nLab
instanton

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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. Specifically for gauge fields which mathematically are represented by principal connections, an instanton is a nontrivial underlying principal bundle (or similarly non-trivial associated vector bundle).

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”.

Instantons enter the axial anomaly/chiral anomaly in the standard model of particle physics which is thought to be a source of baryogenesis in the early universe. Generally the QCD vacuum state is argued to consist of a superposition of all possible instanton sectors, see at QCD instanton.

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.

Examples

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

References

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 September 16, 2014 09:55:15 by Urs Schreiber (185.26.182.33)