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relation between 5d Maxwell theory and self-dual 3-forms in 6d -- section
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 κ } \{x^\kappa\} , { x α } \{x^\alpha\} , respectively, adapated to an isometric embedding
ℝ 4 , 1 ↪ ι 5 ℝ 5 , 1 κ = 0 , 1 , 2 , 3 , 4 , α = 0 , 1 , 2 , 3 , 4 , 5
\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 H_3 on ℝ 5 , 1 \mathbb{R}^{5,1} decomposes as
for unique differential forms of the form
F ^ = 1 2 F ^ κ 1 κ 2 ( x κ , x 5 ) d x κ 1 ∧ d x κ 2
\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
H ^ = 1 3 ! H ^ κ 1 κ 2 κ 3 ( x κ , x 5 ) d x κ 1 ∧ d x κ 2 ∧ d x κ 3 .
\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 H_3 has vanishing Lie derivative along the x 5 x^5 -direction,
(2) ℒ 5 H 3 = 0
\mathcal{L}_5 H_3 \;=\; 0
then also these components forms do not depend on x 5 x^5 are actualls pullbacks of differential forms on ℝ 4 , 1 \mathbb{R}^{4,1} .
In terms of this decomposition, the 6d Hodge dual of H 3 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) ⋆ 6 H 3 = ( ⋆ 5 H ^ ) ∧ d x 5 − ⋆ 5 F ^
\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. )
⋆ 6 ⋆ 6 H 3 = + H 3
\star_6 \star_6 H_3 \;=\; + H_3
we may ask for H 3 H_3 to he Hodge self-dual . By (3) this means equivalently that its 5d components are 5d Hodge duals of each other:
( H 3 = ⋆ 6 H 3 ) ⇔ H 3 = F ^ ∧ d x 5 + H ^ ( H ^ = ⋆ 5 F ^ ) .
\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 x^5 -dependence (2) then the condition that H 3 H_3 be a closed and self-dual 3-form is equivalent to its 5d components F ^ \widehat{F} (H ^ \widehat{H} ) being the (dual) field strength /Faraday tensor satisfying the Maxwell equations of D=5 Maxwell theory (without source current ):
d H 3 = 0 ⋆ 6 H 3 = H 3 } A D=6 self-dual 3-form theory ⇔ ℒ 5 H 3 = 0 H 3 = F ^ ∧ d x 5 + ⋯ { d F ^ = 0 d ⋆ 5 F ^ = 0 A D=5 Maxwell theory
\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 .
Last revised on May 4, 2020 at 11:54:52.
See the history of this page for a list of all contributions to it.