(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

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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 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...