LECTURE 2
Spin and the Seiberg-Witten Equations
As we said at the beginning of the last lecture, the data for the Seiberg-Witten
equations are a connection on a certain line bundle and a section of a certain C^2
bundle. Having explained connections in the last lecture, we now need to specify
the line bundle and the C^2 bundle. At the end of the lecture we will finally be able
to write down the Seiberg-Witten equations.2.1. Principal bundles and associated bundles
We will use the following general procedure for constructing interesting vector bun-
dles.Definition 2.1. Let X be a smooth manifold and Ga Lie group. A principal G
bundle on Xis a manifold P, a surjective map 7r: P--+ X , and a (right) G action
on P such that:- The G action respects 7r, i.e. the diagram
PxG~P
~ln
x
commutes.- The G action is free and transitive on each fiber.
- Over open balls U C X there exist G-equivariant, fiber-preserving diffeo-
morphisms
Plu ~ U x G.
The canonical example is the frame bundle Fr of a smooth manifold X. If
X is n-dimensional, oriented, and has a Riemannian metric, then Fr is a principal
SO(n) bundle on X. The fiber over x E X consists of all linear maps ]Rn --+ TxX
sending the standard metric and orientation on ]Rn to the metric and orientation
on TxX. The group SO(n) acts by composition on the right.
Out of a principal bundle we can construct many vector bundles as follows.
Let V be a vector space and p: G--+ Aut(V) a representation of G. We define the
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