Graduate Seminar on Differential Geometry (S4D1)
Hauptseminar Geometrie (S2D1)
University of Bonn, Winter semester 2015/16
Instructors: Christian Blohmann [3], Saskia Voß
Time/venue: Wednesday 14:15-16:00, Max Planck Institute for Mathematics, seminar room
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.
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.
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.
If you are interested in participating you can send an email to blohmann@mpim-bonn.mpg.de [4] 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.
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).
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 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
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]
[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
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/41
[2] http://www.mpim-bonn.mpg.de/node/4234
[3] http://people.mpim-bonn.mpg.de/blohmann/
[4] mailto:blohmann@mpim-bonn.mpg.de
[5] http://www.johno.dk/mathematics/fiberbundlestryk.pdf