!AS/Park City Mathematics Series
Volume 7, 1999An Introduction to the Seiberg-Witten
Equations on Symplectic Manifolds
Michael Hutchings and Clifford Henry Taubest
Introduction
The Seiberg-Witten equ ations are defined on any smooth 4-manifold. By ap-
propriately counting the solutions to the equations, one obtains smooth 4-manifold
invariants. On a symplectic 4-manifold , these invariants have a symplectic in-
terpretation as a count of pseudoholomorphic curves. This allows us to transfer
information between the smooth and symplectic categories in four dimensions.
In the following lectures, we will try to explain this story. In the first two
lectures, we review some background from differential geometry which will allow us
to write down t he Seiberg-Witten equations at t he end of t he second lecture. In the
third lecture we define the Seiberg-Witten invariants and discuss their most basic
properties. In t he fourth lecture we compute the simplest of the Seiberg-Witten
invariants on a symplectic 4-manifold. In the fifth lecture we relate the remaining
Seiberg-Witten invariants in t he symplectic case to pseudoholomorphic curves.(^1) Dept. of Mathematics, Stanford University, Stanford, CA, 94305.
tDept. of Mathematics, Harvard University, Cambridge, MA 02138.
E-mail address: hutchings©gauss. stanf ord. edu, tchtaubes©math. harvard. edu.
© 1999 American Mathematical Society
105