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25
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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 ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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
Classification of Solutions of a Toda System in R²
, 2001
"... 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 = ( ..."
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Cited by 12 (2 self)
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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 .
Nahm transform for doublyperiodic instantons
"... This work concerns the study of certain finiteenergy solutions of the antiselfdual YangMills equations on Euclidean 4dimensional space which are periodic in two directions, socalled doublyperiodic instantons. We establish a circle of ideas involving equivalent analytical and algebraicgeometri ..."
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Cited by 11 (6 self)
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This work concerns the study of certain finiteenergy solutions of the antiselfdual YangMills equations on Euclidean 4dimensional space which are periodic in two directions, socalled doublyperiodic instantons. We establish a circle of ideas involving equivalent analytical and algebraicgeometric 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 doublyperiodic instantons. By combining differentialgeometric and holomorphic methods, we show in chapters 3 through 5 that doublyperiodic instantons correspond bijectively to certain singular Higgs pairs, i.e. meromorphic solutions of
Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions
, 2005
"... In this paper, we continue to consider the 2dimensional (open) Toda system (Toda lattice) for SU(N + 1) N∑ ..."
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Cited by 10 (6 self)
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In this paper, we continue to consider the 2dimensional (open) Toda system (Toda lattice) for SU(N + 1) N∑
Generalised connections and curvature
 MR2177174 (2006j:53029), Zbl 1098.46057
"... 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 ..."
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Cited by 9 (6 self)
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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 YangMills theory are given. 1
Zero and Infinite Curvature Limits of Hyperbolic Monopoles.
, 1997
"... 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. ..."
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Cited by 8 (6 self)
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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.
Antiselfdual instantons with Lagrangian boundary conditions I: Elliptic theory
 Bubbling, Comm. Math. Phys
"... Abstract: We study nonlocal Lagrangian boundary conditions for antiselfdual instantons on 4manifolds with a spacetime splitting of the boundary. We establish the basic regularity and compactness properties (assuming L pbounds on the curvature for p> 2) as well as the Fredholm theory in a compac ..."
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Cited by 6 (5 self)
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Abstract: We study nonlocal Lagrangian boundary conditions for antiselfdual instantons on 4manifolds with a spacetime splitting of the boundary. We establish the basic regularity and compactness properties (assuming L pbounds 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 3manifolds with boundary. The present paper is part of a program proposed by Salamon for the proof of the AtiyahFloer conjecture for homology3spheres. 1.
Compactification of Hyperbolic Monopoles
 Nonlinearity
, 1996
"... 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 twosphere. The homeomorphism extends to a map of the natural compactifications of the two spaces. We also show that the scattering methods use ..."
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Cited by 5 (5 self)
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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 twosphere. 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 selfmaps of the twosphere. AMS classification: 81T13, 53C07, 55P10 1 Introduction. It has long been known that moduli spaces of monopoles and holomorphic maps of the twosphere 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 selfmaps of the twosphere. His approach was to identify hyperbolic monopoles as instantons over the foursphere invariant under ...
Periodic Instantons and the Loop Group
"... 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 Int ..."
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Cited by 4 (1 self)
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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 antiselfdual 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 threedimensional 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 circlevalued `. Nahm studied periodic instantons, calling them calorons [17]. Later, Garland and Murray studied perio...
Degenerating Metrics and Instantons on the FourSphere.
, 1996
"... We give a direct proof of Atiyah's theorem relating instantons over the foursphere with holomorphic maps from the twosphere to the loop group. Our approach uses the nonlinear heat flow equation for Hermitian metrics as used in the study of Kahler manifolds. The proof generalises immediately to a ..."
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Cited by 4 (3 self)
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We give a direct proof of Atiyah's theorem relating instantons over the foursphere with holomorphic maps from the twosphere to the loop group. Our approach uses the nonlinear heat flow equation for Hermitian metrics as used in the study of Kahler manifolds. The proof generalises immediately to a larger class of fourmanifolds. AMS classification: 81T13, 53C07, 55P10 1 Introduction. It is interesting to both mathematicians and physicists to relate gaugetheoretic constructions over fourmanifolds to spaces of holomorphic curves into related manifolds. In physical terms, this amounts to relating the instantons of fourdimensional and twodimensional theories. One of the earliest results of this type is a theorem of Atiyah that relates YangMills instantons over the foursphere to holomorphic maps of the twosphere 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 YangMi...