1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

Lecture 3. The Seiberg-Witten Invariants

Let X be a closed oriented smooth 4-manifold. Let g be a Riemannian metric

on X and let μ be a self-dual 2-form. Let s E Sx be a Spine structure and let

8+, 8_, L be the corresponding vector bundles. If A is a connection on L and 'ljJ is

a section of 8+, then (A, 'ljJ) satisfy the Seiberg-Witten equations when

(

DA'l/J )

F(A,'l/J) = FJ-q('l/J) -iμ

vanishes. Let us introduce

ms, g ,μ = {(A,'l/J): F(A,'ljJ) = O}

to denote the space of solutions of the Seiberg-Witten equ ations. In this lecture we
will discuss how to get information about X by studying topological properties of
ms ,g ,μ that do not depend on g and μ. This information constitutes the Seiberg-
Witten invariants of X. For more details on the construction of these invariants,
see [13, 9 ].

3.1. Gauge transformations

To begin with, the space m (we usually suppress the subscripts s, g, μ) has an

infinite-dimensional group action on it, which we want to mod out by to get a finite

dimensional space. The group in question is C^00 (X, 81 ). A function h : X --+ 81

(a gauge transformation) acts on m by

h(A, 'l/J) = (A - 2h-^1 dh, h'l/J).

Exercise 3.1. 1. Check that this action sends m to itself.

2. Show that this action is free, except that a point with 'ljJ = 0 has stabilizer
   31.

Define

M = m/C^00 (X,8^1 ) ,

M^0 = m/{¢ E C^00 (X, 81 ): ¢(*) = 1}.

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