonic oscillator**The Stone-von Neumann Theorem**(18 Nov 2015)

Heisenberg algebra and Heisenberg group; abelian group extensions; stabilizer group of characters (little groups); induced representations; constructing representations of semidirect products by induction; the Stone-von Neumann Theorem**Polarizations**(25 Nov 2015), Chair: Saskia

Maximal sets of commuting operators and lagrangian subspaces; definition of real and complex polarizations; polarizations with smooth quotient; holonomy of a foliation; Bott connection of a foliation; local standard forms of lagrangian foliations**Geometric quantization with polarized sections**(9 Dec 2015)

The subspace of polarized section; polarization condition for operators; the Bohr-Sommerfeld set; the Hilbert space; geometric quantization of symplectic vector spaces and of cotangent bundles**Covering groups and projective representations**(16 Dec 2015)

Covering groups; normal subgroups of the fundamental group; projective representations and central group extensions; reduction and extension of the structure group of a fiber bundle; the fundamental groups of unitary and symplectic groups**Metalinear and metaplectic structures**(13 Jan 2015), Chair: Saskia

The metalinear group; the metaplectic group; presentations of the metaplectic group; metalinear and metaplectic structures**The Hilbert space of polarized half-densities**(20 Jan 2016) Chair: Néstor

Half-densities on vector spaces; the square root of the canonical bundle; the scalar product and Hilbert space of half-densites; half-densities on cotangent bundles**Quantization with half-densities**(27 Jan 2016)

The polarization condition for half-densities; the case of cotangent bundles; the corrected Bohr-Sommerfeld condition**Pairing maps**(3 Feb 2016)

Relation between geometric quantizations for different polarizations; Fourier transform as example- Backup meeting (10 Feb 2016)

Remarks: The dates of the session are subject to changes and not strict since some topics take more time than an entire meeting and some less. Due to the dies academicus there will be no seminar on Dec 2, 2015.

#### Talks

Talks marked with * are more difficult.

1: none

2: Proof of the Ambrose-Singer theorem. **Lory Aintablian**

3*: The Chern-Weil homomorphism (Sec. 9 in [4]) **Kaan Öcal**

4a: Prequantization of cotangent bundles and symplectic vector spaces (Sec. 22.2 in [1]) **Oleg Hamm**

4b: Prequantization engergy spectrum of the harmonic oscillator (Prop. 22.6 in [1])

5: Computing the unitary operators of the Stone-von Neumann theorem (Exercise 3 on p. 302 of [1]) **Donald Youmans**

6a: The Bott connection of a lagrangian foliation (Sec. 4.7 in [2]) **Sofia Amontova**

6b: Local standard forms of a polarization (Sec. 4.7 in [2]) **Elisa Atza**

7: Geometric quantization of symplectic vector spaces (Prop. 22.11 and Prop. 22.12 in [1])

8: Computation of the fundamental groups of $\mathrm{SU}(n)$, $\mathrm{U}(n)$, and $\mathrm{Sp}(n)$ (Ch. IV §2 in [6]) **Max Körfer**

9a: Presentations of the metalinear and metaplectic groups (Ch. V §4, pp. 251-253 and §5, pp. 261-263 in [6]) **Giuseppe Gentile**

9b*: Metalinear structures induced by short exact sequences (Prop. 4.2 Ch. V §4 in [6]) **Elba Garcia Falide**

10: The Hilbert space of half-densities on cotangent bundles (Sec. 23.6.4 in [1]) **Paul Hege**

11: The Bohr-Sommerfeld set of $T^* S^1$ with and without half-densities (Example 23.29 and Exercise 9 on p.524 in [1]) **Rígel Juárez**

12a: Fourier transform as pairing map (Example 7.20 in [8])

12b*: Computing the pairing map (Example on pp. 297-302 of [6]) **Ksenia Fedosova**

#### Literature

The seminar will basically follow chapters 22 and 23 in [1], which is a concise and modern exposition of geometric quantization. [2] is the standard text on geometric quantization which is much more comprehsive.

**Textbooks on geometric quantization**

[1] Brian C. Hall, *Quantum Theory for Mathematicians*, Springer 2013

[2] N. M. J. Woodhouse, *Geometric Quantization*, 2nd ed., Oxford University Press 1998

[3] Jedrzej Sniaticky, *Geometric Quantization and Quantum Mechanics*, Springer 1980

**Additional literature**

[4] Johan Dupont, Fibre Bundles and Chern-Weil Theory, lecture notes (unpublished), Aarhus Universitet 2003, available at: http://www.johno.dk/mathematics/fiberbundlestryk.pdf

[5] B. Kostant, Quantization and unitary representations, in: R. M. Dudley et al., *Lectures in Modern Analysis and Applications*, Lecture Notes in Mathematics 170, Springer 1970, pp. 87–208

[6] Victor Guillemin, Shlomo Sternberg, *Geometric Asymptotics*, revised ed., Amer. Math. Soc. 1991

[7] Robert J. Blattner, Quantization and representation theory, in:* Harmonic analysis on homogeneous spaces*, Amer. Math. Soc. 1973, pp. 147-165

[8] Sean Bates, Alan Weinstein, *Lectures on the Geometry of Quantization*, Berkeley Mathematics Lecture Notes 8, Amer. Math. Soc. 1997

[9] Jean-Luc Brylinski, *Loop Spaces, Characteristic Classes and Geometric Quantization,* Birkhäuser 1993

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