Kahn-Priddy theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The Kahn-Priddy theorem characterizes a comparison map between [[cohomology]] with [[coefficients]] in the [[suspension spectrum]] of the infinite [[real projective space]] $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$ and [[stable cohomotopy]].

## Statement

Write $\mathbb{R}P^\infty \in Ho(Top)$ for the [[homotopy type]] of [[real projective space]] (an object in the [[classical homotopy category]]), and write $\Sigma^\infty \mathbb{R}P^\infty_+ \in Ho(Spectra)$ for its [[suspension spectrum]] regarded as an [[H-group ring spectrum]] in the [[stable homotopy category]].

For each $n$ there is a canonical inclusion (see [Whitehead 83, p. 20](#Whitehead83)).

$$
  \mathbb{R}P^{n-1}
     \hookrightarrow
  O(n)^+
$$

due to Hopf 35, which is compatible with the inclusions as $n$ varies

$$
  \array{
    \mathbb{R}P^{n-1}
      &\hookrightarrow&
    O(n)
    \\
    \downarrow && \downarrow
    \\
    \mathbb{R}P^n
       &\hookrightarrow&
    O(n+1)
  }
$$

and hence induces an inclusion

$$
  \mathbb{R}P^\infty \hookrightarrow O
$$

> check

Composing this with the [[J-homomorphism]] gives a map

$$
  \phi \;\colon\; \Sigma^\infty \mathbb{R}P^\infty \longrightarrow \mathbb{S}
$$

from the [[H-group ring spectrum]] of infnite [[real projective space]] to the [[sphere spectrum]].

Then for $X$ a connected 2-primary finite [[CW-complex]], the function that takes stable maps into this H-group ring spectrum to maps to the [[sphere spectrum]], hence to the [[stable cohomotopy]] of $X$

$$
  \phi_X
    \;\colon\;
  [\Sigma^\infty X_+ , \Sigma^\infty \mathbb{R}P^\infty_+ ]
    \overset{surj.}{\longrightarrow}
  [\Sigma^\infty X_+, \mathbb{S}]
$$

is [[surjective function|surjective]].

In this form this is stated in [Adams 73, lemma 3.1](#Adams73) (see the notation introduced below lemma 2.2: for $p = 2$ then Adams's $L$ is $\mathbb{R}P^\infty$).

## Analog for complex projective space

The statement was announced in the above  form in [Segal 73, prop. 2](#Segal73), where the analogous statement for [[complex projective space]] and [[topological K-theory]] is proven (see [this prop.](complex+projective+space#HGroupRingSpectrumSurjectsOntoTopologicalKTheory)):

$$
  [\Sigma^\infty X_+ , \Sigma^\infty \mathbb{C}P^\infty_+ ]
    \overset{surj.}{\longrightarrow}
  K_{\mathbb{C}}(X)
  \,.
$$

Notice that [[Snaith's theorem]] asserts that this map becomes in fact an [[isomorphism]] to [[reduced K-theory]] after quotienting out the [[Bott generator]] $\beta \in \Sigma^\infty \mathbb{C}P^\infty_+$.

## References

The original formulation is due to

* {#KahnPriddy72} [[Daniel Kahn]], [[Stewart Priddy]], _Applications of the transfer to stable homotopy theory_, Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 981-987 ([Euclid](https://projecteuclid.org/euclid.bams/1183534135))

A strengthening was obtained in

* {#Adams73} [[John Frank Adams]], _The Kahn-Priddy theorem_, Mathematical Proceedings of the Cambridge Philosophical Society, Mathematical Proceedings of the Cambridge Philosophical Society 55, 1973

Review is in

* {#Whitehead83} [[George Whitehead]], pages 20, 21 of _Fifty years of homotopy_, Bulletin of the AMS, Volume 8, Number 1, 1983 ([pdf](http://www.ams.org/journals/bull/1983-08-01/S0273-0979-1983-15072-3/S0273-0979-1983-15072-3.pdf))

The analogous statement for [[complex projective space]] and complex [[topological K-theory]] is due to

* {#Segal73} [[Graeme Segal]], _The stable homotopy of complex of projective space_, The quarterly journal of mathematics (1973) 24 (1): 1-5. ([[Segal72.pdf:file]], [doi:10.1093/qmath/24.1.1]( https://doi.org/10.1093/qmath/24.1.1))


Kai Behrend

* [website](http://www.math.ubc.ca/~behrend/)

category: people


Kai Keller

## Thesis

* {#Keller10} [[Kai Keller]], _Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization_, PhD thesis ([arXxiv:1006.2148](https://arxiv.org/abs/1006.2148))

on the formulation of [[causal perturbation theory]] in terms of [[dimensional regularization]] via [[Feynman diagram]] expansion

## related $n$Lab entries

* [[Feynman diagram]]

* [[causal perturbation theory]]

category: people


Kakeya Conjecture

A Kakeya Set is a set that contains a unit line segment for every direction. For example, a ball of radius one half is a Kakeya set.

The Kakeya Set Conjecture asserts that every compact Kakeya set $E\subset\mathbb{R}^n$ has [[Hausdorff dimension]] $n$.

## X-Ray Transform

The X-ray transform is the operator
$$\operatorname{X} f(l) := \int_l f\,d\lambda(x),$$
where $l\subset\mathbb{R}^n$ is a line, and $\lambda$ the Lebesgue measure on the line. The manifold $M_n$ can be represented as
$$
M_n = \{(\sigma, x)\in \mathbb{P}^{n-1}\times\mathbb{R}^n \mid \langle x,\sigma\rangle = 0\};
$$
here $\mathbb{P}^{n-1} := S^{n-1}/\{\pm\}$ is the space of lines in $\mathbb{R}^n$ that pass through the origin, and $(\sigma,x)$ represents a line that passes through $x$ with direction $\sigma$. We endow $M_n$ with a measure $\mu$ invariant under rigid motions. It is an open problem to determine the exponents $1\le p,q,r\le\infty$ for which the following inequality holds:
$$
\left\Vert \operatorname{X} f\right\Vert_{L^q(\sigma\mapsto L^r_x(\mathbb{R}^{n-1}))} \le C\left\Vert f\right\Vert_{L^p(\mathbb{R}^n)}.
$$
For brevity we will write $L^q_\sigma L^r_x := L^q(\sigma\mapsto L^r_x(\mathbb{R}^{n-1}))$.

We can write the X-ray transform of a function $f\in C_c^\infty(\mathbb{R}^n)$ as
$$
\operatorname{X} f(l) = \int f\delta_{l}(x)\,dx
$$
where $\delta_{l} := \lim_{\varepsilon\to 0}\varepsilon^{-n+1}1_{\{x\in\mathbb{R}^n\mid \;l\cap B_\varepsilon(x)\neq\emptyset\}}$. If $(\sigma,x)$ is a line $l\in M_n$, and $P_\sigma$ the projection to the plane $\{y\mid \langle y,\sigma\rangle = 0\}$, then $\delta_l$ is the [[pullback of a distribution|pullback]] $P_\sigma^*\delta_x = \delta_x\circ P_\sigma$, where $\delta_x$ is the Dirac delta centered at $x\in\mathbb{R}^{n-1}$.

The operator dual to $\operatorname{X}$, _i.e._ $\int \operatorname{X}h\,f\,d\mu = \int h\,\operatorname{X}^*f\,dx$, acts on functions $f$ in $M_n$ as
$$
\operatorname{X}^*f(x) = \int f\delta_{\{x\}}(l)\,d\mu(l);
$$
here $\delta_{\{x\}} := \lim_{\varepsilon\to 0}\varepsilon^{-n+1}1_{\{l\in M_n\mid \;l\cap B_\varepsilon(x)\neq\emptyset\}}$  ---this is a distribution in $M_n$, do not confuse with the usual Dirac delta $\delta_x$ centered at $x$. The distribution $\delta_{\{x\}}$ is the restriction of the measure $\mu$ in $M_n$ to the set of lines passing through $x\in\mathbb{R}^n$.

The first bound we can get for the X-ray transform is
$$
\int \vert \operatorname{X} f\vert\,d\mu(l) \le \int \operatorname{X}\vert f\vert\,1\,d\mu(l) = \int \vert f\vert\,\operatorname{X}^*1\,dx = \int \vert f\vert\,dx,
$$
since $\operatorname{X}^*1$ is constant. Alternatively, we can use the fact that the integral of $\operatorname{X}\vert f\vert$ along all the lines with the same direction equals $\left\Vert f\right\Vert_1$, which also implies that $\operatorname{X}:L^1(\mathbb{R}^n)\to L^1(M_n)$. Any other bound of the X-ray transform can be interpolated with this bound to get further inequalities.

Besides $L^p$-spaces, we need to consider spaces of regular functions.

+-- {: .num_defn #SobolevSlobodeckij}
###### Definition
**([[Sobolev space]])

Let $\varphi$ be a smooth function with Fourier transform supported in $\vert\xi\vert\sim 1$, and let $\tilde{\varphi}$ be a smooth function with Fourier transform supported in $\vert\xi\vert\lesssim 1$. Suppose that
$$
1 = \sum_{k\ge 0}\varphi_k,
$$
where $\varphi_0 := \tilde{\varphi}$ and $\varphi_k(\xi) := \varphi(\xi/2^k)$, for $k\ge 1$. Let $P_k$ be the projection $(P_k f)^\wedge := \varphi_k \hat{f}$.

The Sobolev-Slobodeckij spaces $W^{s,p}(\mathbb{R}^n)$ are, for $1\lt p\lt \infty$ and $-\infty\lt s\lt \infty$, defined as
$$
\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)} := \Big[\sum_{k\ge 0}(2^{sk}\Vert P_k f\Vert_{L^p(\mathbb{R}^n)})^p\Big]^\frac{1}{p}\quad if s \neq integer,
$$
and
$$
\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)} := \Big[\sum_{\vert\alpha\vert\le s}\Vert \partial^\alpha f\Vert_{L^p(\mathbb{R}^n)}^p\Big]^\frac{1}{p} \quad if s = integer.
$$
See ([Triebel 1978, Sec. 2.3.1](#triebel1978)).

=--

The relation between that X-ray transform and the Kakeya set conjecture is the content of the following Theorem

+-- {: .num_theorem #ThmXRayKakeya}
###### Theorem
**([Bourgain 1991, Lemma 2.15](#bourgain1991))

Let $n-sp\gt 0$. If for every function $f$ such that $supp\,f\subset K$, for $K$ compact, it holds that
$$
\label{eqThmXRayKakeya}
\Vert \operatorname{X} f\Vert_{L^1_\sigma L^\infty_x(M_n)}\le C_K\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)},
$$
then the Hausdorff dimension of a Kakeya set is at least $n-sp$.

=--

+-- {: .proof}
###### Proof

To compute the Hausdorff dimension we can use either balls $B_r(x)$ or dyadic cubes $Q_k := [l2^k,(l+1)2^k]^n$, for $k$ integer. In fact, we can cover a cube $Q_k$ with a ball of radius $\sqrt{n}2^{k-1}$, and conversely we can cover a ball of radius $r$ with a few cubes $Q_k$, for $2^{k-1}\lt r\le 2^k$.

Let $E\subset\mathbb{R}^n$ be a Kakeya set. Give any covering $\mathcal{C} = \{Q_k\}$ of $E$ at scale $0\lt\delta\le 1$, i.e. a covering such that for every cube $Q_k\in\mathcal{C}$ its side-length is $2^k\le\delta$, the goal is to show that if $0\le d\lt n-sp$ then
$$
\sum_{Q_k\in\mathcal{C}}l(Q_k)^d \to \infty \quad as\;\delta\to 0,
$$
where $l(Q_k) = 2^k$ is the side-length of the cube.

Let $\mathcal{C}$ be a covering of $E$ at scale $\delta$. We denote by $\mathcal{C}_k$ the collection of all the cubes in $\mathcal{C}$ with side-length $2^k$, and we denote by $A_k$ the union of all the cubes in $\mathcal{C}_k$; therefore,
$$
1_E \le \sum_k 1_{A_k} := \sum_k \Big(\sum_{Q_k\in\mathcal{C}_k}1_{Q_k}\Big).
$$
Since our hypotheses involve the spaces $W^{s,p}$, we should mollify $1_{A_k}$. We take a suitable smooth function $\varphi$ with compact support,  define the dilations $\varphi_k(x) := 2^{-nk}\varphi(x/2^k)$, and replace $1_{A_k}$ by $\varphi_k*1_{A_k}$.

Since $E$ contains a unit line segment in every direction $\sigma$, then for every $\sigma\in \mathbb{P}^{n-1}$ it holds that $\Vert \operatorname{X}1_E(\cdot,\sigma)\Vert_{L^\infty} \ge 1$, and then that
$$
1\le \Vert \operatorname{X}1_E\Vert_{L^1_\sigma L^\infty_x}\le C\sum_{2^k\le\delta}\Vert \operatorname{X}\big(\varphi_k*1_{A_k}\big)\Vert_{L^1_\sigma L^\infty_x}.
$$
By (eq:eqThmXRayKakeya) we have that
$$
1\le C\sum_{2^k\le\delta} \Vert \varphi_k*1_{A_k}\Vert_{W^{s,p}}.
$$
In general, the $W^{s,p}$-norm of $\varphi_k*f$ is
$$
\Vert \varphi_k*f\Vert_{W^{s,p}}\le C2^{-ks}\Vert f\Vert_p;
$$
see Lemma \ref{LemmSobolevConv} below. Hence,
$$
1\le C\sum_{2^k\le\delta} \Vert \varphi_k*1_{A_k}\Vert_{W^{s,p}} \le C\sum_{2^k\le\delta}2^{-ks+\frac{nk}{p}}\vert \mathcal{C}_k\vert^\frac{1}{p}
$$
where $\vert \mathcal{C}_k\vert$ denotes the number of cubes in $\mathcal{C}_k$.

By hypothesis $n-sp\gt 0$, then for every $0\le d \lt n-sp$ we can apply Hölder inequality to get
$$
1 \le \Big(\sum_{2^k\le\delta} 2^{p'k(\frac{n-d}{p}-s)}\Big)^\frac{1}{p'}\Big(\sum_k 2^{kd}\vert\mathcal{C}_k\vert\Big)^\frac{1}{p} \le C\delta^\alpha\Big(\sum_{Q_k\in\mathcal{C}}l(Q_k)^d\Big)^\frac{1}{p},
$$
where $\alpha = \frac{n-d}{p}-s\gt 0$. The statement of the Theorem follows.
=--

+-- {: .num_lemma #LemmSobolevConv}
###### Lemma

If $\psi$ is a smooth function and $\psi_l(x) := 2^{nl}\psi(2^l x)$, for $l\ge 0$, then
$$
\label{LemmSobolevConv}
\Vert \psi_l*f\Vert_{W^{s,p}(\mathbb{R}^n)}\le C2^{ls}\Vert f\Vert_p,
$$
where $C$ depends on $\psi$.
=--

+-- {: .proof}
###### Proof

If $s$ is an integer, then (eq:LemmSobolevConv) follows from $\partial^\alpha(\psi_l*f) = (\partial^\alpha \psi_l)*f$ and Young Inequality for convolutions.

If $s\neq$ integer, then we have to estimate the norm of $P_k f$; recall that $(P_k f)^\wedge := \varphi_k \hat{f}$, where $\varphi_k(\xi) := \varphi(\xi/2^k)$. If $k\le l$ then by Young Inequality for convolutions
$$
\Vert P_k(\psi_l*f)\Vert_p \le \Vert \check{\varphi}_k\Vert_1\Vert \psi_l\Vert_1\Vert f\Vert_p \le C\Vert f\Vert_p.
$$
If $k\gt l$ then fix a number $2N\gt s$. We estimate the norm of $P_k f$ as
$$
\Vert P_k(\psi_l*f)\Vert_p \le \Vert (\vert \xi\vert^{-2N}\varphi_k)^\vee\Vert_1 \Vert \Delta^N(\psi_l*f)\Vert_p.
$$
As before we get $\Vert \Delta^N (\psi_l*f)\Vert_p\le C2^{2Nl}\Vert f\Vert_p$. For the other term, since $(\vert \xi\vert^{-2N}\varphi_k)^\vee(x) = 2^{(n-2N)k}(\vert \xi\vert^{-2N}\varphi)^\vee(2^k x)$ then we get
$$
\Vert (\vert \xi\vert^{-2N}\varphi_k)^\vee\Vert_1 = 2^{-k2N}\Vert (\vert \xi\vert^{-2N}\varphi)^\vee\Vert_1;
$$
we conclude so that $\Vert P_k(\psi_l*f)\Vert_p \le C2^{2N(l-k)}\Vert f\Vert_p$. Therefore, the $W^{s,p}$-norm of $\psi_l*f$ is
$$
\begin{aligned}
\Vert \psi_l*f\Vert_{W^{s,p}(\mathbb{R}^n)}^p &\le \sum_{0\le k\le l}2^{pks}\Vert P_k(\psi_l*f)\Vert_p^p + \sum_{k\gt l}2^{pks}\Vert P_k(\psi_l*f)\Vert_p^p \\
&\le C\Big(\sum_{0\le k \le l}2^{pks} + 2^{p2Nl}\sum_{k\gt l}2^{p(s-2N)k}\Big)\Vert f\Vert_p^p \\
&= C2^{psl}\Vert f\Vert_p^p,
\end{aligned}
$$
which concludes the proof.
=--

For a fixed direction $\sigma$, the X-ray transform maps $f$ to a function $x\mapsto \operatorname{X} f(x,\sigma)$ in $\mathbb{R}^{n-1}$. The function $\operatorname{X} f(\cdot,\sigma)$ turns out to be more regular than $f$.

+-- {: .num_theorem}
###### Theorem

If $f\in L^2(\mathbb{R}^n)$ has support $supp f\subset K$, for $K$ compact, then
$$
\label{eqThmL2Bound}
\left\Vert \vert \langle D\rangle^\frac{1}{2}\operatorname{X} f\right\Vert_{L^2(M_n)} \le C_K\left\Vert f\right\Vert_{L^2(\mathbb{R}^n)}.
$$

=--

###### Remark

The Japanese bracket $\langle\cdot\rangle$ is defined as $\langle \xi\rangle := (1+\vert \xi\vert^2)^\frac{1}{2}$, and then $\langle D\rangle^2 = I-\Delta$.

+-- {: .proof}
###### Proof

We set $g := \langle D\rangle^\frac{1}{2} f$, and for a fixed direction $\sigma\in \mathbb{P}^{n-1}$ we compute the Fourier transform of the function $x\in\mathbb{R}^{n-1}\mapsto \operatorname{X}g(x,\sigma)$. By rotational symmetry we can assume that $\sigma = e_n$, so
$$
\operatorname{X}g(x,e_n) = \int g(x,t)\,dt.
$$
The Fourier transform is $(\operatorname{X}g(\cdot,e_n))^\wedge(\eta) = \hat{g}(\eta,0)$. If $\delta_{\perp\sigma} = \lim_{\varepsilon\to 0}\varepsilon^{-1}1_{\{\xi\mid \vert\langle \xi,\sigma\rangle\vert\le\varepsilon\}}$ is the Dirac delta of the plane normal to $\sigma$, then we may write
$$
\Vert \operatorname{X}g(\cdot,\sigma)\Vert_{L^2(\mathbb{R}^{n-1})}^2 = \int \vert \hat{g}\vert^2\delta_{\perp\sigma}\,d\xi.
$$
Hence,
$$
\Vert \operatorname{X}g(\cdot,\sigma)\Vert_{L^2(M_n)}^2 = \int\vert \hat{g}\vert^2\Big(\int_{S^{n-1}}\delta_{\perp\sigma}(\xi)\,dS(\sigma)\Big)\,d\xi = \int \langle \xi\rangle\vert\hat{f}\vert^2\frac{d\xi}{\vert \xi\vert}.
$$
For high frequencies we have that
$$
\int_{\vert\xi\vert\ge 1} \langle \xi\rangle\vert\hat{f}\vert^2\frac{d\xi}{\vert \xi\vert} \le C\Vert f\Vert_2^2.
$$
For low frequencies we have that
$$
\int_{\vert\xi\vert\le 1} \langle \xi\rangle\vert\hat{f}\vert^2\frac{d\xi}{\vert \xi\vert} \le C\Vert f\Vert_1^2 \le C_K\Vert f\Vert_2^2.
$$
The statement of the Theorem follows.
=--

Sobolev embedding Theorem $\Vert h\Vert_{L^\infty(\mathbb{R}^{n-1})}\le C\Vert h\Vert_{W^{s,2}(\mathbb{R}^{n-1})}$, for $s\gt \frac{n-1}{2}$, and the inequality (eq:eqThmL2Bound) allow us to conclude that for every $f$ such that $supp\,f\subset K$, for $K$ compact, it holds that
$$
\Vert \operatorname{X} f\Vert_{L^2_\sigma L^\infty_x}\le C_K \Vert f\Vert_{W^{s,2}(\mathbb{R}^n)},
$$
for $s\gt \frac{n}{2}-1$. Hence, by Theorem \ref{ThmXRayKakeya} the Hausdorff dimension of a Kakeya set $E\subset\mathbb{R}^n$ is at least 2. This result is the best possible in $\mathbb{R}^2$ ([Córdoba 1977](#cordoba1977)), but  far away from the expected dimension for $n\ge 3$.

+-- {: .num_defn #LorentzSpace}
###### Definition
**(Lorentz Spaces)

Let $(X,\nu)$ be a [[measure space]]. The quasinorm of the space $L^{p,q}(\nu)$, for $1\le p\lt\infty$ and $1\le q \lt\infty$, is
$$
\left\Vert f\right\Vert_{p,q} := \Big(p\int_0^\infty s^q\nu\{\vert f\vert\ge s\}^\frac{q}{p}\frac{ds}{s}\Big)^\frac{1}{q},
$$
and, for $1\le p\lt\infty$ and $q=\infty$, is
$$
\left\Vert f\right\Vert_{p,\infty} := \sup_{s\gt 0} (s\nu\{\vert f\vert\ge s\}^\frac{1}{p})
$$
=--

+-- {: .num_theorem}
###### Theorem
**([Drury 1983](#drury1983))

If $E\subset \mathbb{R}^n$ is a measurable set, and $1_E$ is the characteristic function of $E$, then
\[
\label{eqThmDrury}
\left\Vert \operatorname{X}(1_E)\right\Vert_{L^{n+1,\infty}(M_n)} \le C\left\vert E\right\vert^\frac{2}{n+1}.
\]

=--

###### Remark

This inequality corresponds to the restricted weak version of the point $(p,q,r) = (\frac{n+1}{2}, n+1, n+1)$.

+-- {: .proof}
###### Proof

Let $f = 1_E$ and define the set $F := \{l\in M_n\mid \operatorname{X} f(l)\ge\lambda\}$, then
$$
\label{eqProofDruryLowerBush}
\lambda\vert F\vert \le \int_F \operatorname{X} f\,d\mu = \int f\operatorname{X}^*1_F \,dx \le \left\Vert \operatorname{X}^*1_F\right\Vert_\infty\left\Vert f\right\Vert_1.
$$
This establishes a lower bound for $\left\Vert \operatorname{X}^*1_F\right\Vert_\infty$, and we will get now an upper bound.

Choose a point $x_0$ such that $\operatorname{X}^*1_F(x_0)\ge \frac{1}{2}\left\Vert \operatorname{X}^*1_F\right\Vert_\infty$ ---$x_0$ is the center of the _Bourgain's bush_. By translation symmetry we can assume that $x_0 = 0$. If $\delta_0$ is the Dirac delta centered at the origin, then
$$
\label{eqProofDruryLower}
\int 1_F \operatorname{X} f\,\operatorname{X}\delta_0\,d\mu(l) \ge \lambda\int_F \operatorname{X}\delta_0\,d\mu(l)\ge \frac{\lambda}{2}\left\Vert \operatorname{X}^*1_F\right\Vert_\infty.
$$
Here we can think of $\operatorname{X}\delta_0$ as the distribution we defined earlier $\delta_{\{0\}} := \lim_{\varepsilon\to 0}\varepsilon^{-n+1}1_{\{l\in M_n\mid \;l\cap B_\varepsilon(0)\neq\emptyset\}}$ supported over the lines passing through the point $0$. We can also justify the notation $\operatorname{X}\delta_0$ by a limiting process since $\delta_0 := \lim_{\varepsilon\to 0}\varepsilon^{-n}1_{B_\varepsilon(0)}$.

On the other hand, we can also estimate the left side of (eq:eqProofDruryLower) as
$$
\int 1_F \operatorname{X} f\,\operatorname{X}\delta_0\,d\mu(l) = \int f\operatorname{X}^*(1_F \operatorname{X}\delta_0)\,dx \le  \left\Vert f\right\Vert_{L^{n,1}} \left\Vert \operatorname{X}^*(1_F \operatorname{X}\delta_0)\right\Vert_{L^{\frac{n}{n-1},\infty}},
$$
The function $\operatorname{X}^*(1_F \operatorname{X}\delta_0)$ can be computed explicitly
$$
\operatorname{X}^*(1_F \operatorname{X}\delta_0)(x) = \frac{1}{\vert x\vert^{n-1}}1_F(l(\tilde{x}, 0)),
$$
where $\tilde{x} := x/\vert x\vert$ and $l(\tilde{x}, 0)$ is the line that passes through $0$ and $x$. Then, we can estimate the norm of $\operatorname{X}^*(1_F \operatorname{X}\delta_0)$ as
$$
\left\Vert \operatorname{X}^*(1_F \operatorname{X}\delta_0)\right\Vert_{L^{\frac{n}{n-1},\infty}} = c_n\vert \operatorname{X}^*1_F(0)\vert^\frac{n-1}{n}\le c_n\left\Vert \operatorname{X}^*1_F\right\Vert_\infty^\frac{n-1}{n}.
$$
Then we get the bound
$$
\label{eqProofDruryUpper}
\int 1_F \operatorname{X} f\,\operatorname{X}\delta_0\,d\mu(l) \le c_n \left\Vert f\right\Vert_{L^{n,1}}\left\Vert \operatorname{X}^*1_F\right\Vert_\infty^\frac{n-1}{n}.
$$
By (eq:eqProofDruryLower) and (eq:eqProofDruryUpper) we get the desired upper bound for $\left\Vert \operatorname{X}^*1_F\right\Vert_\infty$:
$$
c_n\lambda \left\Vert \operatorname{X}^*1_F\right\Vert_\infty^\frac{1}{n} \lesssim \left\Vert f\right\Vert_{L^{n,1}}.
$$
This and (eq:eqProofDruryLowerBush) allow us to conclude that
$$
\vert F\vert \lesssim \frac{1}{\lambda^{n+1}}\left\Vert f\right\Vert_1\left\Vert f\right\Vert_{L^{n,1}}^n = \frac{1}{\lambda^{n+1}}\vert E\vert^2,
$$
which implies the restricted weak inequality (eq:eqThmDrury).

=--

Drury's Theorem and the bound $L^1\to L^1$ imply that
$$
\Vert \operatorname{X} f\Vert_{L^q(M_n)}\le C\Vert f\Vert_{L^p(\mathbb{R}^n)},
$$
for $1\le p\lt \frac{n+1}{2}$ and $\frac{n-1}{q}+1 = \frac{n}{p}$. By Sobolev embedding Theorem $\Vert h\Vert_{L^\infty(\mathbb{R}^{n-1})} \le \Vert h\Vert_{W^{s,q}(\mathbb{R}^{n-1})}$, for $s\gt \frac{n-1}{q}$, we get further
$$
\Vert \operatorname{X} f\Vert_{L^q_\sigma L^\infty_x(M_n)}\le C\Vert f\Vert_{W^{s,p}(\mathbb{R}^n)},
$$
for $1\le p\lt \frac{n+1}{2}$, $\frac{n-1}{q}+1 = \frac{n}{p}$ and $s\gt \frac{n}{p}-1$. Hence, by Theorem \ref{ThmXRayKakeya} the Hausdorff dimension of a Kakeya set $E\subset\mathbb{R}^n$ is at least $\frac{n+1}{2}$.

## References

* {#bourgain1991} [[Bourgain, J.]], Besicovitch type maximal function operators and applications to Fourier Analysis, Geom. Funct. Anal., 1, 1991

* {#cordoba1977} [[Córdoba, A.]], The Kakeya Maximal Function and the Spherical Summation Multipliers, Amer. J. Math., 99, 1977

* [[Christ, M.]], Estimates for the $k$-plane transform, Indiana Univ. Math. J., 33, 1984

* {#drury1983} [[Drury, S. W.]], $L^p$ estimates for the X-ray transform, Illinois J. Math., 27, 1983

* {#triebel1978} [[Triebel, H]], Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company, 1978


Kalb-Ramond field

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **Kalb-Ramond field**  or **B-field** is the [[higher U(1)-gauge field]] that generalizes the [[electromagnetic field]] from point [[particles]] to [[string theory|strings]].

Its dual incarnation in [[KK-compactifications]] of [[heterotic string theory]] to 4d is a candidate for the hypothetical [[axion]] field ([Svrcek-Witten 06, p. 15](#SvrcekWitten06)).

Recall that the [[electromagnetic field]] is modeled as a [[cocycle]] in degree 2 [[ordinary differential cohomology]] and that this mathematical model is fixed by the fact that charged particles that trace out 1-dimensional trajectories couple to the electromagnetic field by an [[action functional]] that sends each trajectory to the holonomy of a $U(1)$-[[connection on a bundle|connection]] on it.

When replacing particles with 1-dimensional trajectories by [[string theory|strings]] with 2-dimensional trajectories, one accordingly expects that they may couple to a higher degree background field given by a degree 3 [[ordinary differential cohomology]] cocycle.

In [[string theory]] this situation arises and the corresponding background field appears, where it is called the _Kalb-Ramond field_  .

Often it is also simply called the **$B$-field** , after the standard symbol used for the 2-forms $(B_i \in \Omega^2(U_i))$ on patches $U_i$ of a [[cover]] of spacetime when the differential cocycle is expressed in a [[Cech cohomology]] realization of [[Deligne cohomology]].

This is the analog of the local 1-forms $(A_i \in \Omega^1(U_i))$ in a Cech cocycle presentation of a [[line bundle]] with [[connection on a bundle|connection]] encoding the [[electromagnetic field]].

The [[field strength]] of the Kalb-Ramond field is a [[differential forms|3-form]] $H \in \Omega$. On each patch $U_i$ it is given by

$$
  H|_{U_i} = d B_i
  \,.
$$

And just as a degree 2 [[Deligne cohomology|Deligne cocycle]] is equivalently encoded in a $U(1)$-[[principal bundle]] [[connection on a bundle|with connection]], the degree 3 differential cocycle is equivalently encoded in

* a degree 3 [[Deligne cohomology|Deligne cocycle]];

* a $\mathbf{B}U(1)$-[[principal 2-bundle]] with connection;

* a $U(1)$-[[bundle gerbe]] with connection.

The study of [[bundle gerbe]]s was largely motivated and driven by the desire to understand the Kalb-Ramond field.

The next higher degree analog of the electromagnetic field is the [[supergravity C-field]].

## Mathematical model from (formal) physical input

The derivation of the fact that the Kalb-Ramond field that is locally given by a 2-form is globally really a degree 3 cocycle in the [[Deligne cohomology]] model for [[ordinary differential cohomology]] proceeds in in entire analogy with the corresponding discussion of the [[electromagnetic field]]:

* **classical background** The [[field strength]] 3-form $H \in \Omega^3(X)$ is required to be closed, $d H_3 = 0$.

* **quantum coupling** The gauge interaction with the quantum string is required to yield a well-defined surface holonomy in $U(1)$ from locally integrating the 2-forms $B_i \in \Omega^2(U_2)$ with $d B_i = H|_{U_i}$ over its 2-dimensional trajectory.

  $$
    hol(\Sigma)
    =
    \textstyle{\prod}_{f}
    \exp(i \textstyle{\int}_f \Sigma^* B_{\rho(f)})
    \textstyle{\prod}_{e \subset f}
    \exp(i \textstyle{\int}_{e} \Sigma^* A_{\rho(f) \rho(e)})
    \textstyle{\prod}_{v \subset e \subset f}
    \exp(i \lambda_{\rho(f) \rho(e) \rho(v)})
    \,.
  $$

  That this is well defined requires that

  $$
    \lambda_{i j k} - \lambda_{i j l} +
    \lambda_{i k l} - \lambda_{j k l}
    = 0 \;mod \, 2\pi
    \,,
  $$

  which says that $(B_i, A_{i j}, \lambda_{i j k})$ is indeed a degree 3 [[Deligne cohomology|Deligne cocycle]].

## Over D-branes

The restriction of the Kalb-Ramond field in the 10-dimensional [[spacetime]] to a [[D-brane]] is a twist (as in [[twisted cohomology]]) of the [[gauge field]] on the D-brane: its 3-class is [[magnetic charge]] for the [[electromagnetic field]]/[[Yang-Mills field]] on the D-brane. See also [[Freed-Witten anomaly cancellation]] or the discussion in ([Moore](#Moore)).

## Related concepts

* [[discrete torsion]]

* [[h1-meson]], [[b1-meson]]

* [[WZW model]], [[basic bundle gerbe]]

[[!include table of branes]]

## References {#References}

The name goes back to:

* {#KalbRamond} M. Kalb, [[Pierre Ramond]], *Classical direct interstring action*, Phys. Rev. D. **9** (1974) 2273-2284 [[doi:10.1103/PhysRevD.9.2273](https://doi.org/10.1103/PhysRevD.9.2273)]

The interpretation as a 4d [[axion]]:

* {#SvrcekWitten06} Peter Svrcek, [[Edward Witten]], _Axions In String Theory_, JHEP 0606:051 (2006) [[arXiv:hep-th/0605206](http://arxiv.org/abs/hep-th/0605206)]

The interpretatin of the B-field as a 3-[[cocycle]] in [[Deligne cohomology]] is due to

* [[Krzysztof Gawędzki]], *Topological Actions in two-dimensional Quantum Field Theories*, in: _Nonperturbative quantum field theory_, Nato Science Series B **185**, Springer (1986) [[spire:257658](http://inspirehep.net/record/257658), [doi:10.1007/978-1-4613-0729-7_5](https://doi.org/10.1007/978-1-4613-0729-7_5), [[Gawedzki-TopologicalActions.pdf:file]]]

picked up in

* {#FreedWitten99} [[Daniel Freed]], [[Edward Witten]], _Anomalies in String Theory with D-Branes_, Asian J. Math. **3** 4 (1999) 819-852 [[arXiv:hep-th/9907189](http://arxiv.org/abs/hep-th/9907189), [InSpire:2655986](https://inspirehep.net/literature/2655986)]

The equivalent formulation in terms of [[connections on bundle gerbes]] originates with

* {#GawędzkiReis02} [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]

* {#CareyJohnsonMurray02} [[Alan Carey]], Stuart Johnson, [[Michael Murray]], _Holonomy on D-Branes_, Journal of Geometry and Physics **52** 2 (2004) 186-216 [[arXiv:hep-th/0204199](http://arxiv.org/abs/hep-th/0204199), [doi:10.1016/j.geomphys.2004.02.008](https://doi.org/10.1016/j.geomphys.2004.02.008)]

See also:

* {#BonoraRuffinoSavelli08} [[Loriano Bonora]], [[Fabio Ferrari Ruffino]], [[Raffaele Savelli]], *Classifying A-field and B-field configurations in the presence of D-branes*, JHEP 0812:078 (2008) [[arXiv:0810.4291](https://arxiv.org/abs/0810.4291), [doi:10.1088/1126-6708/2008/12/078](https://doi.org/10.1088/1126-6708/2008/12/078)]

* [[Fabio Ferrari Ruffino]], *Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes*, Nuclear Physics, Section B **858** (2012) 377-404 [[arXiv:1104.2798](https://arxiv.org/abs/1104.2798), [doi:10.1016/j.nuclphysb.2012.01.013](https://doi.org/10.1016/j.nuclphysb.2012.01.013)]

A more refined discussion of the [[differential cohomology]] of the Kalb-Ramond field and the [[RR-fields]] that it interacts with:

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_, Surveys in Differential Geometry **7** (2002) 129-194 [  [arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)]

In fact, in full generality the Kalb-Ramond field on an [[orientifold]] background is not a plain bundle gerbe, but a _[[Jandl gerbe]]_, a connection on a nonabelian $AUT(U(1))$-[[principal 2-bundles]] for the [[automorphism 2-group]] $AUT(U)(1))$ of $U(1)$:

for the bosonic string this is discussed in

* {#SSW05} [[Urs Schreiber]], [[Christoph Schweigert]], [[Konrad Waldorf]], _Unoriented WZW models and Holonomy of Bundle Gerbes_, Communications in Mathematical Physics **274** 1 (2007) 31-64 [[arXiv:hep-th/0512283](http://arxiv.org/abs/hep-th/0512283), [doi:10.1007/s00220-007-0271-x](https://doi.org/10.1007/s00220-007-0271-x)]

and for the refinement to the [[superstring]] in

* [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Precis_, in: [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics **83**, AMS (2011) [[arXiv:0906.0795](http://arxiv.org/abs/0906.0795)]

* {#DistlerFreedMooreII} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Spin structures and superstrings_, Surveys in Differential Geometry, Volume 15 (2010) ([arXiv:1007.4581](http://arxiv.org/abs/1007.4581), [doi:10.4310/SDG.2010.v15.n1.a4](http://dx.doi.org/10.4310/SDG.2010.v15.n1.a4))

See at _[[orientifold]]_ for more on this; also at *[[discrete torsion]]*.

The role of the KR field in [[twisted K-theory]] (see *[[K-theory classification of D-brane charge]]*) is discussed a bit also in

* {#Moore} [[Greg Moore]], _K-theory from a physical perspective_ [[arXiv:hep-th/0304018](http://arxiv.org/abs/hep-th/0304018)]

In relation to [[first-order formulation of gravity|Einstein-Cartan theory]]:

* Richa Kapoor, _A review of Einstein Cartan Theory to describe superstrings with intrinsic torsion_ ([arXiv:2009.07211](https://arxiv.org/abs/2009.07211))

* Tanmoy Paul, _Antisymmetric tensor fields in modified gravity: a summary_ ([arXiv:2009.07732](https://arxiv.org/abs/2009.07732))

In the context of [[cosmology]] with the Kalb-Ramond field as a [[dark matter]]-candidate (cf, *[[axion]]* and *[[fuzzy dark matter]]*):

* Christian Capanelli, Leah Jenks, Edward W. Kolb, Evan McDonough, *Cosmological Implications of Kalb-Ramond-Like-Particles* [[arXiv:2309.02485](https://arxiv.org/abs/2309.02485)]

See also:

* [[Peter D. Jarvis]], [[Jean Thierry-Mieg]], *Antisymmetric tensor fields: actions, symmetries and first order Duffin-Kemmer-Petiau formulations* [[arXiv:2311.01675](https://arxiv.org/abs/2311.01675)]

* [[Jean Thierry-Mieg]], [[Peter D. Jarvis]], *Conformal invariance of antisymmetric tensor field theories in any even dimension* [[arXiv:2311.01701](https://arxiv.org/abs/2311.01701)]

[[!redirects Kalb-Ramond fields]]

[[!redirects Kalb--Ramond field]]
[[!redirects Kalb–Ramond field]]
[[!redirects B-field]]
[[!redirects B-fields]]

[[!redirects B2-field]]


Kaluza-Klein mechanism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Kaluza-Klein mechanism_, named after [[Theodor Kaluza]] and [[Oskar Klein]], is the observation that pure [[gravity]] on a [[Cartesian product|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]] (a "[[graviphoton]]" [[field (physics)|field]], but in addition also a [[dilaton]]) -- [[Einstein-Maxwell theory]].

When applied to [[supergravity]] then this mechanism produces not just [[force]] fields but also [[fermion|fermionic]] [[matter]] in the resulting [[effective field theory]], hence yields [[Einstein-Yang-Mills-Dirac theory]], see the examples below at _[KK-reduction of 11d Supergravity](#KKReductionOf11dSupergravity)_. As such the KK-mechanism is not unlike the idea of [[geometrodynamics]], according to which fundamentally there is only [[dynamics|dynamical]] [[spacetime]] geometry in the form of [[gravity]] with all matter and other forces being special incarnations of this.

Since in [[general relativity]] also the size and shape of the fiber $F$ is dynamical, generically [[effective field theories]] arising from KK-compactification contain spurious fields parameterizing the geometry of $F$. In the simplest case this is just the [[dilaton]], encoding the total [[volume]] of $F$, more generally these fields are often called the _[[moduli]]_ fields. Since these moduli fields are not observed in [[experiment]], naive KK-models are generically [[phenomenology|phenomenologically]] unviable. However, in variants of gravity such as higher dimensional [[supergravity]] there are possibilities for the moduli to obtain masses and hence for the KK-models to become viable after all. This is the problem of _[[moduli stabilization]]_.

### The mechanism
 {#IdeaTheMechanism}

More precisely, the [[Einstein-Hilbert action]] functional on [[pseudo-Riemannian metrics]] on a [[Cartesian product|product]] [[spacetime]] $X \times F$

$$
  g_{X \times F}
   \;\mapsto\;
   S_{EH}(g_{X \times F})
   \;\coloneqq\;
   \int_{X \times F} R(g_{X \times F}) \, dvol(g_{X \times F})
$$

when restricted to [[pseudo-Riemannian metrics]] of the special form

$$
  \left(g^{KK}_{X \times F} \left(x,f\right) \right)
   \;\in \;
  Sym^2\left(\Gamma\left(T^* X \times F\right)\right)
   \;=\;
   \left(
    \array{
      g_F(f) & A^a(x)k_a
      \\
      A^a(x) k_a & g_X(x)
    }
  \right)
  \,,
$$

where $\{k_a \in \Gamma(T F)\}$ are a [[basis]] for the [[Killing vector field]]s of $(F,g_F)$, is equivalently rewritten as

$$
  S_{EH}\left(g^KK_{X \times F}\right)
   =
  S_{EH}\left(g_X\right) + S_{YM}(A) + S_{mod}\left(g_F\right)
  \,,
$$

where

* the first term is the [[Einstein-Hilbert action|EH action]] of $g_X$ on $X$

* and the second term is the [[action functional]] of [[Yang-Mills theory]] for the [[connection on a bundle|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 object|infinitesimal]] transformation of [[Riemannian metric|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:

$$
  \frac{d}{d \epsilon} A  = d_X \lambda^a + [\lambda,A]
  \,.
$$

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 [[Cartesian product|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 exert on test particles. For instance the trajectory of a [[charged particle]] subject to the [[Lorentz force]] exerted 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](#Bleecker81)) or ([Bartlett 13](#Bartlett13)) 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.).

### 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 _[[mass|massive]] [[field (physics)|fields]]_ in the [[effective QFT|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 (physics)|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]] [[field (physics)|fields]] $g_F$ do not acquire effective [[masses]] on $X$ this way. Therefore [[model (physics)|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]] [[field (physics)|fields]], which however are not being seen in actual accelerator [[experiments]].

Therefore if in a [[model (physics)|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 (physics)|model]] need to be introduced that serve to equip the [[moduli]] [[field (physics)|fields]] with an effective [[potential]] with a [[positive number|positive]] [[minimum]], such that these [[field (physics)|fields]] to acquire an effective [[mass]] on $X$.

In [[physics]] [[model building]] the problem of constructing such a more general KK-[[model (physics)|model]] is called the **[[moduli stabilization]] problem** . (See also at _[[landscape of string theory vacua]]_.)

### Formalization
 {#Formalization}

KK-compactification along trivial fibrations is closely related to forming [[mapping stacks]]: if $\mathbf{Fields}_n$ is the [[moduli stack]] of [[field (physics)|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]]

$$
  \mathbf{Fields}_{n-k} \coloneqq [\Sigma_k, \mathbf{Fields}]
$$

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 configuration over an $(n-k)$-dimensional [[spacetime]] $X_{n-k}$

$$
  \phi \colon X_{n-k} \longrightarrow \mathbf{Fields}_{n-k}
$$

is equivalently a field configuration of the $n$-dimensional field theory

$$
  X_{n-k} \times \Sigma_k \longrightarrow \mathbf{Fields}_n
$$

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 eventually retain the full information.

One example of KK-reduction where the full mapping stack appears is the reduction of [[topological twist|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](#Witten08)). 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]]).

## Examples

### Reductions of pure gravity with realistic gauge groups
 {#ReductionsOfPureGravity}

The [[gauge group]] of the [[experiment|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](#Witten81)) 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-[[dimension|dimensional]] [[manifolds]] $F$ that do yield the desired [[gauge group]] of the [[standard model of particle physics|standard model]], [Witten 81](#Witten81) also shows that for none of them does the remaining [[field (physics)|field]] content of the [[standard model of particle physics|standard model]] -- the [[fermions]] and the [[Higgs field]] -- come out correctly, due to lack of [[chiral fermions]].

Largely due to this result, the original pure Kaluza-Klein ansatz that starts with just pure [[Einstein gravity]] with no other [[field (physics)|fields]]) is nowadays regarded as non-viable route towards the [[standard model of particle physics]]. But one can further play with the idea and consider more flexible [[model (physics)|models]] that still exhibit the essence of KK-reduction in parts.

Notably the [[11-dimensional supergravity]] mentioned above contains more [[field (physics)|fields]] than just the field of [[gravity]]. Specifically it contains the [[supergravity C-field]] which is a [[higher gauge theory|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 [[model (physics)|models]] which come at least very close to the [[standard model of particle physics]] when considered on [[orbifold]] spacetimes. This we come to [below](#KKReductionOf11dSupergravity), see also at _[[duality between M-theory and type IIA string theory]]_.

### Reductions of type II supergravity
 {#ReductionsOfTypeIISupergravity}

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 physics 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 inhomogeneous 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

$$
  (g^KK, F) \mapsto S_{EH}(g_X) + S_{YM}(A) + S_{mod}(g_F, \mathcal{F})
$$

_does_ produce the previously missing [[positive number|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

* _[[supergravity and Calabi-Yau manifolds]]_,

* _[[string phenomenology]]_

* _[string theory FAQ -- What does it mean to say that string theory has a "landscape" of solutions?](string%20theory%20FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)_.

### KK-Reduction of 11d supergravity
 {##KKReductionOf11dSupergravity}

...[[duality between M-theory and type IIA string theory]]...

The lift of the [above](#ReductionsOfTypeIISupergravity) reduction of [[type II supergravity|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 cohomology|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 [[model (physics)|physics]], see for instance the _[[G2-MSSM]]_ [[model (physics)|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]]_:

[[!include F-theory compactifications -- table]]

### High codimension compactification of $D=10$ super Yang-Mills

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]].

### Cascades of KK-reductions
 {#CascadedOfKKReductionsFromHolographicBoundaries}

Apart from possibly producing [[phenomenology|phenomenologically]]
interesting [[model (physics)|models]] from [[UV-completion|UV-complete]] fundamental [[theory (physics)|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 [[supersymmetry|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 field theory|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 D=4 super Yang-Mills theory|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 field theory|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 [[topologically twisted D=4 super Yang-Mills theory|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 [[phenomenology|phenomenological]] viable [[model (physics)|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.

[[!include gauge theory from AdS-CFT -- table]]

### Infinite-temperature thermal field theory
 {#InfiniteTemperatureThermalFieldTheory}

With the formulation of [[thermal field theory]] via [[Wick rotation]] as a [[Euclidean field theory]] on a [[Riemannian manifold]] $X_3 \times S^1_\beta$ whose compact/periodic Euclidean time runs along a [[circle]] $S^1_\beta$ of [[length|circumference]] $\beta = 1/T$, the limit of infinite temperature corresponds to the [[limit of a sequence|limit]] $\beta \to 0$ in which this [[circle]] [[fiber]] shrinks away.

In terms of the thermal [[Euclidean field theory]] on $X_3 \times S^1_\beta$ therefore the [[infinite-temperature thermal field theory|infinite-temperature limit]] is given by Kaluaza-Klein-type dimensional reduction along this circle [[fiber]] to a 3-dimensional [[Euclidean field theory]] on $X_3$ ([Ginsparg 80](#Ginsparg80), [Appelquist-Pisarski 81](#AppelquistPisarski81), [Nadkarni 83](#Nadkarni83), [Jourjine 84](#Jourjine84), [Nadkarni 88](#Nadkarni88)).

As usual with [[KK-reduction]], some care must be exercised to ensure that the compactified theory is itself still a [[local field theory]]. Depending on how exactly one proceeds this may be subtle ([Landsman 89](#Landsman89)), but there exist robust approaches ([Reisz 92](#Reisz92), [Kajantie-Laine-Rummukainen-Shaposhnikov 96](#KajantieLaineRummukainenShaposhnikov96)).

## Related concepts

* [[spontaneously broken symmetry]]

* **Kaluza-Klein mechanism**

  * [[Kaluza-Klein monopole]]

  * [[double dimensional reduction]]

  * [[moduli stabilization]]

  * [[landscape of string theory vacua]]

    * [[flux compactification]]

    * [[supersymmetry and Calabi-Yau manifolds]]

    * [string theory FAQ -- What does it mean to say that string theory has a "landscape" of solutions?](string%20theory%20FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)

* [[hadron Kaluza-Klein theory]]

* [[Randall-Sundrum model]]

* KK-compactification of the [[time]]-direction, combined with [[Wick rotation]] (when it applies), yield [[thermal quantum field theory]]

* [[Bubble of nothing]]

[[!include KK-compactifications of M-theory -- table]]

## References
 {#References}

### General

Brief reviews:

* D. Bailin and A. Love, _Kaluza-Klein theories_, Rept. Prog. Phys. 50 (1987) 1087–1170 ([spire:253064](https://inspirehep.net/literature/253064), [doi:10.1088/0034-4885/50/9/001](https://iopscience.iop.org/article/10.1088/0034-4885/50/9/001))

* [[Matthias Blau]], chapter 43 of _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 1.4 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

Textbook accounts:

* [[Thomas Appelquist]], A. Chodos, [[Peter Freund]], *Modern Kaluza-Klein Theories*, Frontiers in Physics, Addison-Wesley (1987) [[spire:255996](https://inspirehep.net/literature/255996)]

* [[Paul Wesson]], _Space-Time-Matter: Modern Kaluza-Klein Theory_, World Scientific 1989 ([doi:10.1142/3889](https://doi.org/10.1142/3889))

* [[Paul Wesson]], James M. Overduin, _Principles of Space-Time-Matter: Cosmology, Particles and Waves in Five Dimensions_, World Scientific 2018 ([doi:10.1142/10871](https://doi.org/10.1142/10871))

Review specifically for [[supergravity]]:

* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Kaluza-Klein supergravity_, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 ([spire:229417](https://inspirehep.net/record/229417), <a href="https://doi.org/10.1016/0370-1573(86)90163-8">doi:10.1016/0370-1573(86)90163-8</a>)

and here specifically for [[sphere fiber bundles]]:

* [[Federico Bonetti]], [[Ruben Minasian]], Valentí Vall Camell, Peter Weck, *Consistent truncations from the geometry of sphere bundles* [[arXiv:2212.08068](https://arxiv.org/abs/2212.08068)]

Review for [[string theory]]:

* [[Michael R. Douglas]], [[Liam McAllister]], *Compactification of Superstring Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2310.20118](https://arxiv.org/abs/2310.20118)]

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

* {#Duff94} [[Michael Duff]], *Kaluza-Klein Theory in Perspective*, Proc. of the Symposium: *The Oskar Klein Centenary*, World Scientific, Singapore (1994) [[arXiv:hep-th/9410046](http://arxiv.org/abs/hep-th/9410046), [doi:10.1142/2816](https://doi.org/10.1142/2816)]

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

* {#Witten81} [[Edward Witten]], _Search for a realistic Kaluza-Klein theory_, Nuclear Physics B Volume 186, Issue 3, 10 August 1981, Pages 412-428 ([spire:10244](inspirehep.net/record/10244), <a href="https://doi.org/10.1016/0550-3213(81)90021-3">doi:10.1016/0550-3213(81)90021-3</a>)

A brief discussion aimed at mathematicians of the KK-reduction of [[gauge fields]] is in

* [[Daniel Freed]], _Lecture 5 of [[Five lectures on supersymmetry]]_.

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](http://www.math.ias.edu/qft))

In

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fre]], _[[Supergravity and Superstrings - A Geometric Perspective]]_

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](http://arxiv.org/abs/0705.1162))

A formalization of Kaluza-Klein compactification in [[perturbation theory]] formalized by [[factorization algebras]] with values in [[BV-complexes]] is in section 19 of

* {#Costello11} [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

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:

* {#Bleecker81} David Bleecker, _Gauge theory and variational principles_, Dover publications, 1981.

An expository talk of the above material from Bleecker can be found in this talk:

* {#Bartlett13} [[Bruce Bartlett]], _The geometry of force_, Sept 2013 ([pdf](https://math.sun.ac.za/bbartlett/assets/The%20geometry%20of%20force.pdf)).

Discussion of KK-compactification on [[SU(3)]] equipped with a [[Riemannian metric|metric]] that is [[invariant metric|left invariant]] but not right invariant, thereby inducing an extra term akin to a [[Higgs field]] (see also at *[[Connes-Lott model]]*):

* Joao Baptista, *Higher-dimensional routes to the Standard Model bosons* ([arXiv:2105.02899](https://arxiv.org/abs/2105.02899))

* Joao Baptista, *Higher-dimensional routes to the Standard Model fermions* ([arXiv:2105.02901](https://arxiv.org/abs/2105.02901))

[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]

### In Wick-rotated thermal field theory
 {#ReferencesInWickrotatedThermalFieldTheory}

Discussion of Kaluza-Klein-style dimensional reduction in [[Wick rotation|Wick rotated]] [[thermal field theory]] on a [[circle]] of compact/periodic Euclidean time $S^1_\beta$ of [[length|circumference]] $\beta = 1/T$ (see at _[[infinite-temperature thermal field theory]]_):

* {#Ginsparg80} [[Paul Ginsparg]], _First and second order phase transitions in gauge theories at finite temperature_, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 (<a href="https://doi.org/10.1016/0550-3213(80)90418-6">doi:10.1016/0550-3213(80)90418-6</a>)

* {#AppelquistPisarski81} Thomas Appelquist, [[Robert Pisarski]], _High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics_, Phys. Rev. D 23, 2305 (1981) ([doi:10.1103/PhysRevD.23.2305](https://doi.org/10.1103/PhysRevD.23.2305))

* {#Nadkarni83} Sudhir Nadkarni, _Dimensional reduction in finite-temperature quantum chromodynamics_, Phys. Rev. D 27, 917 (1983) ([doi:10.1103/PhysRevD.27.917](https://doi.org/10.1103/PhysRevD.27.917))

* {#Nadkarni88} Sudhir Nadkarni, _Dimensional reduction in finite-temperature quantum chromodynamics. II_, Phys. Rev. D 38, 3287 (1988) ([doi:10.1103/PhysRevD.38.3287](https://doi.org/10.1103/PhysRevD.38.3287))

* {#Jourjine84} Alexander N Jourjine, _Quantum field theory in the infinite temperature limit_, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 (<a href="https://doi.org/10.1016/0003-4916(84)90003-4">doi:10.1016/0003-4916(84)90003-4</a>)

* {#Landsman89} [[Klaas Landsman]], _Limitations to dimensional reduction at high temperature_, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 (<a href="https://doi.org/10.1016/0550-3213(89)90424-0">doi:10.1016/0550-3213(89)90424-0</a>)

* {#Reisz92} T. Reisz, _Realization of dimensional reduction at high temperature_, Z. Phys. C - Particles and Fields (1992) 53: 169 ([doi:10.1007/BF01483886](https://doi.org/10.1007/BF01483886))

* {#Braaten95} Eric Braaten, _Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories_, Phys. Rev. Lett. 74, 2164 (1995) ([doi:10.1103/PhysRevLett.74.2164](https://doi.org/10.1103/PhysRevLett.74.2164))

* {#KajantieLaineRummukainenShaposhnikov96} K. Kajantie, M. Laine, K. Rummukainen, M. Shaposhnikov, _Generic rules for high temperature dimensional reduction and their application to the standard model_, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 (<a href="https://doi.org/10.1016/0550-3213(95)00549-8">doi:10.1016/0550-3213(95)00549-8</a>)

and specifically with an eye to discussion of the [[quark-gluon plasma]] in

* {#BlaizotIancuRebhan03} Jean-Paul Blaizot, Edmond Iancu, [[Anton Rebhan]], section 2.2.4 of _Thermodynamics of the high temperature quark gluon plasma_, Quark–Gluon Plasma 3, pp. 60-122 (2004) ([arXiv:hep-ph/0303185](https://arxiv.org/abs/hep-ph/0303185), [spire:615570](http://inspirehep.net/record/615570))

* {#Blaizot04} Jean-Paul Blaizot, around p. 17 of _Thermodynamics of the high temperature Quark-Gluon Plasma_, AIP Conf. Proc. 739, 63-96 (2004) ([doi:10.1063/1.1843592](https://doi.org/10.1063/1.1843592))

### On higher bundles

Discussion in [[higher differential geometry]] of Kaluza-Klein compactification along [[principal ∞-bundles]], relating to [[double field theory]], [[T-folds]], [[non-abelian T-duality]], [[type II geometry]], [[exceptional geometry]]:

* [[Luigi Alfonsi]], *Global Double Field Theory is Higher Kaluza-Klein Theory*, Fortschr. Phys. **68** (2020) 2000010 [[arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010)]

* [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_ ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969))

[[!redirects Kaluza-Klein theory]]

[[!redirects dimensional reduction]]
[[!redirects moduli stabilization problem]]

[[!redirects Kaluza-Klein reduction]]
[[!redirects Kaluza-Klein reductions]]

[[!redirects Kaluza-Klein compactification]]
[[!redirects Kaluza-Klein compactifications]]

[[!redirects KK-reduction]]
[[!redirects KK-reductions]]

[[!redirects KK reduction]]
[[!redirects KK reductions]]

[[!redirects KK-compactification]]
[[!redirects KK-compactifications]]

[[!redirects KK compactification]]
[[!redirects KK compactifications]]

[[!redirects KK-mechanism]]
[[!redirects KK-mechanisms]]

[[!redirects KK mechanism]]
[[!redirects KK mechanisms]]

[[!redirects Kaluza-Klein]]


Kaluza-Klein monopole

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

In the context of [[Kaluza-Klein theory]] where an [[electromagnetic field]] in [[Einstein-Maxwell theory]] in [[dimension]] $d$ is modeled by a configuration of pure Einstein [[gravity]] in dimension $d+1$, a _Kaluza-Klein monopole_ is a configuration of gravity in dimension $d+1$ which in dimension $d$ looks like a magnetic [[monopole]] ([Sorkin 83](#Sorkin83), [Gross-Perry 83](#GrossPerry83)).

### In supergravity

This situation is of particular interest in the reduction of [[11-dimensional supergravity]] (or [[M-theory]], where one also speaks of the [[M-brane|MK6-brane]]) where the Kaluza-Klein magnetic monopole charge is interpreted as [[D6-brane]] charge under [[duality between M-theory and type IIA string theory]].

The _Kaluza-Klein monopole_ ([Han-Koh 85](#HanKoh85)) is one type of solution of the [[equations of motion]] of [[11-dimensional supergravity]]. It is given by the [[product]] $N_4\times \mathbb{R}^{11-5,1}$ of Euclidean [[Taub-NUT spacetime]] with [[Minkowski spacetime]]. Upon [[Kaluza-Klein compactification]] this looks like a [[monopole]], whence the name. (For discussion as an [[ADE-singularity]] see [IMSY 98, section 9](#IMSY98), [Asano 00, section 3](#Asano00).)

Upon [[KK-compactification]] on a 6-dimensional [[fiber]], with the 11d KK-monopole / [[D6-brane]] completely [[wrapped brane|wrapping]] the fiber, the KK-monopole in [[11d supergravity]] becomes the KK-monopole in [[5d supergravity]]. Further compactifying on a circle leads to a [[black hole]] in 4d, incarnated as a D0/D6 bound state (e.g. [Nelson 93](#Nelson93)).

[[!include KK-monopole geometries -- table]]

## Properties

(For the moment the following is all about the KK-monopoles in [[11d supergravity]]/[[M-theory]].)

### Far-horizon geometry
 {#NearHorizonGeometry}

We discuss the [[far horizon geometry]] of coincident MK6-branes.

The [[metric tensor]] of $N$ coincident [[KK-monopoles]] in [[11-dimensional supergravity]] in the limit that $\ell_{th} \coloneqq N \ell_P \to 0$ is

\[
  \label{SmallNMK6}
  g_{MK6}
    \;=\;
  g_{\mathbb{R}^{6,1}}
  +
  (d y)^2
  +
  y^2
  \big(
    (d \theta)^2
    +
    (\sin \theta)^2 (d \varphi)^2
    +
    (\cos \theta)^2 (d \phi)^2
  \big)
\]

subject to the identification

\[
  \label{OrbifoldIdentificationForKKMonopole}
  (\varphi, \phi)
  \;\sim\;
  (\varphi, \phi) + (2\pi/N ,2\pi/N)
  \,.
\]

This is equation (47) in [IMSY 98](#IMSY98), which applies subject to the condition

$$
  U/\left(\frac{N}{g^{2/3}_{YM}}\right)
  \;=\;
  U/\left(\frac{N}{(2\pi)^{4/3} \ell_P}\right)
  \;\gg\;
  1
$$

from a few lines above. Inserting this condition into the definition $y^2 \coloneqq 2 N \ell^3_P U $ right above (47) shows that

$$
  \begin{aligned}
  y^2
  & =
  2 N \ell^3_P U
  \\
  & =
  2(2\pi)^{-4/3} N^2 \ell_P^2
  \;
  \underset{
    \gg 1
  }{
  \underbrace{
     \left(U/\left(\frac{N}{ (2 \pi)^{4/3} \ell_P}\right)\right)
  }}
  \end{aligned}
$$

hence that the distance $y$ from the locus of the MK6-brane is large in units of

$$
  \ell_{th} \;=\; \sqrt{2} (2\pi)^{-2/3} N \ell_P
  \,.
$$

The identification (eq:OrbifoldIdentificationForKKMonopole) means that this is the [[orbifold]] [[metric cone]] $\mathbb{R}^{6,1} \times \left( \mathbb{R}^4/(\mathbb{Z}_N)\right)$, hence an [[ADE classification|
A-type]] [[ADE-singularity]]. To make this more explicit, introduce the complex coordinates

$$
  v \;\coloneqq\; y \, e^{i \varphi} \sin \theta
  \;\;\;
  w \;\coloneqq\; y \, e^{i \phi} \cos \theta
$$

on $\mathbb{R}^4 \simeq \mathbb{C}^2$, in terms of which (eq:SmallNMK6) becomes

$$
  g_{MK6}
  \;\coloneqq\;
  d v d \overline v
  +
  d w d \overline w
$$

and which exhibit the identification (eq:OrbifoldIdentificationForKKMonopole) as indeed that of the [[ADE classification|A-type]] $\mathbb{Z}_N$-action ([Asano 00, around (18)](#Asano00)).

### Relation to the D6-brane in type IIA string theory
 {#RelationToTheD6Brane}

Under the relation between [[M-theory]] and [[type IIA superstring theory]] an [[ADE orbifold]] of the 11d KK-monopole  corresponds to [[D6-branes]] combined with [[O6-planes]] ([Townsend 95, p. 6](#Townsend95), [Atiyah-Witten 01, p. 17-18](#AtiyahWitten01) see also e.g. [Berglund-Brandhuber 02, around p. 15](#BerglundBrandhuber02)).

By ([Townsend 95, (1)](#Townsend95), [Sen 97c (1)-(4)](#Sen97c)) the 11d [[spacetime]] describing the KK-monopole lift of a plain single D6 brane is   $\mathbb{R}^{6,1}\times \mathbb{R}^3\times S^1$ with [[metric tensor]] away from the origin of the $\mathbb{R}^3$-factor (which is the locus of the lifted D6/monopole) being

$$
  d s_{11}^2 =
  - d t^2
  + d s_{\mathbb{R}^6}^2
  + (1+\mu/r) d s_{\mathbb{R}^3}^2
  + (1+ \mu/r)^{-1} (d x^{11} - A_i d x^i)^2
  \,,
$$

where

* $A = A_i d x^i$ is any 3-form on $\mathbb{R}^3$ satisfying $d_{\mathbb{R}^3} A = \star d (1-\mu/r)$;

* $r$ denotes the distance in $\mathbb{R}^3$ from the origin.

* $\mu$ is the [[charge]] of the monopole.

Away from $\{0\} \in \mathbb{R}^3$ this gives a [[circle bundle]] with [[first Chern class]] measured by the integral of $R_0 \coloneqq d A$ (the [[RR-field]] of the [[D0-brane]]) over any [[sphere]] surrounding the singular locus in $\mathbb{R}^3 - \{0\}$. By the [[electric-magnetic duality]] between [[D0-branes]] and [[D6-branes]] (due to the [[self-dual higher gauge theory|self-duality]] of the [[RR-field]]) this means, from the 10-dimensional perspective, that at $0 \in \mathbb{R}^3$ a [[D6-brane]] is located.

> graphics grabbed from [Acharya-Gukov 04](#AcharyaGukov04)

Notice that the circle bundle away from its degeneration locus as a bundle over $S^2 \hookrightarrow \mathbb{R}^3 -\{0\}$ is necessarily of the form $S^3 \to S^2$, a multiple of the [[complex Hopf fibration]] (see also [Atiyah-Maldacena-Vafa 00, p. 10](#AtiyahMaldacenaVafa00)).

Discussion as an [[ADE singularity|A-type ADE singularity]] is in ([Sen 97c, section 2](#Sen97c)).  Generalization to [[ADE singularity|D-type singularities]] and hence D6-branes in [[orientifolds]] is in ([Sen 97c ,section 3](#Sen97c)).
Discussion as the [[fixed point]] of the [[circle group]]-[[action]] on the M-theory circle fibers is in ([Townsend 95, p. 6](#Townsend95), [Atiyah-Witten 01, pages 17-18](#AtiyahWitten01)). Witten emphasizes that it is important that the location of the D6 is not just a [[cyclic group]] [[orbifold]] singularity but really a [[circle group]]-[[action]] [[fixed point]] [[conical singularity]]:

> {#Witten01OnConicalSingularity} Chiral fermions arise when the locus of A−D−E singularities passes through isolated points at which X has an isolated conical singularity that is not just an orbifold singularity ([Witten 01, p. 2](#Witten01)).

Discussion dealing carefully with the perspective where the locus of the [[D6-brane]] is not remived: [Gaillard & Schmude 09](#GaillardSchmude09).

For more on this see at _[D6-brane -- Relation to other branes](D6-brane#RelationToOtherBranes)_ and at _[[M-theory lift of gauge enhancement on D6-branes]]_.

### Relation to the D7-brane in type IIB string theory/F-theory

Under further [[T-duality]] to [[type IIB superstring theory]]/[[F-theory]] these D6-branes  become the [[D7-branes]].

[[!include F-branes -- table]]

### Other Brane charges (?)

In ([Hull 97](#Hull97)) it was argued that the KK-monopole in [[11-dimensional supergravity]] is the object which carries the 6-form charge [[Poincaré duality|Poincaré dual]] to the time-component of the 5-form charge of the [[M5-brane]] as appearing in the [[M-theory super Lie algebra]] via

$$
  \wedge^5 (\mathbb{R}^{10,1})^\ast
  \simeq
  \wedge^5 (\mathbb{R}^{10})^\ast \oplus \wedge^6 \mathbb{R}^{10}
  \,.
$$

(The same kind of relation identifies the time-component of the [[M2-brane]] charge with the charge of the [[M9-brane]], see there.)

## Related concepts

* [[M-brane]]

* [[Taub-NUT spacetime]]

* [[gravitational instanton]]

## References

### In 5d gravity

Original articles:

* {#Sorkin83} [[Rafael Sorkin]], _Kaluza-Klein monopole_, Phys. Rev. Lett. 51 (1983) 87 ([doi:10.1103/PhysRevLett.51.87](https://doi.org/10.1103/PhysRevLett.51.87))

* {#GrossPerry83} [[David Gross]], [[Malcolm Perry]], _Magnetic Monopoles in Kaluza-Klein Theories_,  Nucl. Phys. B226 (1983) 29 (doi:10.1016/0550-3213(83)90462-5)

* {#Ruback86} P. J. Ruback, _The motion of Kaluza-Klein monopoles_, Comm. Math. Phys. Volume 107, Number 1 (1986), 93-102 ([euclid:1104115933](https://projecteuclid.org/euclid.cmp/1104115933))

* A. L. Cavalcanti de Oliveira, E. R. Bezerra de Mello, _Kaluza-Klein Magnetic Monopole in Five-Dimensional Global Monopole Spacetime_, Class. Quant. Grav. 21 (2004) 1685-1694 ([arXiv:hep-th/0309189](http://arxiv.org/abs/hep-th/0309189))

Review includes

* Emre Sakarya, _Kaluza-Klein monopole_, 2007 ([pdf](https://inspirehep.net/record/875942/files/875942.pdf))

Discussion of [[topological T-duality]] for KK-monopoles is in

* {#Pande06} Ashwin S. Pande, _Topological T-duality and Kaluza-Klein Monopoles_, Adv. Theor. Math. Phys., 12, (2007), pp 185-215 ([arXiv:math-ph/0612034](https://arxiv.org/abs/math-ph/0612034))

### In 11d supergravity

#### Original articles

* {#HanKoh85} Seung Kee Han, I. G. Koh, _$N = 4$ Remaining Supersymmetry in Kaluza-Klein Monopole Background in D=11 Supergravity Theory_, Phys.Rev. D31 (1985) 2503, in [[Michael Duff]] (ed.), _[[The World in Eleven Dimensions]]_ 57-60 ([doi:10.1103/PhysRevD.31.2503](https://doi.org/10.1103/PhysRevD.31.2503), [spire:204521](http://inspirehep.net/record/204521))

* {#Townsend95} [[Paul Townsend]], _The eleven-dimensional supermembrane revisited_, Phys. Lett. B **350** (1995) 184-187 [[arXiv:hep-th/9501068](http://arxiv.org/abs/hep-th/9501068)]

* {#Hull97} [[Chris Hull]], _Gravitational Duality, Branes and Charges_, Nucl.Phys. B509 (1998) 216-251 ([arXiv:hep-th/9705162](http://arxiv.org/abs/hep-th/9705162))

* {#Sen97a} [[Ashoke Sen]], _Kaluza-Klein Dyons in String Theory_, Phys. Rev. Lett. 79:1619-1621, 1997 ([arXiv:hep-th/9705212](http://arxiv.org/abs/hep-th/9705212))

* {#Sen97b} [[Ashoke Sen]], _Dynamics of Multiple Kaluza-Klein Monopoles in M- and String Theory_, Adv. Theor. Math. Phys. 1:115-126, 1998 ([arXiv:hep-th/9707042](https://arxiv.org/abs/hep-th/9707042))

* {#Sen97c} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

* {#IMSY98} Nissan Itzhaki, [[Juan Maldacena]], Jacob Sonnenschein, Shimon Yankielowicz, _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 1998 ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

* {#Asano00} Masako Asano, _Compactification and Identification of Branes in the Kaluza-Klein monopole backgrounds_ ([arXiv:hep-th/0003241](https://arxiv.org/abs/hep-th/0003241))

* {#AtiyahMaldacenaVafa00} [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], _An M-theory Flop as a Large N Duality_, J.Math.Phys.42:3209-3220,2001 ([arXiv:hep-th/0011256](http://arxiv.org/abs/hep-th/0011256))

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

* {#Witten01} [[Edward Witten]], _Anomaly Cancellation On Manifolds Of $G_2$ Holonomy_ ([arXiv:hep-th/0108165](http://arxiv.org/abs/hep-th/0108165))

* {#BerglundBrandhuber02} Per Berglund, Andreas Brandhuber, _Matter From $G_2$ Manifolds_, Nucl.Phys. B641 (2002) 351-375 ([arXivLhep-th/0205184](http://arxiv.org/abs/hep-th/0205184))

* {#GaillardSchmude09} [[Jérôme Gaillard]], [[Johannes Schmude]], _The lift of type IIA supergravity with D6 sources: M-theory with torsion_, JHEP 1002:032,2010 ([arXiv:0908.0305](http://arxiv.org/abs/0908.0305))

Discussion for the [[massive type IIA string theory|massive]] case, [[KK-reduction|reducing]] to the [[D6-brane]] in [[massive type IIA string theory]]:

* [[Eric Bergshoeff]], [[Eduardo Eyras]], [[Yolanda Lozano]], *The massive Kaluza-Klein monopole*, Physics Letters B
**430** 1–2 (1998) 77-86 [doi:10.1016/S0370-2693(98)00501-2, [arXiv:hep-th/9802199](https://arxiv.org/abs/hep-th/9802199)]

Discussion in terms of [[G-structures]]:

* [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Adolfo Guarino, _KK-monopoles and $G$-structures in M-theory/type IIA reductions_, JHEP 1502 (2015) 096 ([arXiv:1411.0575](https://arxiv.org/abs/1411.0575))

Discussion of a [[BFSS matrix model|BFSS-like]] [[matrix model]] for [[MK6-branes]]:

* [[Amihay Hanany]], Gilad Lifschytz, _M(atrix) Theory on $T^6$ and a m(atrix) Theory Description of KK Monopoles_, Nucl. Phys. B519:195-213, 1998 ([arXiv:hep-th/9708037](https://arxiv.org/abs/hep-th/9708037))

Relation to [[massive type IIA string theory]]:

* Eduardo Eyras, [[Yolanda Lozano]], *The Kaluza-Klein Monopole in a Massive IIA Background*, Nucl. Phys. B**546** (1999) 197-218 [[arXiv:hep-th/9812188](https://arxiv.org/abs/hep-th/9812188), doi:10.1016/S0550-3213(99)00098-X]

#### Review

* [[Clifford Johnson]], section 10.5 of _D-brane primer_ ([arXiv:hep-th/0007170](http://arxiv.org/abs/hep-th/0007170))

* [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 333 of _String Theory and M-Theory: A Modern Introduction_, 2007

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 6.3.3 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], p. 45 of: _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys. Rept. **392** (2004) 121-189 [[arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191)]

### Relation to black holes

Relation to [[black holes in string theory]]

* {#Nelson93} William Nelson, _Kaluza-Klein Black Holes in String Theory_, Phys. Rev. D49:5302-5306, 1994 ([arXiv:hep-th/9312058](http://arxiv.org/abs/hep-th/9312058))

[[!redirects Kaluza-Klein monopoles]]

[[!redirects KK monopole]]
[[!redirects KK monopoles]]

[[!redirects KK-monopole]]
[[!redirects KK-monopoles]]

[[!redirects MK6]]
[[!redirects MK6-brane]]
[[!redirects MK6-branes]]