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An introduction to gauge theory and its applications. Paper from the 25th Brazilian mathematics colloquium – 25\(^\circ\) colóquio Brasileiro de matemática, Rio de Janeiro, Brazil July 24–29, 2005. (English) Zbl 1078.81058

Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA) (ISBN 85-244-0235-0/pbk). iv, 35 p. (2005).
These notes contain a very short non-technical introduction to recent developments into the theory of four-manifolds. First, very briefly vector bundles, connections and Chern classes are described. Then the Yang-Mills equations in 4 dimensions are discussed, in particular the structure of moduli spaces of anti-self-dual solutions or instantons, including the ADHM construction is outlined. Next the topology of smooth four-manifolds is sketched, Donaldson’s non-existence theorems and examples of exotic \({\mathbb R}^4\) are given. Finally the Hitchin-Kobayashi correspondence between the classes of irreducible anti-self-dual connections and stable holomorphic structures in a hermitian vector bundle over a Kähler surface is described. The notes are completed with the basic explanation of the origin of the Bogomolny, Hitchin and Nahm equations and with a brief introduction to Hitchin’s integrable systems.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
58J28 Eta-invariants, Chern-Simons invariants
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
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