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=2 Yang-Mills theory

• D=4 Yang-Mills theory

• D=5 Maxwell theory

• D=5 super Yang-Mills theory

• Skyrme model

References

General

• Yuji Tachikawa, Section 6.4 of: $\mathcal{N}=2$ supersymmetric dynamics for pedestrians, Lecture Notes in Physics, vol. 890, 2014 (arXiv:1312.2684, doi:10.1007/978-3-319-08822-8, web version)

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)

Further discussion of this approximation:

• Josh Cork, Chris Halcrow, ADHM skyrmions (arXiv:2110.15190)

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) [doi:10.1103/PhysRevD.97.014024, arXiv:1711.03873]

• Lorenzo Bartolini, Stefano Bolognesi, Sven Bjarke Gudnason, Tommaso Rainaldi, Mass and Isospin Breaking Effects in the Skyrme Model and in Holographic QCD [arXiv:2312.15404]

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)

and further developed in the nuclear matrix model:

• Koji Hashimoto, Norihiro Iizuka, Piljin Yi, A Matrix Model for Baryons and Nuclear Forces, JHEP 1010:003, 2010 (arXiv:1003.4988)

• Si-wen Li, Tuo Jia, Matrix model and Holographic Baryons in the D0-D4 background, Phys. Rev. D 92, 046007 (2015) (arXiv:1506.00068)

• Koji Hashimoto, Yoshinori Matsuo, Takeshi Morita, Nuclear states and spectra in holographic QCD, JHEP12 (2019) 001 (arXiv:1902.07444)

• Yasuhiro Hayashi, Takahiro Ogino, Tadakatsu Sakai, Shigeki Sugimoto, Stringy excited baryons in holographic QCD, Prog Theor Exp Phys (2020) (arXiv:2001.01461)

In relation to Yang-Mills monopoles:

• Stefano Bolognesi, Solitons, Large $N$ and Holography, 2015 (pdf)

Discussion, in this context, of D-term effects affecting hadron stability:

• Mitsutoshi Fujita, Yoshitaka Hatta, Shigeki Sugimoto, Takahiro Ueda, Nucleon D-term in holographic QCD $[$arXiv:2206.06578$]$

More on baryons in the Sakai-Sugimoto model of holographic QCD:

• Salvatore Baldino, Lorenzo Bartolini, Stefano Bolognesi, Sven Bjarke Gudnason, Holographic Nuclear Physics with Massive Quarks, Phys. Rev. D 103 126015 (2021) [doi:10.1103/PhysRevD.103.126015, arXiv:2102.00680]

More on mesons in holographic QCD:

• Johanna Erdmenger, Nick Evans, Ingo Kirsch, Ed Threlfall, Mesons in Gauge/Gravity Duals - A Review, Eur. Phys. J. A 35 (2008) 81-133 [arXiv:0711.4467, doi:10.1140/epja/i2007-10540-1]

• Daniel Ávila, Leonardo Patiño, Melting holographic mesons by cooling a magnetized quark gluon plasma (arXiv:2002.02470)

• Xuanmin Cao, Hui Liu, Danning Li, Pion quasiparticles and QCD phase transitions at finite temperature and isospin density from holography, Phys. Rev. D 102, 126014 (2020) (arXiv:2009.00289)

• Xuanmin Cao, Songyu Qiu, Hui Liu, Danning Li, Thermal properties of light mesons from holography (arXiv:2102.10946)

• Artur Amorim, Miguel S. Costa, Matti Järvinen, Regge theory in a Holographic dual of QCD in the Veneziano Limit (arXiv:2102.11296)

• Roldão da Rocha, Information in AdS/QCD: mass spectroscopy of isovector mesons, Phys. Rev. D 103 106027 (2021) [arXiv:2103.03924, doi:10.1103/PhysRevD.103.106027]

• Shahin Mamedov, Narmin Nasibova, Temperature dependence of $\rho$ meson-nucleon coupling constant from the soft-wall model (arXiv:2103.10494)

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 783 (2018), 222-226 [doi:10.1016/j.physletb.2018.06.052, arXiv:1805.07241]

with related discussion in:

• Tatiana Ivanova, Olaf Lechtenfeld, Alexander Popov, Non-Abelian sigma models from Yang-Mills theory compactified on a circle, Physics Letters B 781 (2018) 322-326 [doi:10.1016/j.physletb.2018.04.013arXiv:1803.07322]

• Olaf Lechtenfeld, Alexander D. Popov, Skyrme-Faddeev model from 5d super-Yang-Mills, Phys. Lett. B 786 (2018) 39-44 [doi:10.1016/j.physletb.2018.09.028, arXiv:1807.11803]

• Olaf Lechtenfeld, Alexander D. Popov, Skyrme and Faddeev models in the low-energy limit of 4d Yang-Mills-Higgs theories, Nuclear Physics B 945 (2019) 114675 [doi:10.1016/j.nuclphysb.2019.114675, arXiv:1808.08972]

See also:

• Y. H. Ahn, Sin Kyu Kang, Hyun Min Lee, Towards a Model of Quarks and Leptons (arXiv:2112.13392)