nLab instanton



Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks (qq)
up-typeup quark (uu)charm quark (cc)top quark (tt)
down-typedown quark (dd)strange quark (ss)bottom quark (bb)
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)


in grand unified theory

minimally extended supersymmetric standard model




dark matter candidates


auxiliary fields

Topological physics



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.


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

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

Relation to Skyrmions, calorons, monopoles

The construction of Skyrmions from instantons is due to

The relation between skyrmions, instantons, calorons, solitons and monopoles is usefully reviewed and further developed in

  • Josh Cork, Calorons, symmetry, and the soliton trinity, PhD thesis, University of Leeds 2018 (web)

  • Josh Cork, Skyrmions from calorons, J. High Energ. Phys. (2018) 2018: 137 (arXiv:1810.04143)

In string theory

In string theory (for D-branes).

The study of M-brane instantons originates around

Specifically membrane instantons are further discussed in

and 5-brane instantons in

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

Last revised on September 20, 2022 at 23:48:26. See the history of this page for a list of all contributions to it.