Contents

Idea

The Kaluza-Klein mechanism is the observation that pure gravity on a product spacetime $X\times F$ with fixed metric $g_F$ on $F$ looks on $X$ like gravity coupled to Yang-Mills theory for gauge group $G$ the Lie group of isometries of $(F,g_F)$.

The mechanism

More precisely, the Einstein-Hilbert action functional

restricted to metrics of the special form

where $\{k_a\in \Gamma (TF)\}$ are a basis for the Killing vector fields of $(F,g_F)$ is equivalently rewritten as

where

• the first term is the EH action of $g_X$ on $X$

• and the second term is the action functional of Yang-Mills theory for the connection Lie algebra-valued 1-form $A$ with values in the Lie algebra of the isometry Lie group $G$ of $(F,g_F)$.

• The last term is regarded as a functional on the moduli space of this ansatz: the space of metrics on the fiber.

More generally, one observes that the infinitesimal transformation of metrics $g_{X\times F}$ of the above form under diffeomorphisms generated by a Killing vector field $(x,f)\mapsto {\lambda }^a(x)k_a(f)$ is on the $A$-component of the form

This is the infinitesimal form of a gauge transformation: an isomorphism in the groupoid of Lie algebra-valued forms. This means than every functional $(g_{X\times F})\mapsto S(g_{X\times F})$ that is invariant under diffeomorphisms will restrict on metrics of the above form to something that looks like an action functional for a gauge theory of the gauge field $A$.

All this of course remains true if the product $X\times F$ is generalized to an associated bundle $E\to X$ with fiber $F$ to a $G$-principal bundle $P\to X$ ($E=P{\times }_{G}F$), in which case the decomposition of the metric applies locally.

A pseudo-Riemannian manifold of this form $(E,g_E^{\mathrm{KK}})$ for fixed moduli $g_F$ is called a Kaluza-Klein compactification of the spacetime $E$. One also speaks of the effective spacetime $X$ as being obtained by dimensional reduction from the spacetime $E$.

Its application in theoretical physics

Various evident generalizations of this ansatz can and are being considered.

Most notably for actual model building in physics it is of interest to consider the case where $g_X$ and $A$ are not necessarily constant along the fiber $F$. Typically in applications these fields are expanded in terms of Fourier-modes on $F$. The coefficients of the higher modes appear as massive fields in the effective KK-action functional. The are called the higher Kaluza-Klein modes . These masses are inversely proportional to the metric volume of $F$. For physical model building this volume is therefore chosen to be very small, such that it implies that the model does not predict the observation of the quanta of these massive modes in existing accelerator experiements.

On the other hand, the extra moduli fields $g_F$ do not acquire effective masses on $X$ this way. Therefore plain Kaluza-Klein theory is trivially ruled out by experiment: for any choice of $F$ it predicts the observation on $X$ of these massless moduli fields, which however are not being seen in actual accelerator experiments.

Therefore if in a model for fundamental physics the Kaluza-Klein mechanism is invoked as a way to explain the existence of the standard model of particle physics Yang-Mills theory from pure gravity the setup needs to be further generalized: other ingredients of the model need to be introduced that serve to equip the moduli fields with an effective potential with a positive minimum, such that these fields to acquire an effective mass on $X$.

In physics model building the problem of constructing such a more general KK-model is called the moduli stabilization problem .

Examples

Reductions of pure gravity with realistic gauge groups

The gauge group of the experimentally verified standard model of particle physics is a quotient of the product of the special unitary groups $\mathrm{SU}(3)$ and $\mathrm{SU}(2)$ and the circle group $U(1)$.

In (Witten) it was observed that the minimal dimension of a fiber $F$ for the KK-reduction on $F$ to yield gauge group $\mathrm{SU}(3)\times \mathrm{SU}(2)\times U(1)$ is 7 . This may be a meaningless numerical coincidence, but might be – and was regarded as being – remarkable: because it means that the minimum total dimension of a KK-compactification $X\times F$ that could yield a realistic model of observed physics is $4+7=11$. This is the uniquely specified dimensional of the maximal supergravity model: 11-dimensional supergravity.

While there are many 7-dimensional manifolds $F$ that do yield the desired gauge group of the standard model, (Witten) also shows that for none of them does the remaining field content of the standard model – the fermions and the Higgs field – come out correctly.

Largely due to this result the pure Kaluza-Klein ansatz is regarded nowadays as a non-viable way to reproduce the standard model of particle physics from a theory pure gravity. But one can further play with the idea and consider more flexible models that still exhibit the essence of KK-reduction in parts.

Reductions of type II supergravity

Motivation for further variants of the KK-ansatz has to a large extent come from models in string theory. During the end of the 20th and the beginning of the 21st century, the widely dominant ansatz followed in the higher energy phyisics community is to study 10-dimensional type II supergravity models KK-reduced on 6-dimensional Calabi-Yau spaces $F$. The advantage of these type II models is that they naturally involve a further gauge field, called the RR-field. This is modeled by cocycles in differential K-theory which means that its field strength $ℱ$ is an inhomogenous closed differential form of even or of odd degree. Moreover, restricted to configurations $ℱ$ of these forms with specified cycles in $F$, the moduli part of the KK-reduced action functional

does produce the previously missing positive potentials for $g_F$ proportional to these cycles of $ℱ$. So KK-reduction of 10-dimensional supergravities can – for a suitable ansatz – cure the old problem of moduli stabilization in KK-theory.

This means that physical model building using the specific ansatz of KK-reduction of type II supergravities on Calabi-Yau fibers reduces to a noteworthy enumerative problem in complex geometry: classify all real 6-dimensional Calabi-Yau spaces with given isometries and given cycles.

While interesting, there are few tools known for performing this classification. The only thing that seems to be clear is that the classification is not sparse: there are many points in this space of choices. Since all this is relevant in model building in string theory, the space of these choices has been termed the landscape of string theory vacua.

See supergravity and Calabi-Yau manifolds for more.

Reduction of 11d supergravity

The lift of the above reduction of type II supergravity on Calabi-Yau manifolds to M-theory is the reduction of 11-dimensional supergravity on G2-manifolds $X_7$. See at M-theory on G2-manifolds for more details.

For non-vanishing field strength (“flux”) of the supergravity C-field in the 4d space this is the Freund-Rubin compactification yielding weak G2 holonomy on $X_7$.

Related concepts

• spontaneously broken symmetry

• Kaluza-Klein mechanism

  • double dimensional reduction

  • moduli stabilization

  • flux compactification

  • landscape of string theory vacua

References

A textbook account is in

• T. Applequist, A. Chodos and P.G.O. Freund, Modern Kaluza-Klein Theories, Addison-Wesley Publ. Comp., (1987)

A survey of the history of the role of the KK-mechanism in theoretical physics is

• Michael Duff, Kaluza-Klein Theory in Perspective (arXiv)

The seminal analysis of the semi-realistic KK-reductions is in

• Edward Witten, Search for a realistic Kaluza-Klein theory , Nuclear Physics B Volume 186, Issue 3, 10 August 1981, Pages 412-428

A textbook discussion in the context of supergravity is in

• Andrew Strominger (notes by John Morgan), Kaluza-Klein compactifications, Supersymmetry and Calabi-Yau spaces , volume II, starting on page 1091 in

  Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. , Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

In

• Leonardo Castellani, Riccardo D'Auria, Pietro Fre, Supergravity and Superstrings - A Geometric PerspectiveCompactification in first order gravity

the mechanism is discussed around Section V.3.3., page 1186 in volume 2.

The discussion in the first order formulation of gravity is given in

• Rodrigo Arosa, Mauricio Romob and Nelson Zamorano, Compactification in first order gravity J.Phys.Conf.Ser.134:012013,2008 (arXiv:0705.1162)