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Perry-Schwarz action

Contents

Context

String theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A Lagrangian density/action functional for the self-dual higher gauge field in 6d and/or the M5-brane Green-Schwarz sigma model, after KK-compactification to 5 worldvolume dimensions.

Details

For the self-dual field and trivial target space metric

We review the definitions from Perry-Schwarz 96, for the worldvolume Lagrangian density of just the 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

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

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

(1)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

BΩ 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

(2) \,

Write

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

for the vector field which reflects the infinitesimal circle group-action on (2). We will write

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 (2), hence a differential 1-form

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

such that

(4)ι v 5θ 5=1AAandAA 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 5ι 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 5ι v 5θ 5ι v 5=θ 5ι 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:=(idθ 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 (3) to be a spacelike isometry. This means that

(5)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))

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

and (PS 96 (6))

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

With this notation the self-duality condition (1) is equivalently (PS 96 (9), see (9) below):

(8)=H˜ \mathcal{F} \;=\; \tilde H

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

(9)(H=H) ( andι v 5H=ι v 5H andθ 5H=θ 5H) (ι v 5H=ι v 5H) (=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 (5) and the third step invokes the definitions (6) and (7) and the fourth step the equality (11).

The gauge field

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

(10)Aι v 5B A \;\coloneqq\; - \iota_{v^5} B

With this we have

B=Aθ 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))

(11) ι v 5H =ι v 5dB =dι v 5B+[ι v 5,d]B =dA+ 5B =F+θ 5ι v 5dA+ 5B hor+ 5θ 5ι v 5B=θ 5 v 5A =F+ 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 (4) and (10).

Hence in terms of FF and B horB^{\mathrm{hor}} the self-duality condition (1), (8) is equivalently expressed as on the right of the following

(12)(H=H)(H˜=F+ 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 5d((dA) hor) =θ 5d(dAθ 5ι 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 (12)

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

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

(13)(H=H)θ 5d(H˜ 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 islooking now for a Lagrangian density whose Euler-Lagrange equations are (13):

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

(14)L12(H˜ 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 (5) the Lagrangian (14) becomes

(15)L =12(H˜ v 5B hor)Hθ 5 =12(ι v 5H v 5B hor)Hθ 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 (7).

Notice that (15) is the quadratic part of the following form-valued bilinear form on 2-form fields:

(B,B )12(ι v 5(dB) 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 (14), 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 (15) may be computed from twice the variation of just the second factor

δL sd =2(12(ι v 5H v 5B hor)d(δB)θ 5) =(d(ι v 5H 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 (13):

(16)θ 5d(ι v 5H 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 (1) on the Perry-Schwarz Lagrangian (15) it becomes

(17)L sd=12(ι v 5H v 5B hor)Hθ 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

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 (11) becomes

=F \mathcal{F} = F

and so the self-duality condition (12) now becomes

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

which means that

H=Fθ 5+ 5F H = F \wedge \theta^5 + \star_5 F

(check relative sign)

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

(dH=0) (d(Fθ 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 M5-brane sigma model

(…)

(Schwarz 97, APPS 97)

(…)

References

The Perry-Schwarz action is due to

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

The covariant version via a scalar auxiliary field is due to

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

Last revised on October 30, 2019 at 05:36:43. See the history of this page for a list of all contributions to it.