In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

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• Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these equations have come from mathematical physics. Gauge theory as a tool for studying topological properties of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980's. Since the birth of the subject, it has retained its close connection with symplectic topology, a subject whose intricate structure was illuminated by Mikhail Gromov's introduction of pseudo-holomorphic curve techniques, also introduced in the early 1980's.
  • Clay Mathematics Institute. Summer School (2006). Floer Homology, Gauge Theory, and Low-dimensional Topology: Proceedings of the Clay Mathematics Institute 2004 Summer School, Alfréd Rényi Institute of Mathematics, Budapest, Hungary, June 5-26, 2004. American Mathematical Society. p. 7. ISBN 978-0-8218-3845-7. 

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