nLab hadron Kaluza-Klein theory



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



The idea of Kaluza-Klein theory has traditionally been applied mostly to “color physics”, such as in attempts to realize the color charges of quantum chromodynamics (quarks and gluons) as a Kaluza-Klein compactification of heterotic supergravity (see for instance at string phenomenologyheterotic models). The success of this approach remains somewhat elusive (see also at landscape of string theory vacua).

Alternatively, Kaluza-Klein theory may be considered for “flavor physics” to produce the charges of flavor-“hidden local symmetries”, namely the baryons and mesons, respectively, hence the hadrons of quantum hadrodynamics. In terms of geometric engineering of QFT via intersecting D-brane models this means to consider gauge theory on flavor branes (instead of on color branes), such as in the Witten-Sakai-Sugimoto model of holographic QCD.

color chargeflavor charge
gauge bosonsgluons
(gauge group-local symmetry)
(flavor-hidden local symmetry)

Indeed, the experimentally observed mesons appear in towers of increasing mass (“higher resonances”), which may usefully be identified as a Kaluza-Klein tower of the single gauge boson of an SU(2)-D=5 Yang-Mills theory (Son-Stephanov 03).

Moreover, the pion field appears as the gauge 0-mode of this tower, right away in its solitonic incarnation as the Skyrmion-excitation in 4d, hence reflecting baryons. (This phenomenon is secretly the old theorem of Atiyah-Manton 89, as explained from the modern perspective of holographic QCD in Sutcliffe 10, Sutcliffe 15).

Various qualitative phenomena of the phenomenology of quantum hadrodynamics find a natural explanation in hadron Kaluza-Klein theory this way, notably:

  1. hidden local symmetry itself (by the very KK-reduction of a gauge theory)

  2. vector meson dominance (as discussed there)

  3. QCD sum rules (…)

  4. (…)

In terms of string phenomenology, the flavor brane-D=5 Yang-Mills theory which gives quantum hadrodynamics this way naturally arises on D4/D8-brane intersections in the Witten-Sakai-Sugimoto model (Sakai-Sugimoto 04, Sakai-Sugimoto 05) or else on M5-branes wrapped on a closed interval (Ivanova-Lechtenfeld-Popov 18)

Already to first approximation, this produces for instance baryon mass spectra with moderate quantitative agreement with experiment (HSSY 07):

graphics grabbed from Sugimoto 16

An extensive review of hadron Kaluza-Klein theory may be found in Rho et al 16.

Strikingly, the experimentally observed hadron-spectrum also exhibits supersymmetry: see at hadron supersymmetry.


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:

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:

with a hyperbolic space-variant in:

Further discussion of this approximation:

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:

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:

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

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

and further developed in the nuclear matrix model:

In relation to Yang-Mills monopoles:

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

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

More on mesons in holographic QCD:

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:


See also:

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


The late Michael Atiyah, following up on his visionary early work in Atiyah-Manton 89, saw the relevance of further develop hadron Kaluza-Klein theory, and suggested using advanced tools of complex geometry for this purpose; for a reminiscence see

This led to a sequence of visionary but speculative articles, including the following:

The idea here is to try to match patterns in the characteristic classes (Chern classes) of complex surfaces to properties of nuclei.

Created on May 8, 2020 at 12:00:05. See the history of this page for a list of all contributions to it.