Related papers: Edges, Orbifolds, and Seiberg-Witten Theory

We prove an additivity property for the normalized Seiberg-Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots.…

We propose a double quantization of four-dimensional ${\cal N}=2$ Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter content. This can be understood as a set of certain non-perturbative Schwinger-Dyson…

Let $M$ a compact connected orientable 4-manifold. We study the space $\Xi$ of $Spin^c$-structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on $M$. In order to study…

Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a…

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion $\spinc$ structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are…

We survey some aspects of the current state of research on Einstein metrics on compact 4-manifolds. A number of open problems are presented and discussed.

Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian…

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin-Thorpe inequality and study the case of equality. We use the link with…

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context…

This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of…

Starting with an $n$-dimensional oriented Riemannian manifold with a Spin-c structure, we describe an elliptic system of equations which recover the Seiberg-Witten equations when $n=3,4$. The equations are for a U(1)-connection $A$ and…

The Seiberg-Witten solution plays a central role in the study of N=2 supersymmetric gauge theories. As such, it provides a proving ground for a wide variety of techniques to treat such problems. In this review we concentrate on the role of…

In this paper, we study the coupled Einstein constraint equations on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. This is physically well-motivated by standard cosmological…

We consider compact complex surfaces with Hermitian metrics which are Einstein but not Kaehler. It is shown that the manifold must be CP2 blown up at 1,2, or 3 points, and the isometry group of the metric must contain a 2-torus. Thus the…

A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant of a non-K\"{a}hler complex surface, which…

We introduce the notion of a special monopole class on a four-manifold. This is used to prove restrictions on the smooth structures of Einstein manifolds. As an application we prove that there are Einstein four-manifolds which are simply…

A compact oriented 4-manifold is defined to be of ``superconformal simple type'' if certain polynomials in the basic classes (constructed using the Seiberg-Witten invariants) vanish identically. We show that all known 4-manifolds of…

This paper was withdrawn by the author due to an error in the proof of the main result; essentially the parameter R used in the proof may depend on the manifold (M, g), not just on dimension and pinching constant.

This article explores solutions to a generalised form of the Seiberg--Witten equations in higher dimensions, first introduced by Fine and the author. Starting with an oriented $n$ dimensional Riemannian manifold with a…