# nLab instanton

Contents

### Context

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# 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.

## References

### In quantum field theory and specifically Yang-Mills theory

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.