Lecture 1. Background from Differential Geometry
The data for the Seiberg-Witten equations are a connection A on a certain line
bundle, and a section 'I/; of a certain e^2 bundle. In this lecture we will review bundles
and connections. We assume familiarity with the tangent bundle and differential
forms. (The material in this lecture can be found in almost any modern book on
differential geometry; see e.g. [8].)Vector Bundles
Let X be a smooth manifold. (In this lecture all spaces will be smooth manifolds.)
A vector bundle on X is essentially a family of vector spaces parametrized by X.
More precisely:Definition 1.1. Let n E {1, 2, ... }. A (complex) vector bundle on X of rank n
is a space E with an action e x E --) E of e and a surjective map 7r : E --) X such
that:- 7r commutes with the e action,
• e* acts freely,
- E is locally isomorphic to a product, i.e. for each x E X , there is a neigh-
borhood U c X of x and a diffeomorphism ~ : Elu = n-^1 (U) --) U x en,
such that the diagramE
Elu -----U x en~/
u
commutes, and such that ~ intertwines the e action with the standard e
action on en. ( ~ is called a local trivialization.)If n = 1, we call E a line bundle. One similarly defines a real vector bundle by
replacing e above with JR. A section of E is a smooth choice of a vector in each
fiber, i.e. a smooth map s : X--) E such that ns =identity. We denote the space
of sections of E by C^00 ( E).
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