Contents

# Contents

## Idea

The Maxwell equations were originally formulated on 4-dimensional Minkowski spacetime, but in their formulation where the Faraday tensor is regarded as a differential 2-form $F$ with Hodge dual $\star F$ and de Rham differentials

$R e \array{ d F & = 0 \\ \star d F & = J }$

they immediately generalize to make sense on Minkowski spacetime of any dimension $D$, and in fact to any pseudo-Riemannian manifold. The case of $D =5$-Maxwell theory (hence D=5 Yang-Mills theory with U(1)-gauge group) plays a role mostly as a subsector of D=5 Einstein-Maxwell theory and/or of D=5 super Yang-Mills theory, as well as a pre-image under Kaluza-Klein compactification of massive Yang-Mills theory in 4-dimensions.

## Properties

### Relation between 5d Maxwell theory and self-dual 3-forms in 6d

Consider 5d- and 6d-dimensional Minkowski spacetime equipped with global orthonormal coordinate charts $\{x^\kappa\}$, $\{x^\alpha\}$, respectively, adapated to an isometric embedding

$\array{ & \mathbb{R}^{4,1} &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}& \mathbb{R}^{5,1} \\ \kappa = & 0, 1, 2, 3, 4\phantom{,} \\ \alpha = & 0, 1, 2, 3, 4, && 5 }$

With this notation, the pullback of differential forms along this embedding is notationally implicit.

Now any differential 3-form $H_3$ on $\mathbb{R}^{5,1}$ decomposes as

(1)$H_3 \;=\; \widehat{F} \wedge d x^{5} + \widehat{H}$

for unique differential forms of the form

$\widehat F \;=\; \tfrac{1}{2}\hat F_{\kappa_1 \kappa_2}(x^\kappa, x^5) d x^{\kappa_1} \wedge d x^{\kappa_2}$

and

$\widehat{H} \;=\; \tfrac{1}{3!} \widehat{H}_{\kappa_1 \kappa_2 \kappa_3}(x^\kappa, x^5) d x^{\kappa_1} \wedge d x^{\kappa_2} \wedge d x^{\kappa_3} \,.$

In the case that $H_3$ has vanishing Lie derivative along the $x^5$-direction,

(2)$\mathcal{L}_5 H_3 \;=\; 0$

then also these components forms do not depend on $x^5$ are actualls pullbacks of differential forms on $\mathbb{R}^{4,1}$.

In terms of this decomposition, the 6d Hodge dual of $H_3$ is equivalently given by the 5d Hodge duals of these components as (best seen by the relation to Hodge pairing according to this Prop.)

(3)$\star_6 H_3 \;=\; \big( \star_5 \widehat{H}\big) \wedge d x^{5} - \star_5 \widehat{F}$

Since the Hodge star operator squares to unity in the special case that it is applied to differential 3-forms on 6d Minkowski spacetime (by this Prop.)

$\star_6 \star_6 H_3 \;=\; + H_3$

we may ask for $H_3$ to he Hodge self-dual. By (3) this means equivalently that its 5d components are 5d Hodge duals of each other:

$\big( H_3 \;=\; \star_{6} H_3 \big) \;\;\; \overset{ H_3 = \widehat{F} \wedge d x^5 + \widehat{H} }{ \Leftrightarrow } \;\;\; \big( \widehat{H} = \star_5 \widehat{F} \big) \,.$

It follows that if there is no $x^5$-dependence (2) then the condition that $H_3$ be a closed and self-dual 3-form is equivalent to its 5d components $\widehat{F}$ ($\widehat{H}$) being the (dual) field strength/Faraday tensor satisfying the Maxwell equations of D=5 Maxwell theory (without source current):

$\underset{ \color{blue} { {\phantom{A}} \atop {\text{D=6 self-dual 3-form theory}} } }{ \left. \array{ & d H_3 = 0 \\ & \star_6 H_3 = H_3 } \right\} } \;\;\; \overset{ {\mathcal{L}_{5} H_3 = 0} \atop {H_3 = \widehat{F}\wedge d x^5 + \cdots} }{ \Leftrightarrow } \;\;\; \underset{ \color{blue} { {\phantom{A}} \atop \text{D=5 Maxwell theory} } }{ \left\{ \array{ & d \widehat{F} = 0 \\ & d \star_5 \widehat{F} = 0 } \right. }$

This may be summarized as saying that the massless part of the Kaluza-Klein reduction of self-dual 3-form theory from 6d to 5d is D=5 Maxwell theory.

Essentially this relation underlies the formulation of the M5-brane via the Perry-Schwarz Lagrangian.

### Relation between 5d Maxwell theory and massive vector mesons in 4d

In an abelian version of hadron Kaluza-Klein theory, hence the way the Skyrmion-model, coupled to the tower of vector mesons, appears in holographic QCD, the Kaluza-Klein reduction of D=5 Maxwell theory to 4d yields a theory of a tower of massive vector particles which may be interpreted as vector mesons (Sutcliffe 10, Sec. 3):

#### Flat space model

Following Sutcliffe 10, Sec. 3, we first consider a toy example where the 5d metric is that of flat Minkowski spacetime.

Consider 4d- and 5d-dimensional Minkowski spacetime equipped with global orthonormal coordinate charts $\{x^\mu\}$, $\{x^\kappa\}$, respectively, adapated to an isometric embedding

$\array{ & \mathbb{R}^{3,1} &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}& \mathbb{R}^{4,1} \\ \mu = & 0, 1, 2, 3\; \\ \kappa = & 0, 1, 2, 3, && 4 }$

With this notation, the pullback of differential forms along this embedding is notationally implicit.

Assuming that the 5d vector potential $\widehat A \in \Omega^1(\mathbb{R}^{4,1})$ is square-integrable along the $x^4$-drection, we may expand it in a linear basis $\{h_n\}_{n \in \mathbb{N}}$ of Hermite functions as a sequence of vector potentials $V^{(n)} \in \Omega^1(\mathbb{R}^{4,1})$ that do not depend on $x^4$ anymore.

(4)$\widehat{A} \big( \{x^\mu\}, x^4 \big) \;=\; \underset{n}{\sum} h_n(x^4) \, V^{(n)}(\{x^\mu\})$

We assume now (not following Sutcliffe 10 in this) that $V^{(n)}_4 = 0$, hence that the $V^{(n)}$ are pulled back from 4d spacetime, and we take these 4d vector potentials to be in 4d Lorenz gauge

(5)$d \star_4 V^{(n)} \;=\; 0 \,.$

By the differential recursion relation of Hermite functions (see there)

(6)$\frac{d}{ d (x^4)^n} h_n(x^4) \;=\; \sqrt{ \tfrac {n} {2} } h_{n-1}(x^4) - \sqrt{ \tfrac {n-1} {2} } h_{n+1}(x^4)$

it follows that the corresponding field strength is expanded as

(7)$d \widehat{A} \;=\; \underset{n}{\sum} h_n \Big( d V^{(n)} + \big( \sqrt{\tfrac{n+1}{2}} V^{(n+1)} - \sqrt{\tfrac{n}{2}} V^{(n-1)} \big) \wedge d x^4 \Big) \,.$

The Hodge dual of that is:

$\star_5 d \widehat{A} \;=\; \underset{n}{\sum} h_n \Big( \star_4 d V^{(n)} \wedge d x^4 + \sqrt{\tfrac{n}{2}} \star_4 V^{(n-1)} - \sqrt{\tfrac{n+1}{2}} \star_{4} V^{(n+1)} \Big) \,.$

(Notice the sign reversal of the last two terms. A quick way to see this is to notice the Hodge pairing-relation (see there) and the fact that commuting the 3-form $\star_4 V^{(n)}$ past the 1-form $d x^4$ produces a sign.)

Now using once more the differential recursion relation (6) of the Hermite functions, the de Rham differential of this Hode dual field strength is

(8)\begin{aligned} & d \star_5 d \widehat{A} \\ & \begin{aligned} = \underset{n}{\sum} h_n \Big( & d \star_4 d V^{(n)} \wedge d x^4 + \sqrt{\tfrac{n}{2}} \underset{ = 0 }{ \underbrace{ d \star_4 V^{(n-1)} } } - \sqrt{\tfrac{n+1}{2}} \underset{ = 0 }{ \underbrace{ d \star_{4} V^{(n+1)} } } \\ & + \;\; \sqrt{\tfrac{n+1}{2}} \big( \star_4 d V^{(n+1)} \wedge d x^4 + \sqrt{\tfrac{n+1}{2}} \star_4 V^{(n)} - \sqrt{\tfrac{n+2}{2}} \star_{4} V^{(n+2)} \big) \wedge d x^4 \\ & - \;\; \sqrt{\tfrac{\;n\;}{2}} \big( \star_4 d V^{(n-1)} \wedge d x^4 + \sqrt{\tfrac{n-1}{2}} \star_4 V^{(n-2)} - \sqrt{\tfrac{\;n\;}{2}} \star_{4} V^{(n)} \big) \wedge d x^4 \Big) \end{aligned} \\ & = \underset{n}{\sum} h_n \Big( d \star_4 d V^{(n)} \wedge d x^4 + \left( n + \tfrac{1}{2} \right) \star_{4} V^{(n)} - \sqrt{\tfrac{n+2}{2}\tfrac{n+1}{2}} \star_{4} V^{(n+2)} - \sqrt{\tfrac{n}{2}\tfrac{n-1}{2}} \star_4 V^{(n-2)} \Big) \wedge d x^4 \end{aligned}

Here the terms over the braces vanish because we have chosen Lorenz gauge in (5).

Hence under the expansion (4) the 5d Maxwell equations $d \star_5 d \widehat A = 0$ become equivalently the following system of coupled equations of massive Yang-Mills theory in 4d (using, with this Prop. that the Hodge star operator on 3-forms in 4d squares to unity):

$\star_4 d \star_4 d V^{(n)} \;=\; - \left( n + \tfrac{1}{2} \right) V^{(n)} + \sqrt{\tfrac{n+2}{2}\tfrac{n+1}{2}} V^{(n+2)} + \sqrt{\tfrac{n}{2}\tfrac{n-1}{2}} V^{(n-2)} \Big)$

Truncating this to the first term yields the Klein-Gordon equation for a field of mass $1/\sqrt{2}$ (by this Prop.):

$\partial^\mu \partial_\mu V^{(1)} \;=\; \tfrac{1}{2} V^{(1)} \,.$

More generally, truncating at some higher number $N$ of modes, one may find a linear transformation among the fields $V^{(n)}$ for $1 \leq n \leq N$ which diagonalizes (8) to obtain a collection of $N$ Klein-Gordon equations for $N$ massive fields.

#### Sakai-Sugimoto model

We discuss 5d Maxwell theory on the 5d spacetime that appears in the Sakai-Sugimoto model (from D4/D8-brane intersections), following Bolognesi-Sutcliffe 13, Sutcliffe 15.

Consider 4d- and 5d-dimensional smooth manifolds diffeomorphic to Cartesian space and equipped with global orthonormal coordinate charts $\{\{x^\mu\}, z\}$, , respectively, adapated to an isometric embedding

(9)$\array{ & \mathbb{R}^{3,1} &\overset{\;\;\;\iota_5\;\;\;}{\hookrightarrow}& \mathbb{R}^{4,1} \\ \mu = & 0, 1, 2, 3\; \\ \kappa = & 0, 1, 2, 3, && z }$
###### Definition

(Sakai-Sugimoto metric)

On the 5d manifold (9) the Sakai-Sugimito metric is the metric tensor

(10)$g \;\coloneqq\; H^{+1} \, \eta_{\mu \nu} d x^\mu \otimes d x^\nu + H^{-1} d z \otimes d z$

where

• $(\eta_{\mu \nu}) = diag(-1,+1,+1,+1)$ is the metric of 4d Minkowski spacetime,

• $H$ is the smooth function given by

$H(\{x^\mu\}, z) \;\coloneqq\; H(z) \;\coloneqq\; \left( 1 + \frac{z^2}{L^2} \right)^{2/3}$

for some real number $L$.

This metric is highlighted in this form in (Bolognesi-Sutcliffe 13, (2.1), (2.2) Sutcliffe 15, (39), (40)).

###### Proposition

(Sakai-Suimoto equation)

Let $A \in \Omega^1(\mathbb{R}^{4,1})$ be a differential 1-form on the spacetime (9) with vanishing $z$-component and in 4d Lorenz gauge:

(11)$A_z = 0$
(12)$\partial_\mu \eta^{\mu \nu} A_\nu = 0 \,.$

Then the 5d vacuum Maxwell equations for the vector potential $A$ with respect to the metric tensor (10) are equivalent to

(13)\begin{aligned} & \star d \star d A \;=\; 0 \\ \Leftrightarrow \;\;\; & \eta^{\mu \nu}\partial_\mu \partial_\nu A \;+\; H^{1/2} \partial_z \left( H^{3/2} \partial_z A \right) \;=\; 0 \end{aligned} \,.

This equation is highlighted in Bolognesi-Sutcliffe 13, (4.14), Sutcliffe 15, (48).

###### Proof

Here is a somewhat lengthy computation:

First notice the general component formula for the operator $\star d \star d$ applied to 1-forms, akin to, but different from, the component formula for the Laplace-Beltrami operator applied to 0-forms/functions (see there)

\begin{aligned} \star d \star d A & = \star d \star (\partial_{j_1} f) A_{j_2} d x^{j_1} \wedge d x^{j_2} \\ & = \star d \left( \tfrac{1}{ 2 \color{green} (D-2)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \, \epsilon_{ i_1 i_2 {\color{green} k_3 \cdots k_{D} } } d x^{ \color{green} k_3 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \star \partial_{ \color{magenta} k_2} \left( \tfrac{1}{ 2 \color{green} (D-2)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2} ) \, \epsilon_{ i_1 i_2 {\color{green} k_3 \cdots k_{D} } } d x^{ \color{magenta} k_2 } \wedge d x^{ \color{green} k_3 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \underset{ = \det\big( (g_{i j})^{-1} \big) \tfrac{1}{2} \delta^{ \color{magenta} k_1 k_2 }_{i_1 i_2} }{ \underbrace{ \tfrac{ (-1)^{D-1} }{ { \color{orange} (D-1)! } 2 { \color{green} (D-2)! } } \epsilon_{ l_1 { \color{orange} l_2 \cdots l_{D} } } g^{ { \color{orange} l_1 } { \color{magenta} k_1 } } g^{ { \color{orange} l_2 } { \color{magenta} k_2 } } g^{ { \color{orange} l_3 } { \color{green} k_3 } } \cdots g^{ { \color{orange} l_D} { \color{green} k_D } } \epsilon_{ i_1 i_2 {\color{green} k_3 \cdots k_{D} } } } } \, \partial_{ \color{magenta} k_2 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \right) \, g_{ {\color{magenta} k_1} r } d x^{ \color{magenta} r } \\ & = \frac{ (-1)^{D-1} }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \tfrac{1}{2} \delta^{ \color{magenta} k_1 k_2 }_{i_1 i_2} \partial_{ \color{magenta} k_2 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \right) \, g_{ {\color{magenta} k_1 } r } d x^{ r } \end{aligned}

Now specialize this to the the case at hand, where

1. $A = A_\mu d x^{\mu} A_z d^z$

2. $A_z = 0$

3. $\partial_\mu \eta^{\mu \nu} A_\nu = 0$

4. $g_{\mu \nu} = H \eta_{\mu \nu}$

5. $\partial_\mu H = 0$.

We get:

\begin{aligned} & \frac{ (-1)^{D-1} }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \tfrac{1}{2} \delta^{ \color{magenta} k_1 k_2 }_{i_1 i_2} \partial_{ \color{magenta} k_2 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \right) \, g_{ {\color{magenta} k_1 } r } d x^{ r } \\ & = \frac{ (-1)^{D-1} }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \tfrac{1}{2} \delta^{ \color{magenta} \mu_1 \mu_2 }_{i_1 i_2} \partial_{ \color{magenta} \mu_2 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \right) \, g_{ {\color{magenta} \mu_1 } r } d x^{ r } \\ & \phantom{=}\; + \frac{ (-1)^{D-1} }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \tfrac{1}{2} \delta^{ \color{magenta} \mu_1 z }_{i_1 i_2} \partial_{ \color{magenta} z } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \right) \, g_{ {\color{magenta} \mu_1 } r } d x^{ r } \\ & \phantom{=}\; + \frac{ (-1)^{D-1} }{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \tfrac{1}{2} \delta^{ \color{magenta} z \mu_2 }_{i_1 i_2} \partial_{ \color{magenta} \mu_2 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{ i_1 j_1 } g^{ i_2 j_2 } (\partial_{j_1} A_{j_2}) \right) \, g_{ {\color{magenta} z } r } d x^{ r } \\ & = \tfrac{ (-1)^{D-1} }{2} H^{-2} \big( \underset{ = 0 }{ \underbrace{ \partial_{\mu_2} \partial_{\mu_1} A^{\mu_2} } } \big) H^1 d x^{\mu_1} - (-1)^{D-1} \tfrac{1}{2} H^{-2} \big( \partial_{\mu_2} \partial^{\mu_2} A^{\mu_1} \big) H^1 d x^{\mu_1} \\ & \phantom{=}\; (-1)^{D-1} \tfrac{1}{2} H^{-3/2} \partial_z \big( H^{3/2} \partial_{\mu_1} \underset{ = 0}{ \underbrace{ A_{z} } } \big) H^1 d x^{\mu_1} - (-1)^{D-1} \tfrac{1}{2} H^{-3/2} \partial_z \left( H^{3/2} \partial_{z} A_{\mu_1} \right) H^1 d x^{\mu_1} \\ & \phantom{=}\; + (-1)^{D-1} \tfrac{1}{2} H^{-3/2} \underset{ = 0 }{ \underbrace{ \partial_{\mu_2} \left( H^{3/2} \partial_z A^{\mu_2} \right) } } H^{-1} d z - (-1)^{D-1} \tfrac{1}{2} H^{-3/2} \partial_{\mu_2} \big( H^{3/2} \partial_{\mu_2} \underset{ = 0 }{ \underbrace{ A_{z} } } \big) H^{-1} d z \\ & = - (-1)^{D-1} \tfrac{1}{2} H^{-1} \, \Big( \eta^{\mu \nu}\partial_\mu \partial_\nu A + H^{1/2} \partial_z \left( H^{3/2} \partial_z A \right) \Big) \end{aligned}

Here the terms over the braces vanish, alternatingly, by the assumptions (11) and (12).

(…)

General:

• Paul Wesson, Chapter 5 of: Space-Time-Matter: Modern Kaluza-Klein Theory, World Scientific 1989 (doi:10.1142/3889)

• Paul Wesson, James M. Overduin, Chapter 6 of: Principles of Space-Time-Matter: Cosmology, Particles and Waves in Five Dimensions, World Scientific 2018 (doi:10.1142/10871)

Relation to vector mesons and the Skyrmion model via the Sakai-Sugimoto model and variants: