nLab instanton

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

• Yang-Mills instanton

  • instanton in QCD

  • BPST-instanton

• Coleman-De Luccia instanton

• S-brane

• wrapped brane

  • membrane instanton

  • D3-brane instanton

  • M5-brane instanton

Related concepts

• Yang-Mills instanton

• instanton sector

• small instanton

• non-perturbative effect, non-perturbative quantum field theory, non-perturbative string theory

• renormalon

• quantum tunneling

• at finite temperature: caloron

  • caloron correspondence

• renormalon

• soliton, vortex

• brane

• Spin(7)-instanton

• worldsheet instanton, membrane instanton

• string theory FAQ – Isn’t it fatal that the string perturbation series does not converge?

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
gauge transformation	equivalence	coboundary
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

References

In quantum field theory and specifically Yang-Mills theory

• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)

• Nicholas Manton, Paul 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)

• David Tong, TASI Lectures on Solitons (arXiv:hep-th/0509216), Lecture 1: Instantons (pdf)

• Mikio Nakahara, Section 10.5.5 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• Roberto Percacci, section 3 of: Non-Perturbative Quantum Field Theory – An Introduction to Topological and Semiclassical Methods, SISSA & ICTP (2024) [doi:10.22323/9788898587056, pdf]

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

• Chris IshamGabor Kunstatter, Phys. Letts. v.102B, p.417, 1981. (doi)

• Chris IshamGabor Kunstatter, J. Math. Phys. v.23, p.1668, 1982. (doi)

A generalization is discussed in

• Edward Frenkel, A. Losev?, Nikita Nekrasov, Instantons beyond topological theory I (arXiv:hep-th/0610149)

• Edward Frenkel, A. Losev?, Nikita Nekrasov, Instantons beyond topological theory II (arXiv:hep-th/0610149)

Expositions and summaries of this are in

• Edward Frenkel, A. Losev?, Nikita Nekrasov, Notes on instantons in topological field theory and beyond (arXiv:hep-th/0702137)

• Jacques Distler, Localized (2006) (blog post)

Relation to Skyrmions, calorons, monopoles

The construction of Skyrmions from instantons is due to

• Michael Atiyah, N S Manton, Skyrmions from instantons, Phys. Lett. B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

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).

• Edward Witten, World-Sheet Corrections Via D-Instantons, JHEP 0002:030, 2000 (arXiv:hep-th/9907041)

• Albion Lawrence, Nikita Nekrasov, Instanton sums and five-dimensional gauge theories, Nucl.Phys. B513 (1998) 239-265 (arXiv:hep-th/9706025)

The study of M-brane instantons originates around

• Katrin Becker, Melanie Becker, Andrew Strominger, Fivebranes, Membranes and Non-Perturbative String Theory, Nucl.Phys.B456:130-152,1995 (arXiv:hep-th/9507158)

Specifically membrane instantons are further discussed in

• Jeffrey Harvey, Gregory Moore, Superpotentials and Membrane Instantons (arXiv:hep-th/9907026)

and 5-brane instantons in

• Edward Witten, Non-Perturbative Superpotentials In String Theory, Nucl.Phys.B474:343-360,1996 (arXiv:hep-th/9604030)

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

• Max Kerstan, Timo Weigand, Fluxed M5-instantons in F-theory (arXiv:1205.4720)