nLab
D=5 Yang-Mills theory

Idea

Yang-Mills theory on spacetimes of dimension $D=5$.

For U(1)-gauge group this reduces to D=5 Maxwell theory. Under KK-compactification this becomes massive Yang-Mills theory in 4 dimensions. Under supersymmetrization it becomes D=5 super Yang-Mills theory.

Properties

Hadrons as KK-modes of 5d Yang-Mills theory

On Skyrmions in 4d as holographic/KK-theoretic reduction of instantons in D=5 Yang-Mills theory:

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

Skyrmions in 4 dimensional spacetime, with vector meson-contributiuons included, are the holographic/KK-theory reduction of instantons in D=5 Yang-Mills theory (Sakai-Sugimoto 04, Section 5.2, Sakai-Sugimoto 05, Section 3.3, reviewed in Sugimoto 16, Section 15.3.4, Bartolini 17, Section 2.

This phenomenon is essentially the theorem of Atiyah-Manton 89, this is highlighted and developed in Sutcliffe 10, Sutcliffe 15.

In this way Skyrmions (and hence baryons and atomic nuclei, see below) appear in the Witten-Sakai-Sugimoto model, which realizes (something close to) non-perturbative QCD as a D4/D8-intersecting D-brane model described by the AdS-QCD correspondence (“holographic QCD”).

This already produces baryon mass spectra with moderate quantitative agreement with experiment (HSSY 07):

graphics from Sugimoto 16

This way, via the equivalence between D4-D8-brane intersections with instantons in the D8-brane-worldvolume, the Skyrme model becomes equivalent to a model of baryons by wrapped D4-branes (Sugimoto 16, 15.4.1):

graphics grabbed from Sugimoto 16

Related concepts

• D=5 Maxwell theory

• D=5 super Yang-Mills theory

• Skyrme model

References

General

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Hadrons as KK-modes of 5d Yang-Mills theory

The suggestion that the tower of observed vector mesons, when regarded as gauge fields of hidden local symmetries of chiral perturbation theory, is reasonably modeled as a Kaluza-Klein tower of D=5 Yang-Mills theory:

• D. T. Son, M. A. Stephanov, QCD and dimensional deconstruction, Phys. Rev. D69 (2004) 065020 (arXiv:hep-ph/0304182, doi:10.1103/PhysRevD.69.065020)

That the pure pion-Skyrmion-model of baryons is approximately the KK-reduction of instantons in D=5 Yang-Mills theory is already due to

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

with a hyperbolic space-variant in

• Michael Atiyah, Paul Sutcliffe, Skyrmions, instantons, mass and curvature, Phys. Lett. B605 (2005) 106-114 (arXiv:hep-th/0411052)

The observation that the result of Atiyah-Manton 89 becomes an exact Kaluza-Klein construction of Skyrmions/baryons from D=5 instantons when the full KK-tower of vector mesons as in Son-Stephanov 03 is included into the Skyrmion model (see also there) is due to

• Paul Sutcliffe, Skyrmions, instantons and holography, JHEP 1008:019, 2010 (arXiv:1003.0023)

• Paul Sutcliffe, Holographic Skyrmions, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 (spire:1383608, doi:10.1142/S0217984915400515)

In the Sakai-Sugimoto model of holographic QCD the D=5 Yang-Mills theory of this hadron Kaluza-Klein theory is identified with the worldvolume-theory of D8-flavour branes intersected with D4-branes in an intersecting D-brane model:

• Tadakatsu Sakai, Shigeki Sugimoto, section 5.2 of Low energy hadron physics in holographic QCD, Prog.Theor.Phys.113:843-882, 2005 (arXiv:hep-th/0412141)

• Tadakatsu Sakai, Shigeki Sugimoto, section 3.3. of More on a holographic dual of QCD, Prog.Theor.Phys.114:1083-1118, 2005 (arXiv:hep-th/0507073)

• Hiroyuki Hata, Tadakatsu Sakai, Shigeki Sugimoto, Shinichiro Yamato, Baryons from instantons in holographic QCD, Prog.Theor.Phys.117:1157, 2007 (arXiv:hep-th/0701280)

• Stefano Bolognesi, Paul Sutcliffe, The Sakai-Sugimoto soliton, JHEP 1401:078, 2014 (arXiv:1309.1396)

• Lorenzo Bartolini, Stefano Bolognesi, Andrea Proto, From the Sakai-Sugimoto Model to the Generalized Skyrme Model, Phys. Rev. D 97, 014024 2018 (arXiv:1711.03873)

Extensive review of this holographic/KK-theoretic-realization of quantum hadrodynamics from D=5 Yang-Mills theory is in:

• Mannque Rho, Ismail Zahed (eds.) The Multifaceted Skyrmion, World Scientific, Second edition, 2016 (doi:10.1142/9710)

  • Shigeki Sugimoto, Skyrmion and String theory, chapter 15 in Mannque Rho, Ismail Zahed (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

Via the realization of D4/D8 brane bound states as instantons in the D8-brane worldvolume-theory (see there and there), this relates also to the model of baryons as wrapped D4-branes, originally due to

• Edward Witten, Baryons And Branes In Anti de Sitter Space, JHEP 9807:006, 1998 (arXiv:hep-th/9805112)

• David Gross, Hirosi Ooguri, Aspects of Large $N$ Gauge Theory Dynamics as Seen by String Theory, Phys. Rev. D58:106002, 1998 (arXiv:hep-th/9805129)

An alternative scenario of derivation of 4d Skyrmions by KK-compactification of D=5 Yang-Mills theory, now on a closed interval, motivated by M5-branes instead of by D4/D8-brane intersections as in the Sakai-Sugimoto model, is discussed in:

• Tatiana Ivanova, Olaf Lechtenfeld, Alexander Popov, Skyrme model from 6d $\mathcal{N}= (2,0)$ theory, Physics Letters B Volume 783, 10 August 2018, Pages 222-226 (arXiv:1805.07241)

following

• Tatiana Ivanova, Olaf Lechtenfeld, Alexander Popov, Non-Abelian sigma models from Yang-Mills theory compactified on a circle, Physics Letters B Volume 781, 10 June 2018, Pages 322-326 (arXiv:1803.07322)