Vector Bundles and an Introduction to Gauge Theory

Who/What/Why

These are the class notes to Math 433 - The Geometry of Vector Bundles and an Introduction to Gauge Theory - during the spring semester of 1998 at the The University of Illinois at Urbana Champaign. The course is being taught by Professor Steven Bradlow. For now, this page will contain links to the just the class notes in dvi,TeX, and PostScript formats. You may download individual lectures or any of three .tar.gz files containing blocks of lectures. As time goes by, more may be added. The notes are being put into TeX by Hung-Jen Hsu and Chris Willett, two graduate students at the University of Illinois. Any complaints, ideas, suggestions, or corrections should be sent to Chris Willett.

The Syllabus and References

Geometry of Vector Bundles and an Introduction to Gauge Theory

The course will cover the following topics:

[1]The Basics
- (a)Vector bundles and principal bundles: definitions and basic constructions.
- (b)Connections, curvature, and gauge groups.
- (c)Characteristic classes and Chern-Weil theory
[2] Some more specialized topics
- (a)Spin bundles and Dirac operators, SpinC bundles.
- (b)Holomorphic bundles and the notion of stability.
- (c)Equations of gauge theory (Yang-Mills, anti-self duality,Hermitian-Einstein, vortex, Seiberg-Witten).
[3]Some big theorems and applications.
- (a) Theorem of Narasimhan and Seshadri, and Hitchin-Kobayashi correspondences.
- (b)Donaldson and Seiberg-Witten invariants.

Lecturer: Steven Bradlow
Time: MWF 10am
Prerequisites:Some facility with the machinery of differential geometry (Math 423 or consent of the instructor)
Useful Texts:

Differential Geometry of Complex Vector Bundles,S. Kobayashi, Princeton, 1987.
Fiber Bundles. D. Husemoller, Springer-Verlag, 2nd Edition, 1975.
Vector Bundles. H. Osborne, 1982.
Characteristic Classes.J.Milnor and J. Stasheff, Princeton, 1974.

Some Other Useful References:

The Topology of Fiber Bundles. Norman Steenrod, Princeton 1974.
Differential Forms in Algebraic Topology. Raoul Bott and Loring Tu, Springer-Verlag.

Individual Class Notes

Below are the class notes from the course in chronological order.

[1]: Examples of Bundles from Geometry(PDF format or dvi format or TeX format or PostScript). Tangent and Normal bundles. The Hopf bundle. Tautological bundles. Homogeneous bundles.
[2]: The Formal Definitions(PDF format or .dvi format or TeX format or PostScript). General bundles. The appropriate categories. Vector bundles. Principal bundles.
[3]: Sections and Maps of Bundles(PDF format or .dvi format or TeX format or PostScript). Sections of a bundle. Bundle maps. Bundle Endo/Automorphisms. Local picture of a bundle. Covers and transition functions.
[4]: Constructing Bundles I(PDF format or .dvi format or TeX format or PostScript). Using covers and transition functions to construct vector bundles.
[5]: Constructing Bundles II(PDF format or .dvi format or TeX format or PostScript). Using covers and transition functions to construct principal bundles. Global descriptions of association bundles. Principal fram bundles.
[6]:Sections of Bundles(PDF format or .dvi format or TeX format or PostScript).Reducing bundles. Local descriptions of sections. Gluing local sections together to get a global section for vector and principal bundles.
[7]:New Bundles from Old I(PDF format or .dvi format or TeX format or PostScript). Detecting trivial bundles. The sum, tensor product, and wedge of two bundles. The dual/hom bundle.
[8]:Metrics on Bundles(PDF format or .dvi format or TeX format or PostScript). Fiberwise metrics on vector bundles. Existence of metrics.
[9]:New Bundles from Old II(PDF format or .dvi format or TeX format or PostScript). A return to the Hom bundle. Kernel,image,cokernel bundles.
[10]:Pulling Back Bundles(PDF format or .dvi format or TeX format or PostScript). Pull backs. Homotopic maps yield isomorphic bundles. The triviality of vector bundles over contractible manifolds.
[11]:Looking for the Universal Bundle(PDF format or .dvi format or TeX format or PostScript). The classification problem. Spanning the fiber by sections. The evaluation map.
[12]:A Digression on the Grassmanian(PDF format or .dvi format or TeX format or PostScript). Definitions and examples of Grassmanians. Frame bundles rerevisited. Grassmanians as smooth manifolds. The universal quotient bundle. Pull backs of bundles over Grassmanians.
[13]:Using the Universal Bundle(PDF format or .dvi format or TeX format or PostScript). The universal bundle for smooth manifolds and vector bundles. Vect_n(B). Constructions for non-compact manifolds and for principal bundles.
[14]:Connections I(PDF format or .dvi format or TeX format or PostScript). Connections on trivial bundles. Connections as a calculus of sections.
[15]:Connections II(PDF format or .dvi format or TeX format or PostScript). Connections as a method for differentiating sections of a vector bundle.
[16]:Connections III(PDF format or .dvi format or TeX format or PostScript). The local form of a connection. Connection 1-forms. Covariant derivatives. Parallel sections. The existence question.
[17]:Connections IV(PDF format or .dvi format or TeX format or PostScript). The existence of connections. Two points of view on global connections from local ones. Parallel transport and partial differential equations.
[18]:Parallel Transport(PDF format or .dvi format or TeX format or PostScript). Horizontal lifts of paths. Parallel Transport. Consequences of solutions of Ds=0. Distributions.
[19]:Integrability(PDF format or .dvi format or TeX format or PostScript). Integrable submanifolds. Obstructions to integrability. Curavture of a connection.
[20]:Curvature(PDF format or .dvi format or TeX format or PostScript). A complex arrising from the curvature of a flat connection. Curvature as a measure of a failure of commutivity.
[21]:Holonomy and Induced Connections(PDF format or .dvi format or TeX format or PostScript). Some causes of holonomy. Homotopic paths induce the same holonomy. The Bianchi identity. Induced connections on sum, tensor.
[22]:Induced Connections II(PDF format or .dvi format or TeX format or PostScript). Induced conections on the dual, Hom, pullbacks.
[23]:Induced Connections III(PDF format or .dvi format or TeX format or PostScript). Pulling back connections over curves. Connections on isomorphic bundles. Compatible connections.
[24]:Connections on Tangent Bundles(PDF format or .dvi format or TeX format or PostScript). The compatibility condition revisited. Connections on tangent bundles. Parallel transport rerevisited. Geodesic curves.
[25]:Consequences the Geodesic Equation(PDF format or .dvi format or TeX format or PostScript). Existence and uniqueness of geodesic curves. The exponential map. The torsion of a connection on a tangent bundle. The Levi-Civita connection.
[26]:Characteristic Classes I(PDF format or .dvi format or TeX format or PostScript). The goal. Two approaches - Universal and concrete. Invariant polynomial functions on matrices.
[27]:Characteristic Classes II(PDF format or .dvi format or TeX format or PostScript) Independence of frame. Characteristic classes are cohomology classes.
[28]:Transgression(PDF format or .dvi format or TeX format or PostScript) The independence of characteristic classes from specific connections. Defining the transgression of two connections.
[29]:Characteristic Classes for Complex Bundles I(PDF format or .dvi format or TeX format or PostScript) Chern-Weil homomorphism. GL(n,C) invariant polynomials. Symmetric functions. Chern classes of a complex bundle.
[30]:Characteristic Classes for Complex Bundles II(PDF format or .dvi format or TeX format or PostScript). Chern classes are integral. Chern characters. Behavior of Chern classes under sum and tensor.
[31]:Characteristic Classes for Complex Bundles III(PDF format or .dvi format or TeX format or PostScript). Relations between Chern classes and characters. Chern characters as a map between bundles and cohomology. The real bundle case.
[32]:Characteristic Classes for Real Bundles(PDF format or .dvi format or TeX format or PostScript). The Todd class. Complexification of real bundles. GL(n,R) invariant polynomials.
[33]:Pontrjagin Classes(PDF format or .dvi format or TeX format or PostScript) O(n)-invariant polynomials. Pontrjagin classes for real bundles. The relationship between Pontrjagin classes and Chern classes of a complexified bundle.
[34]:Orientable Bundles(PDF format or .dvi format or TeX format or PostScript) The underlying real structure of complex bundles. Orientation preserving transformations. Oriented bundles.
[35]: The Euler Class(PDF format or .dvi format or TeX format or PostScript) The Euler class of a real, even dimensional, oriented bundle. The Euler class of the underlying real structure of a complex bundle. Relations to Chern classes.
[36]:Detecting Orientability(PDF format or .dvi format or TeX format or PostScript) Changing the orientation of a bundle. The Euler class as an index theorem. The questioj of orientable bundles.
[37]:Cech Cohomology and Orientability(PDF format or .dvi format or TeX format or PostScript) j-cochains and the coboundary map. The first Stieffel-Whitney class as an obstruction to orientability.
[38]:Spin Structures(PDF format or .dvi format or TeX format or PostScript) The existence of Spin(n). Spin bundles and spin structures. The second Stieffel-Whitney class as an obstruction to spin structures.
[39]:Spin_C I Structures(PDF format or .dvi format or TeX format or PostScript) Spin_C construction from Spin. Obstructions to Spin_C structures.
[40]:Spin_C II(PDF format or .dvi format or TeX format or PostScript) Obstructions to Spin_C structures in terms of the second Stieffel-Whitney class and the first Chern class. Equations on bundles:ASD,Instantons,Yang-Mills.

Blocks

You will need a way to uncompress these files. The easiest way is to do this is on a UniX machine, where you would use tar and gunzip. For example,
gunzip vbas-tex.tar.gz
tar xvf vbas-tex.tar
If you are on a Macintosh, there are share/freeware utilities to do this. If you are on a Microsoft machine, I am sure there are equivalents.

The Basics of Bundles (lectures 1-13).TeX format, .dvi format, or PostScript format.
Connections on Vector Bundles (lectures 14-25). TeX format, .dvi format, or PostScript format.

A text file of notation in TeX can be examined here.
A template for the TeX documents can be obtained here.

Last modified May 8, 1998, by Chris Willett