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In quantum field theory

In quantum field theory an instanton is a field configuration with a “topological twist”: not in the connected component of the trivial field configurations.

The term derives from the special case of instantons on a sphere but modeled as field configurations on a Euclidean space constrained to vanish asymptotically. These look like solutions localized in spacetime: “at an instant”.

In string theory

More generally, in string theory a brane which wraps a completely spacelike cycle in target space is called an instanton, since the worldvolume of such a brane is localized in the time-direction of target space. Under passing to the effective quantum field theory of the string theory, this reproduces many instantons in the sense of quantum field theory above.


gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map


In quantum field theory and specifically Yang-Mills theory

  • Dan Freed, Karen Uhlenbeck, Instantons and four-manifolds, Springer-Verlag, (1991)

  • Nicholas Manton, Paul M. Sutcliffe, Topological solitons, Cambridge Monographs on Math. Physics, gBooks

  • Werner Nahm, Self-dual monopoles and calorons, in Group theoretical methods in physics (Trieste, 1983), pages 189-200. Springer, Berlin (1984) (journal)

See also literature at Yang-Mills instanton.

Yang-Mills instantons on spaces other than just spheres are explicitly discussed in

  • Gabor Kunstatter, Yang-mills theory in a multiply connected three space, Mathematical Problems in Theoretical Physics: Proceedings of the VIth International Conference on Mathematical Physics Berlin (West), August 11-20,1981. Editor: R. Schrader, R. Seiler, D. A. Uhlenbrock, Lecture Notes in Physics, vol. 153, p.118-122 (web)

based on

A generalization is discussed in

Expositions and summaries of this are in

In string theory

In string theory (for D-branes).

The study of M-brane instantons originates around

Specifically membrane/M2-brane instantons are further discussed in

In the context of F-theory and M5-brane instantons:

Revised on May 12, 2014 09:39:57 by Urs Schreiber (