Published online by Cambridge University Press: 25 June 2025
In this article, written at the end of 1996, we survey some of the most important results in Seiberg-Witten Theory which are directly related to Algebraic or Kählerian Geometry. We begin with an introduction to abelian Seiberg-Witten Theory, with special emphasis on the generalized Seiberg-Witten invariants, which take also into account 1-homology classes of the base manifold. The more delicate case of manifolds with b+ = 1 is discussed in detail; we present our universal wall-crossing formula which shows that, crossing a wall in the parameter space, produces jumps of the invariants which are of a purely topological nature.
Next we introduce nonabelian Seiberg-Witten equations associated with very general compact Lie groups, and we describe in detail some of the properties of the moduli spaces of PU(2)-monopoles. The latter play an important role in our approach to prove Witten's conjecture. Then we specialize to the case where the base manifold is a Kahler surface, and we present the complex geometric interpretation of the corresponding moduli spaces of monopoles. This interpretation is another instance of a Kobayashi-Hitchin correspondence, which is based on the analysis of various types of vortex equations. Finally we explain our strategy for a proof of Witten's conjecture in an abstract setting, using the algebraic geometric "coupling principle" and "master spaces" to relate the relevant correlation functions.
Introduction
In October 1994, E. Witten revolutionized the theory of 4-manifolds by introducing the now famous Seiberg-Witten invariants [Witten 1994]. These invariants are defined by counting gauge equivalence classes of solutions of the Seiberg-Witten monopole equations, a system of nonlinear PDE's which describe the absolute minima of a Yang-Mills-Higgs type functional with an abelian gauge group.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.