Types of quantum field thories
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.
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
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
Nicholas Manton, Paul M. Sutcliffe, Topological solitons, Cambridge Monographs on Math. Physics, gBooks
See also literature at Yang-Mills instanton.
Yang-Mills instantons on spaces other than just spheres are explicitly discussed in
A generalization is discussed in
Expositions and summaries of this are in
The study of M-brane instantons originates around