Physics 222

Geometry and Topology for Physicists

    or

Mathematics of Gauge Theories and Gravitation

Spring, 2004

University of California, Berkeley

Instructor:  Robert Littlejohn
Office:  449 Birge
Office Hours:   Friday 1-2pm
Telephone:  642-1229
Email:  physics222@wigner.berkeley.edu
Reader:  Nadir Jeevanjee jeevanje@socrates.berkeley.edu

Lecture:  430 Birge
Time:  TuTh 12:30-2
Text(s):   M. Nakahara, Geometry, Topology and Physics (IOP Pubishing, Bristol,
2003), 2nd edition, and Theodore Frankel, The Geometry of Physics (Cambridge, 1997), recommended.
  
The textbook by Nakahara hasn't changed too much from the first (1990) edition, and you can probably get by with it
if you don't want to buy the new one.

Organization and Logistics

The email address for this course is physics222@wigner.berkeley.edu.   If you wish to be included on the mailing list for course announcements, homework notices, etc., send an email to this address with your name.   (You don't have to be enrolled.)  If you drop the course or don't want to receive any more announcements, send an email to this address with a request to be dropped. 

The course web site (this site) will be used to post lecture notes, special notes and homework assignments, and homework solutions.

This is a variable unit course (2-4 units).  Fewer units means less work, and may be useful if you are busy in research etc.

The grade will be based on weekly homework.  Those taking the course for 2, 3 or 4 units will be expected to make a good attempt on 50%, 75% or 100%, respectively, of the homework.   For some problems it will be possible to get full credit simply by writing "trivial" or "done this before". 

This is by nature a survey course.  The general plan will be to convey the key ideas, a working familiarity with the formalism and terminology, and an intuitive feel for each of the main areas, without going too far into technicalities or difficult proofs.   Certain key results will be stated without proof.  

There are no specific prerequisites beyond graduate standing in Physics.   However, the course is probably more suited to second (and beyond) year students.  This is partly just a matter of experience in dealing with physical problems, although at times we will call on things normally learned in the first year of graduate school, such as how to perform Lorentz transformations on the Dirac equation.

No particular mathematics background is required beyond that usually expected of graduate students in physics (linear algebra, complex analysis, etc), but it will help if you have some familiarity with mathematical notation and ways of thinking.

Lecture notes are my handwritten notes that I prepare for lecture.  I don't guarantee that they will be identical to what is presented in class, but they should be pretty close usually, so you can probably do without taking notes in class if you want to download these.  These notes are available in pdf format only.


Homework assignments will normally (I hope) be made available on this web site by Friday of each week, and will be due one week later, Friday at 5pm, in the envelope outside my office (449 Birge). 




Course Content

1.  Physical context.  Gauge theories, monopoles, instantons, order parameters, general relativity, Berry's phase, classical mechanics, etc.
2.  Mathematical background.  Maps, functions, hierarchy of spaces and manifolds, vector spaces, metrics, pull-backs and adjoints, raising and lowering indices, quotient spaces, group actions etc.
3.  Homology and homotopy.  Abelian groups, simplexes, homology groups, fundamental group, higher homotopy groups, defects in liquid crystals, textures in superfluid  He_3.
4.  Differentiable manifolds, tangent, cotangent spaces and bundles, flows and Lie derivatives, differential forms, Stokes' theorem, differential geometry of group manifolds, Lie algebras, de Rham cohomology, Poincare' lemma, Frobenius theorem.
5.  Riemannian manifolds.  Parallel transport, connection, curvature, holonomy, isometries and Killing vectors, vielbeins, Hodge theory, Cartan structure equations.
6.  Complex manifolds, complex differential forms, Kaehler manifolds (maybe).
7.  Symplectic manifolds, classical mechanics, Poisson brackets, integrable systems, Liouville-Arnold theorem, Lie-Poisson brackets (maybe).
6.  Fiber bundles, vector bundles, principal bundles, spin and Clifford bundles, connections, holonmy, curvature, gauge field theories, general relativity as a gauge theory, geometrical phases.
8.  Characteristic classes, topology of fiber bundles, Chern classes and characters, Chern-Simons forms, Stiefel-Whitney classes.
9.  Index theorems, Atiyah-Singer index theorem, de Rham complex, signature complex, spin complex, Chern-Simons invariants (if time).

This is pretty ambitious for one semester, but the basic aim of the course will be to present some of the differential geometry and topology involved in fiber bundles and gauge theories, which we can do.