(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

This work concerns the study of certain finite-energy solutions of the antiself-dual Yang-Mills equations on Euclidean 4-dimensional space which are periodic in two directions, so-called doubly-periodic instantons. We establish a circle of ideas involving equivalent analytical and algebraic-geometric descriptions of these objects. In the first introductory chapter we provide an overview of the problem and state the main results to be proven in the thesis. In chapter 2, we study the asymptotic behaviour of the connections we are concerned with, and show that the coupled Dirac operator is Fredholm. After laying these foundations, we are ready to address the main topic of the thesis, the construction of a Nahm transform of doubly-periodic instantons. By combining differential-geometric and holomorphic methods, we show in chapters 3 through 5 that doubly-periodic instantons correspond bijectively to certain singular Higgs pairs, i.e. meromorphic solutions of

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given. 1

In this paper, we consider solutions of the following (open) Toda system (Toda lattice) for SU(N + 1) \Gamma 1 2 \Deltau i = N X j=1 a ij e u j in R 2 ; for i = 1; 2; \Delta \Delta \Delta ; N , where K = (a ij ) N \ThetaN is the Cartan matrix for SU(N + 1). We show that any solution u = (u 1 ; u 2 ; \Delta \Delta \Delta ; uN ) with Z R 2 e u i ! 1; i = 1; 2; \Delta \Delta \Delta ; N; can be obtained from a rational curve in C P N .

Abstract: We study nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary. We establish the basic regularity and compactness properties (assuming L p-bounds on the curvature for p> 2) as well as the Fredholm theory in a compact model case. The motivation for studying this boundary value problem lies in the construction of an instanton Floer homology for 3-manifolds with boundary. The present paper is part of a program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres. 1.

We show that the zero curvature limit of the space of hyperbolic monopoles gives the Euclidean monopoles, settling a conjecture of Atiyah. We also study the infinite curvature limit of the space of hyperbolic monopoles and show that the associated rational maps appear explicitly here.

In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for SU(N + 1) N∑

We give a direct proof of Atiyah's theorem relating instantons over the four-sphere with holomorphic maps from the two-sphere to the loop group. Our approach uses the non-linear heat flow equation for Hermitian metrics as used in the study of Kahler manifolds. The proof generalises immediately to a larger class of four-manifolds. AMS classification: 81T13, 53C07, 55P10 1 Introduction. It is interesting to both mathematicians and physicists to relate gaugetheoretic constructions over four-manifolds to spaces of holomorphic curves into related manifolds. In physical terms, this amounts to relating the instantons of four-dimensional and two-dimensional theories. One of the earliest results of this type is a theorem of Atiyah that relates Yang-Mills instantons over the four-sphere to holomorphic maps of the two-sphere to the loop group [2]. Theorem 1 (Atiyah) For any classical group G and positive integer k, the following two spaces are diffeomorphic: (1) the parameter space of Yang-Mi...

We prove that the space of SU(2) hyperbolic monopoles based at the centre of hyperbolic space is homeomorphic to the space of (unbased) rational maps of the two-sphere. The homeomorphism extends to a map of the natural compactifications of the two spaces. We also show that the scattering methods used in the study of monopoles apply to the configuration space for hyperbolic monopoles giving a homotopy equivalence of this space with the space of continuous self-maps of the two-sphere. AMS classification: 81T13, 53C07, 55P10 1 Introduction. It has long been known that moduli spaces of monopoles and holomorphic maps of the two-sphere are intimately related. In [1, 2] Atiyah introduced the space of hyperbolic monopoles showing that for integral mass the space of charge k SU(2) monopoles based at infinity is isomorphic to the space of degree k based holomorphic self-maps of the two-sphere. His approach was to identify hyperbolic monopoles as instantons over the four-sphere invariant under ...

We construct a large class of periodic instantons. Conjecturally we produce all periodic instantons. This confirms a conjecture of Garland and Murray that relates periodic instantons to orbits of the loop group acting on an extension of its Lie algebra. AMS classification: 81T13, 53C07, 55P10 1 Introduction Periodic instantons are solutions of the anti-self-dual equations FB = \Gamma FB for a connection B on a trivial vector bundle with structure group G over S 1 \Theta R 3 . In this paper, G is a compact Lie group with complexification G c equipped with a representation acting on C n that is unitary on G. Put B = A+ \Phid` so F A = dA \Phi \Gamma @ ` A (1) where we use the three-dimensional Hodge star operator and is the reciprocal of the radius of the circle. One can think of the connection and Higgs field as defined over R 3 and dependent on the circle-valued `. Nahm studied periodic instantons, calling them calorons [17]. Later, Garland and Murray studied perio...