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We study a relation between harmonic spheres in loop spaces of compact Lie groups and Yang–Mills fields on the Euclidean four-spaceR4

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Fourteenth International Conference on Geometry, Integrability and Quantization June 8–13, 2012, Varna, Bulgaria Ivaïlo M. Mladenov, Andrei Ludu and Akira Yoshioka, Editors

Avangard Prima, Sofia 2013, pp 11–34 doi: 10.7546/giq-14-2013-11-34

HARMONIC SPHERES AND YANG–MILLS FIELDS

ARMEN SERGEEV

Steklov Mathematical Institute, Gubkina 8, 119991 Moscow, Russia

Abstract. We study a relation between harmonic spheres in loop spaces of compact Lie groups and Yang–Mills fields on the Euclidean four-spaceR4.

CONTENTS

1. Introduction . . . 12

2. Harmonic Maps . . . 13

2.1. Harmonic Self-maps of the Riemann Sphere . . . 13

2.2. General Definition of Harmonic Maps. . . 15

2.3. Harmonic Maps of Almost Complex Manifolds . . . 17

3. Instantons and Yang–Mills Fields . . . 18

3.1. Yang–Mills Equations onR4. . . 18

3.2. Instantons . . . 20

4. Twistor Interpretation of Instantons . . . 21

4.1. Basic Twistor Bundle overS4. . . 21

4.2. Atiyah–Hitchin–Singer Construction and Penrose Twistor Program . 22 4.3. Atiyah–Ward and Donaldson Theorems . . . 24

5. Twistor Interpretation of Harmonic Spheres . . . 24

5.1. Eells–Salamon Theorem . . . 24

5.2. Complex Grassmann Manifolds and Flag Bundles . . . 26

5.3. Harmonic Spheres in Grassmann Manifolds: Burstall–Salamon Theorem . . . 26

6. Atiyah Theorem and Harmonic Spheres Conjecture . . . 27

6.1. Loop Spaces of Compact Lie Groups. . . 27

6.2. Holomorphic Spheres in Loop Spaces: Theorem of Atiyah . . . 28 11

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