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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 $X$, as an effective field theory, like gravity coupled to Yang-Mills theory – Einstein-Yang-Mills theory – for gauge group $G$ the Lie group of isometries of $(F,g_F)$. In particular for $F = S^1$ the circle, it yields electromagnetism coupled to gravity (and a dilaton) – Einstein-Maxwell theory.
More precisely, the Einstein-Hilbert action functional on pseudo-Riemannian metrics on a product spacetime $X \times F$
when restricted to pseudo-Riemannian metrics of the special form
where $\{k_a \in \Gamma(T F)\}$ 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 $(F,g_F)$;
and finally the last term is regarded as a functional on the moduli space of this ansatz: the moduli space of Riemannian 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 given on the $A$-component as follows:
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 $A$.
Moreover, all this of course remains true if the product $X \times F$ – which we may think of as the trivial $F$-fiber bundle over $X$ – is generalized to any associated bundle $E \to X$ with fiber $F$, associated to a $G$-principal bundle $P \to X$ (hence such that $E = P \times_G F$), in which case the above decomposition of the metric applies locally.
Then one finds that the KK-mechanics indeed not only reproduces gauge fields and their correct dynamics from pure gravity in higher dimensions, but also the forces which they excert on test particles. For instance the rajectory of a charged particle subject to the Lorentz force excerted by an electromagnetic field in $d$-dimensional spacetime is in fact a geodesic in the field of pure gravity of the total space of the corresponding KK-un-compactified circle principal bundle. See (Bleecker 81) or (Bartlett 13) for a pedagogical discussion of this effect.
A pseudo-Riemannian manifold of this form $\left(E, g^KK_{E}\right)$ 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 $E$.
(Beware that the term “compactification” here is vaguely related to but rather different from the use of the term in mathematics, as in one-point compactification etc.).
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. They 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 experiments (such as the LHC).
On the other hand, the extra moduli fields $g_F$ do not acquire effective masses on $X$ this way. Therefore models in plain Kaluza-Klein theory are trivially ruled out by experiment: for any choice of $F$ they 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 in Yang-Mills theory from pure gravity, then 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 . (See also at landscape of string theory vacua.)
KK-compactification along trivial fibrations is closely related to forming mapping stacks: if $\mathbf{Fields}_n$ is the moduli stack of fields] for an $n$-dimensional field theory (see at prequantum field theory for more on this), then for $\Sigma_{k}$ a $k$-dimensional manifold with $k \lt n$ the mapping stack
may be thought of as the moduli stack of fields for an $(n-k)$-dimensional field theory. By the definition universal property of the mapping stack, this lower dimensional field theory is then such that a field configiuration over an $(n-k)$-dimensional spacetime $X_{n-k}$
is equivalently a field configuration of the $n$-dimensional field theory
on the product space $X_{n-k}\times \Sigma_k$ (the trivial $\Sigma_{k}$-fiber bundle over $X_{n-k}$).
Traditionally KK-reduction is understood as retaining only parts of $\mathbf{Fields}_{n-k}$ (the “0-modes” of fields on $\Sigma_k$ only) but of course one may consider arbitrary corrections to this picture and eventially retain the full information.
One example of KK-reduction where the full mapping stack appears is the reduction of topologically twisted N=4 D=4 super Yang-Mills theory on a complex curve $C$ as it appears in the explanation of geometric Langlands duality as a special case of S-duality (Witten 08, section 6). Here $\mathbf{Fields}_4 = \mathbf{B}G_{\mathrm{conn}}$ is the universal moduli stack of $G$-principal connections (or rther that of $G$-Higgs bundles).
The gauge group of the experimentally verified standard model of particle physics is a quotient of the product of the special unitary groups $SU(3)$ and $SU(2)$ and the circle group $U(1)$.
In (Witten 81) it was observed that the minimal dimension of a fiber $F$ for the KK-reduction to yield the gauge group $SU(3) \times SU(2) \times U(1)$ is $d_F = 7$ . 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 81) 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 original pure Kaluza-Klein ansatz that starts with just pure Einstein gravity with no other fields) is nowadays regarded as a non-viable to produce the standard model of particle physics. But one can further play with the idea and consider more flexible models that still exhibit the essence of KK-reduction in parts.
Notably the 11-dimensional supergravity mentioned above contains more fields than just the field of gravity. Specifically it contains the supergravity C-field which is a higher analog of the electromagnetic field. KK-reductions of 11-dimensional supergravity and of its conjectured M-theory UV-completion for instance on G2-manifold fibers turn out to be able to yield models which come at least very close to the standard model of particle physics. This we come to below.
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 $F$. (See also at supersymmetry and Calabi-Yau manifolds.)
The advantage of these type II models is that they naturally involve further higher gauge fields, called the B-field and the RR-field. These are modeled by cocycles in ordinary differential cohomology and in differential K-theory, respectively, which means that their field strength $\mathcal{F}$ is an inhomogenous closed differential form of even or of odd degree. Moreover, restricted to configurations $\mathcal{F}$ of these forms with specified cycles in the fiber $F$, the moduli part of the KK-reduced action functional
does produce the previously missing positive potentials for $g_F$ proportional to these cycles of $\mathcal{F}$. 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 manifolds 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.
For more on this see also
The lift of the above reduction of type IIA supergravity on Calabi-Yau manifolds to M-theory is the KK-reduction of 11-dimensional supergravity/M-theory on G2-manifolds $X_7$. The role of the B-field and RR-field is now played by the supergravity C-field, a higher gauge field in twisted ordinary differential cohomology of degree 3+1. 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 $X_7$.
Such KK-compactification of the form M-theory on G2-manifolds turn out to yield very detailed and at least semi-realistic physics, see for instance the G2-MSSM model.
The 1-form gauge field induced by KK-compactification of M-theory on a circle is the potential of the degree-2 part of the RR-field of type IIA string theory which couples to the D0-brane.
Followed by a T-duality operations these compactifications yields type IIB superstring theory compactifications and this route is known as F-theory:
F-theory KK-compactified on elliptically fibered complex analytic fiber $\Sigma$
$dim_{\mathbb{C}}(\Sigma)$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|
F-theory | F-theory on CY2 | F-theory on CY3 | F-theory on CY4 |
The KK-compactification of D=10 super Yang-Mills theory to $D=1$ is related to what is called the BFSS matrix model. Compactification even further down to $D = 0$ gives the IKKT matrix model.
Apart from possibly producing phenomenologically interesting models from UV-complete fundamental theories on higher dimensional spacetimes, KK-reduction generally serves to connect a wide range of quantum field theories with each other in a way that serves to illuminate their general structure and to map the moduli space of all of them.
For instance various deep but rather mysterious properties of 4-dimensional Yang-Mills theory (the central theory of the standard model of particle physics), or at least of its supersymmetric versions N=1 D=4 super Yang-Mills theory/N=2 D=4 sYM/N=4 D=4 sYM, find their natural geometric interpretation by understanding this 4d theory as the KK-compactification of the 6d (2,0)-superconformal QFT on a torus fiber. Notably the Montonen-Olive duality/S-duality of N=2 D=4 super Yang-Mills theory is the conformal transformation remnant of the compactification torus remaining of the conformal invariant of the 6d (2,0)-superconformal QFT (see there for more on this).
At the same time, the 6d (2,0)-superconformal QFT is itself related to a yet higher dimensional theory, namely to a 7d Chern-Simons theory, not by KK-reduction, but by AdS-CFT duality. But that 7d Chern-Simons theory in turn is the KK-reduction of the 11-dimensional Chern-Simons term in 11-dimensional supergravity. Hence all this structure follows again from the maximal dimensional supergravity theory (M-theory) if in addition to KK-reduction one also considers AdS-CFT-boundary relations.
In an analogous way, the supersymmetric version of 4-dimensional Yang-Mills theory with twice that supersymmetry, namely N=4 D=4 sYM, is itself directly related by AdS-CFT to type II supergravity.
And this process continues further down in dimension: N=2 D=4 super Yang-Mills theory and N=4 D=4 sYM themselves have further KK-reductions to 2-dimensional field theories. For $N = 4$ and after passing to the topologically twisted theory, these are an A-model and a B-model topological string TCFT, respectively. Now what used to be S-duality/Montonen-Olive duality in 4d and conformal invariance in 6d and topological invariant in 7d and 11d here becomes geometric Langlands duality in 2d (see there for more on this) and produces Donaldson theory.
Hence by iteratively applying KK-reductions and other dualities and topological twists, one finds long cascades of different quantum field theories that all superficially look very different, but which thereby become closely related as different aspects of one single higher dimensional field theory. Not all of these lower dimensional theories can be phenomenological viable models, but even the superficially “unrealistic” theories such as the 6d (2,0)-superconformal QFT serve, via KK-reduction, to explain and illuminate deep properties of (semi-)realistic theories such as super Yang-Mills theory in 4d.
The following table displays parts of this cascade of field theories which are induced from 11-dimensional supergravity/M-theory under iterative KK-reduction, AdS-CFT duality and topological twists. There is another such cascade which starts instead from AdS-CFT applied to type II supergravity and then proceeds downward in dimension. This is displayed further below.
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$ | compactificationon elliptic fibration followed by T-duality |
7-dimensional supergravity | |
$\;\;\;\;\downarrow$ topological sector | |
7-dimensional Chern-Simons theory | |
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality | |
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-brane worldvolume theory with type IIB S-duality |
$\;\;\;\;\; \downarrow$ topological twist | |
topologically twisted N=2 D=4 super Yang-Mills theory | |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | |
A-model on $Bun_G$, Donaldson theory |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
type II string theory |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$ |
$\;\;\;\; \downarrow$ topological sector |
5-dimensional Chern-Simons theory |
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality |
N=4 D=4 super Yang-Mills theory |
$\;\;\;\;\; \downarrow$ topological twist |
topologically twisted N=4 D=4 super Yang-Mills theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface |
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence |
Kaluza-Klein mechanism
A textbook account is in
A survey of the history of the role of the KK-mechanism in theoretical physics is
The seminal analysis of the semi-realistic KK-reductions is in
A brief discussion aimed at mathematicians of the KK-reduction of gauge fields is in
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
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
A formalization of Kaluza-Klein compactification in perturbation theory formalized by factorization algebras with values in BV-complexes is in section 19 of
A textbook account of the geometry behind the Lorentz force in the Kaluza-Klein mechanism (the idea that geodesics on the gauge bundle project to curved trajectories on the base manifold) can be found in the introduction of chapter 1 and in chapters 9 and 10 of:
An expository talk of the above material from Bleecker can be found in this talk: