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 3H_3 on ℝ 5,1\mathbb{R}^{5,1} decomposes as

(1)H 3=F^∧dx 5+H^ H_3 \;=\; \widehat{F} \wedge d x^{5} + \widehat{H}

for unique differential forms of the form

F^=12F^ κ 1κ 2(x κ,x 5)dx κ 1∧dx 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^=13!H^ κ 1κ 2κ 3(x κ,x 5)dx κ 1∧dx κ 2∧dx 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 3H_3 has vanishing Lie derivative along the x 5x^5-direction,

(2)ℒ 5H 3=0 \mathcal{L}_5 H_3 \;=\; 0

then also these components forms do not depend on x 5x^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 3H_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)⋆ 6H 3=(⋆ 5H^)∧dx 5−⋆ 5F^ \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⋆ 6H 3=+H 3 \star_6 \star_6 H_3 \;=\; + H_3

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

(H 3=⋆ 6H 3)⇔H 3=F^∧dx 5+H^(H^=⋆ 5F^). \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 5x^5-dependence (2) then the condition that H 3H_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):

dH 3=0 ⋆ 6H 3=H 3}AD=6 self-dual 3-form theory⇔ℒ 5H 3=0H 3=F^∧dx 5+⋯{ dF^=0 d⋆ 5F^=0AD=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.

Details

For the free self-dual field and trivial target space metric

We review the definitions from Perry-Schwarz 96, Section 2 “The Free Theory” (following Henneaux-Teitelboim 88), for the worldvolume Lagrangian density of just the free self-dual higher gauge field on a circle principal bundle-worldvolume for would-be target space being Minkowski spacetime.

In doing so, we translate to coordinate-invariant Cartan calculus-formalism and generalized to KK-compactification on possibly non-trivial circle principal bundle:

Worldvolume and self-duality

Let

(Σ 6,g) (\Sigma^6, g)

be a pseudo-Riemannian manifold of dimension 6 and of signature (−,+,+,+,+,+)(-,+,+,+,+,+), to be called the worldvolume.

In this dimension and with this signature, the Hodge star operator squares to +1+1. This allows to consider for a differential 3-form

H∈Ω 3(Σ 6) H \;\in\; \Omega^3\big(\Sigma^6\big)

the condition that it be self-dual (PS 96 (2))

(4)H=⋆H. H \;=\; \star H \,.

We will assume in the following that HH is exact differential form, hence that there exists a differential 2-form

B∈Ω 2(Σ 6) B \in \Omega^2\big( \Sigma^6 \big)

such that (PS 96 (4))

H=dB. H = d B \,.

S 1S^1-compactification

Consider then on Σ 6\Sigma^6 the structure of an S 1=U(1)S^1 = U(1)-principal bundle

(5) \,

Write

(6)v 5∈Γ(TΣ 6) v^5 \in \Gamma( T \Sigma^6 )

for the vector field which reflects the infinitesimal circle group-action on (5). We will write

ℒ v 5=[d,ι v 5]:Ω •(Σ 6)⟶Ω •(Σ 6) \mathcal{L}_{v^5} \;=\; \big[d, \iota_{v^5} \big] \;\colon\; \Omega^\bullet\big( \Sigma^6 \big) \longrightarrow \Omega^\bullet\big( \Sigma^6 \big)

for the Lie derivative of differential forms along v 5v^5, and make use of Cartan's magic formula expressing it as an anti-commutator, as shown.

Next consider an Ehresmann connection on the S 1S^1-bundle (5), hence a differential 1-form

θ 5∈Ω 1(Σ 6) \theta^5 \;\in\; \Omega^1\big( \Sigma^6 \big)

such that

(7)ι v 5θ 5=1AAandAAℒ v 5θ 5=0 \iota_{v^5} \theta^5 = 1 \phantom{AA} \text{and} \phantom{AA} \mathcal{L}_{v^5} \theta^5 = 0

So in particular

θ 5∧ι v 5:Ω •(Σ 6)⟶Ω •(Σ 6) \theta^5 \wedge \iota_{v^5} \;:\; \Omega^\bullet\big( \Sigma^6\big) \longrightarrow \Omega^\bullet\big( \Sigma^6\big)

is a projection operator:

θ 5∧ι v 5∘θ 5∧ι v 5=θ 5∧ι v 5 \theta^5 \wedge \iota_{v^5} \circ \theta^5 \wedge \iota_{v^5} \;=\; \theta^5 \wedge \iota_{v^5}

The complementary projection is that onto horizontal differential forms

(−) hor:=(id−θ 5ι v 5):Ω •(Σ 6)⟶Ω •(Σ 6) (-)^{\mathrm{hor}} := \big(\mathrm{id} - \theta^5 \iota_{v^5}) \;:\; \Omega^\bullet\big( \Sigma^6\big) \longrightarrow \Omega^\bullet\big( \Sigma^6\big)

We require v 5v^5 (6) to be a spacelike isometry. This means that

(8)⋆∘ι v 5=−θ 5∧∘⋆:Ω 3(Σ 6)⟶Ω 4(Σ 6) \star \circ \iota_{v^5} = - \theta^5 \wedge \circ \star \;:\; \Omega^3\big( \Sigma^6\big) \longrightarrow \Omega^4\big( \Sigma^6 \big)

Self-duality after S 1S^1-compactification

Set (PS 96 (5))

(9)ℱ≔ι v 5H \mathcal{F} \;\coloneqq\; \iota_{v^5} H

and (PS 96 (6))

(10)H˜≔ι v 5⋆H \tilde H \;\coloneqq\; \iota_{v^5} \star H

With this notation the self-duality condition (4) is equivalently (PS 96 (9), see (12) below):

(11)ℱ=H˜ \mathcal{F} \;=\; \tilde H

To make this fully explicit, notice that we have the following chain of logical equivalences:

(12)(H=⋆H) ⇔( andι v 5H=ι v 5⋆H andθ 5∧H=θ 5∧⋆H) ⇔(ι v 5H=ι v 5⋆H) ⇔(ℱ=H˜) \begin{aligned} \big( H = \star H \big) & \Leftrightarrow \left( \array{ & \phantom{\text{and}\;} \iota_{v_5} H = \iota_{v^5} \star H \\ & \text{and}\; \theta^5 \wedge H = \theta^5 \wedge \star H } \right) \\ & \Leftrightarrow \big( \iota_{v^5} H = \iota_{v^5} \star H \big) \\ &\Leftrightarrow \big( \mathcal{F} \;=\; \widetilde H \big) \end{aligned}

Here the first step is decomposition of the self-duality equation into components, the second step follows by (8) and the third step invokes the definitions (9) and (10) and the fourth step the equality (14).

The gauge field

Define the vector potential (PS 96 above (4))

(13)A≔−ι v 5B A \;\coloneqq\; - \iota_{v^5} B

With this we have

B=A∧θ 5+B hor. B \;=\; A \wedge \theta^5 + B^{\mathrm{hor}} \,.

Set also (PS 96 above (4))

F≔(dA) hor F \;\coloneqq\; \big( d A \big)^{\mathrm{hor}}

then (PS 96 (5))

(14)ℱ ≔ι v 5H =ι v 5dB =−dι v 5B+[ι v 5,d]B =dA+ℒ 5B =F+θ 5∧ι v 5dA+ℒ 5B hor+ℒ 5θ 5∧ι v 5B⏟=−θ 5∧ℒ v 5A =F+ℒ v 5B hor \begin{aligned} \mathcal{F} &\coloneqq \iota_{v^5} H \\ & = \iota_{v^5} d B \\ & = - d \iota_{v^5} B + [\iota_{v^5}, d] B \\ & = d A + \mathcal{L}_5 B \\ & = F + \theta^5 \wedge \iota_{v^5} d A + \mathcal{L}_5 B^{\mathrm{hor}} + \underset{ = -\theta^5 \wedge \mathcal{L}_{v^5} A }{ \underbrace{ \mathcal{L}_5 \theta^5 \wedge \iota_{v^5} B } } \\ & = F + \mathcal{L}_{v^5} B^{\mathrm{hor}} \end{aligned}

where in the last step under the brace we used (7) and (13).

Hence in terms of FF and B horB^{\mathrm{hor}} the self-duality condition (4), (11) is equivalently expressed as on the right of the following

(15)(H=⋆H)⇔(H˜=F+ℒ v 5B hor) \big( H = \star H \big) \;\Leftrightarrow\; \big( \widetilde H \;=\; F + \mathcal{L}_{v^5} B^{\mathrm{hor}} \big)

Weak self-duality and PS-equations of motion

Notice that

θ 5∧d((dA) hor) =θ 5∧d(dA−θ 5∧ι v 5dA) =θ 5∧(dθ 5)∧ι v 5dA. \begin{aligned} \theta^5 \wedge d \big( (d A)^{hor} \big) & = \theta^5 \wedge d \big( d A - \theta^5 \wedge \iota_{v_5} d A\big) \\ & = \theta^5 \wedge (d \theta^5) \wedge \iota_{v_5} d A \end{aligned} \,.

Hence assume now hat the Ehresmann connection is flat, hence dθ 5=0d \theta^5 = 0.

Then the self-duality condition in the form (15)

H˜−ℒ v 5B hor=(dA) hor \widetilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \;=\; (d A)^{\mathrm{hor}}

implies, after applying θ 5∧d\theta^5 \wedge d to both sides, the second-order equation (PS 96 (16))

(16)(H=⋆H)⇒θ 5∧d(H˜−ℒ v 5B hor)=0 (H = \star H) \;\;\;\Rightarrow\;\;\; \theta^5 \wedge d \big( \widetilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \;=\; 0

This equation by itself is hence a weakened form of the self-duality condition, a kind of “self-duality up to horizontally closed terms”.

The proposal of Perry-Schwarz 96, Sec. 2 is to take this as the relevant equation of motion for the theory on S 1S^1.

Lagrangian density

Therefore one is looking now for a Lagrangian density whose Euler-Lagrange equations are (16):

The Perry-Schwarz-Lagrangian is (PS 96 (17))

(17)L≔−12(H˜−ℒ v 5B hor)∧⋆H˜ L \;\coloneqq\; - \tfrac{1}{2} \big( \tilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge \star \tilde H

With (8) the Lagrangian (17) becomes

(18)L =−12(H˜−ℒ v 5B hor)∧H∧θ 5 =−12(ι v 5⋆H−ℒ v 5B hor)∧H∧θ 5 \begin{aligned} L & = - \tfrac{1}{2} \big( \tilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge H \wedge \theta^5 \\ & = - \tfrac{1}{2} \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge H \wedge \theta^5 \end{aligned}

where in the second line we inserted the definition (10).

Notice that (18) is the quadratic part of the following form-valued bilinear form on 2-form fields:

(B,B ′)↦−12(ι v 5⋆(dB)−ℒ v 5B hor)∧(dB ′)∧θ 5 (B, B^\prime) \;\mapsto\; - \tfrac{1}{2} \big( \iota_{v^5} \star (d B) - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge (d B^\prime) \wedge \theta^5

Moreover, this bilinear form is symmetric up to a total derivative. For the first summand this is manifest from its incarnation in (17), since the Hodge pairing is symmetric, and for the second term this follows by “local integration by parts”.

As a consequence, the Euler-Lagrange equations of the Perry-Schwarz Lagrangian density (18) may be computed from twice the variation of just the second factor

δL sd =2(−12(ι v 5⋆H−ℒ v 5B hor)∧d(δB)∧θ 5) =(d(ι v 5⋆H−ℒ v 5B hor))∧(δB)∧θ 5+d(⋯) \begin{aligned} \delta L_{\mathrm{sd}} & = 2 \Big( - \tfrac{1}{2} \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge d(\delta B) \wedge \theta^5 \Big) \\ & = \Big( d \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \Big) \wedge (\delta B) \wedge \theta^5 + d(\cdots) \end{aligned}

to indeed be (16):

(19)θ 5∧d(ι v 5⋆H−ℒ v 5B hor)=0. \theta^5 \wedge d \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \;=\; 0 \,.

Notice that if we do use the self-duality condition (4) on the Perry-Schwarz Lagrangian (18) it becomes

(20)L sd=−12(ι v 5H−ℒ v 5B hor)∧H∧θ 5AAAAAifH=⋆H L_{\mathrm{sd}} \;=\; - \tfrac{1}{2} \big( \iota_{v^5} H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge H \wedge \theta^5 \phantom{AAAAA} \text{if} \;\; H = \star H

Example: Reduction to 5d Maxwell theory

Consider the special case that

ℒ v 5B=0, \mathcal{L}_{v^5} B = 0 \,,

which corresponds to keeping only the 0-mode under KK-compactification along the circle fiber.

Then (14) becomes

ℱ=F \mathcal{F} = F

and so the self-duality condition (15) now becomes

ι v 5⋆H=F. \iota_{v^5} \star H \;=\; F \,.

which means that

H=F∧θ 5+⋆ 5F H = F \wedge \theta^5 + \star_5 F

(check relative sign)

Since dH=d∘dB=0d H = d \circ d B = 0, this implies

(dH=0) ⇔(d(F∧θ 5+⋆ 5F)=0) ⇔({d 5F =0 d 5⋆ 5F =0) \begin{aligned} \big( d H = 0 \big) & \Leftrightarrow \Big( d \big( F \wedge \theta^5 + \star_5 F \big) = 0 \Big) \\ & \Leftrightarrow \left( \left\{ \array{ d_5 F & = 0 \\ d_5 \star_5 F & = 0 } \right. \right) \end{aligned}

These are of course Maxwell's equations on Σ 5\Sigma^5.

(PS 96 above (16))

For the full interacting M5-brane sigma model

The full interacting PS Lagrangian (PS 96 (63)) has more terms..

(…)

(Schwarz 97, APPS 97)

(…)

D=5 super Yang-Mills theory

References

The Perry-Schwarz action is due to

Malcolm Perry, John Schwarz, Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory, Nucl. Phys. B489 (1997) 47-64 (arXiv:hep-th/9611065)
John Schwarz, Coupling a Self-Dual Tensor to Gravity in Six Dimensions, Phys. Lett. B395:191-195, 1997 (cds:317663, doi:10.1016/S0370-2693(97)00094-4)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, World-Volume Action of the M Theory Five-Brane, Nucl.Phys. B496 (1997) 191-214 (arXiv:hep-th/9701166)

A similar construction but with compactification along the timelike direction is due to

Marc Henneaux, Claudio Teitelboim, Dynamics of chiral (self-dual) pp-forms, Physics Letters B Volume 206, Issue 4, 2 June 1988, Pages 650-654 (doi:10.1016/0370-2693(88)90712-5)

The double dimensional reduction to the Green-Schwarz sigma-model of the D4-brane:

APPS 97, Section 6
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, Section 6 of Dual D-Brane Actions, Nucl. Phys. B496 (1997) 215-230 (arXiv:hep-th/9702133)

The covariant version via a scalar auxiliary field is due to

Paolo Pasti, Dmitri Sorokin, Mario Tonin, On Lorentz Invariant Actions for Chiral P-Forms, Phys.Rev. D55 (1997) 6292-6298 (arXiv:hep-th/9611100)
Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for a D=11 Five-Brane with the Chiral Field, Phys. Lett. B398 (1997) 41 (arXiv:hep-th/9701037)
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys. Rev. Lett. 78 (1997) 4332-4334 (arXiv:hep-th/9701149)

Speculations about non-abelian generalizations (for several coincident M5-branes):

Chong-Sun Chu, A proposal for the worldvolume action of multiple M5-branes, 2013 (pdf)

The above text follows

Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 2 of: Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5 (arXiv:1908.00042)

Last revised on May 29, 2020 at 15:49:28. See the history of this page for a list of all contributions to it.