superalgebra

and

supergeometry

Contents

Idea

The Green-Schwarz action functional is the action functional of a sigma-model that describes the propagation of a fundamental $p$-brane $\Sigma$ on a supermanifold spacetime.

Definition

The Green-Schwarz action functionals are of the standard sigma-model form for target spaces that are super-homogeneous spaces $G/H$ for $G$ a Lie supergroup and $H$ a sub-super-group, and for background gauge fields that are super-WZW-circle n-bundles with connection/bundle gerbes on $G$.

These action functionals were first considered in (Green-Schwarz 84) for superstrings in various dimensions. The full interpretation of the action functional as an higher Wess-Zumino-Witten theory-type action controled by the Lie algebra cohomology of the super Poincaré Lie algebra (or rather of the super translation Lie algebra inside it) is due to (Azcárraga-Townsend89).

Supercoordinates

We briefly review some basics of the canonical coordinates and the super Lie algebra cohomology of the super Poincaré Lie algebra and super Minkowski space, which are referred to below (see for instance Azcárraga-Townsend 89, and see at super Cartesian space and at signs in supergeometry.).

By the general discussion at Chevalley-Eilenberg algebra, we may characterize the super Poincaré Lie algebra $\mathrm{𝔰𝔦𝔰𝔬}\left(D-1,1\right)$ by its CE-algebra $\mathrm{CE}\left(\mathrm{𝔰𝔦𝔰𝔬}\left(D-1,1\right)\right)$ “of left-invariant 1-forms” on its group manifold.

Definition

The Chevalley-Eilenberg algebra $\mathrm{CE}\left(\mathrm{𝔰𝔦𝔰𝔬}\left(d-1,1\right)\right)$ is generated on

• elements $\left\{{e}^{a}\right\}$ and $\left\{{\omega }^{ab}\right\}$ of degree $\left(1,\mathrm{even}\right)$

• and elements $\left\{{\psi }^{\alpha }\right\}$ of degree $\left(1,\mathrm{odd}\right)$

with the differential defined by

${d}_{\mathrm{CE}}{\omega }^{ab}={\omega }^{a}{}_{b}\wedge {\omega }^{bc}$d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
${d}_{\mathrm{CE}}{e}^{a}={\omega }^{a}{}_{b}\wedge {e}^{b}+\frac{i}{2}\overline{\psi }{\Gamma }^{a}\psi$d_{CE} e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
${d}_{\mathrm{CE}}\psi =\frac{1}{4}{\omega }^{ab}{\Gamma }_{ab}\psi \phantom{\rule{thinmathspace}{0ex}}.$d_{CE} \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi \,.

Removing the terms involving $\omega$ here this is the super translation algebra.

In this way the super-Poincaré Lie algebra and its extensions is usefully discussed for instance in (D’Auria-Fré 82) and in (Azcárraga-Townsend 89, CAIB 99). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the coordinates on super Minkowski space.

Remark

The abstract generators in def. 1 are identified with left invariant 1-forms on the super-translation group (= super Minkowski space) as follows.

Let $\left({x}^{a},{\theta }^{\alpha }\right)$ be the canonical coordinates on the supermanifold ${ℝ}^{d\mid N}$ underlying the super translation group. Then the identification is

• ${\psi }^{\alpha }=d{\theta }^{\alpha }$.

• ${e}^{a}=d{x}^{a}+\frac{i}{2}\overline{\theta }{\Gamma }^{a}d\theta$.

Notice that this then gives the above formula for the differential of the super-vielbein in def. 1 as

$\begin{array}{rl}d{e}^{a}& =d\left(d{x}^{a}+\frac{i}{2}\overline{\theta }{\Gamma }^{a}d\theta \right)\\ & =\frac{i}{2}d\overline{\theta }{\Gamma }^{a}d\theta \\ & =\frac{i}{2}\overline{\psi }{\Gamma }^{a}\psi \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,.
Remark

The term $\frac{i}{2}\overline{\psi }{\Gamma }^{a}\psi$ is sometimes called the supertorsion of the supervielbein $e$, because the defining equation

${d}_{\mathrm{CE}}{e}^{a}-{\omega }^{a}{}_{b}\wedge {e}^{b}=\frac{i}{2}\overline{\psi }{\Gamma }^{a}\psi$d_{CE} e^{a } -\omega^a{}_b \wedge e^b = \frac{i}{2}\bar \psi \Gamma^a \psi

may be read as saying that $e$ is torsion-free except for that term. Notice that this term is the only one that appears when the differential is applied to “Lorentz scalars”, hence to object in $\mathrm{CE}\left(\mathrm{𝔰𝔦𝔰𝔬}\right)$ which have “all indices contracted”.

Notably we have

$d\left(\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p}}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{p}}\right)\propto \left(\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p}}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{p-1}}\right)\wedge \left(\overline{\Psi }\wedge {\Gamma }_{{a}_{p}}\Psi \right)\phantom{\rule{thinmathspace}{0ex}}.$d \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p} \right) \propto \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}} \right) \wedge \left( \overline{\Psi} \wedge \Gamma_{a_p} \Psi \right) \,.

This remaining operation ”$e↦{\Psi }^{2}$” of the differential acting on Loretz scalars is sometimes denoted ”${t}_{0}$”, e.g. in (Bossard-Howe-Stelle 09, equation (8)).

This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of $\left(D,p\right)$ a Fierz identity implies that the term

$\left(\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p}}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{p-1}}\right)\wedge \left(\overline{\Psi }\wedge {\Gamma }_{{a}_{p}}\Psi \right)$\left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}} \right) \wedge \left( \overline{\Psi} \wedge \Gamma_{a_p} \Psi \right)

vanishes identically, and hence in these dimensions the term

$\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p}}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{p}}$\overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}

Kinetic term

(…)

kinetic action

${\int }_{\Sigma }⟨{\varphi }^{*}{\Pi }^{a},{\varphi }^{*}{\Pi }^{b}{\eta }_{ab}⟩$\int_\Sigma \langle \phi^\ast\Pi^a, \phi^\ast \Pi^b \eta_{a b}\rangle

(…)

WZW term

Let $\left({e}^{a},{\omega }^{ab},{\psi }^{\alpha }\right)$ be the standard generators of the Chevalley-Eilenberg algebra $\mathrm{CE}\left(\mathrm{𝔰𝔦𝔰𝔬}\left(d,1\right)\right)$ of the super Poincaré Lie algebra, as discussed there.

The part of the Lie algebra cohomology of the super translation Lie algebra that is invariant under the Lorentz transformations is spanned by closed elements of the form

$\mu =\left(d\overline{\theta }{\Gamma }_{{a}_{1},\cdots ,{a}_{p}}\wedge d\theta \right)\wedge {\Pi }^{{a}_{1}}\wedge \cdots \wedge {\Pi }^{{a}_{p}}\phantom{\rule{thinmathspace}{0ex}}.$\mu = (d \bar \theta \Gamma_{a_1, \cdots, a_p} \wedge d \theta) \wedge \Pi^{a_1} \wedge \cdots \wedge \Pi^{a_p} \,.

These exist (are closed) only for certain combinations of $d$ and $p$. The possible values are listed below.

For a bosonic WZW model the background gauge field induced by such a cocycle would be the corresponding Lie integration to a circle n-bundle with connection. Here, since the super translation group is contractible, a Poincaré lemma applies and these circle $n$-connections are simply given by globally defined connection form $\beta$ satisfying

$d\beta =\mu \phantom{\rule{thinmathspace}{0ex}}.$d \beta = \mu \,.

The WZW part of the GS action is then

${S}_{\mathrm{WZW}}:\varphi ↦{\int }_{\Sigma }{\varphi }^{*}\beta$S_{WZW } : \phi \mapsto \int_\Sigma \phi^* \beta

(…)

Properties

Siegel- or $\kappa$-symmetry

The Green-Schwarz action has an extra fermionic symmetry, on top of the genuine supersymmetry, first observed in (Siegel 83) for the superparticle and in (Siegel 84) for the superstring in 3-dimensions, and finally in (GreenSchwarz 84) for the critical superstring in 10-dimensions. This is also called $\kappa$-symmetry. It has a natural interpretation in terms of the super-Cartan geometry of target space (McArthur, GKW).

Dimensions – the brane scan

The Green-Schwarz action functional of a $p$-brane propagating on an $d$-dimensional target spacetimes makes sense only for special combinations of $\left(p,d\right)$, for which there are suitanble super Lie algebra cocycles on the super translation Lie algebra (see above).

The corresponding table has been called the brane scan in the literature, now often called the “old brane scan”, since it has meanwhile been further completed (see below). In (Duff 87) the “old brane scan” is displayed as follows.

In the $D=10$-row we see the critical superstring of string theory and its magnetic dual, the NS5-brane. The top row shows the M2-brane in 11-dimensional supergravity.

Moving down and left the diagonals corresponds to double dimensional reduction.

Remark

The first non-empty column of the table is a reflection of the exceptional isomorphisms of the spin group in low dimensions and the normed division algebras:

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
$3=2+1$$\mathrm{Spin}\left(2,1\right)\simeq \mathrm{SL}\left(2,ℝ\right)$$ℝ$ the real numbers
$4=3+1$$\mathrm{Spin}\left(3,1\right)\simeq \mathrm{SL}\left(2,ℂ\right)$$ℂ$ the complex numbers
$6=5+1$$\mathrm{Spin}\left(5,1\right)\simeq \mathrm{SL}\left(2,ℍ\right)$$ℍ$ the quaternionslittle string
$10=9+1$$\mathrm{Spin}\left(9,1\right){\simeq }_{\mathrm{some}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sense}}\mathrm{SL}\left(2,𝕆\right)$$𝕆$ the octonionsheterotic/type II string
Remark

What is missing in the “old brane scan” are the D-branes in $D=10$ and the M5-brane in $D=11$ (See also BPST). The reason is that the M5 corresponds to a 7-cocycle not on the ordinary super Poincaré Lie algebra, but on its L-infinity algebra extension, the supergravity Lie 3-algebra. The completion in super L-infinity algebra theory is discussed in (FSS 13), as The brane bouquet.

So (with notation as above) we have the following.

The brane scan.

The Green-Schwarz type super $p$-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

$\stackrel{D}{=}$$p=$123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9$*$
8$*$
7M2${}_{\mathrm{top}}$
6F1${}_{\mathrm{little}}$, S1${}_{\mathrm{sd}}$S3
5$*$
4$*$$*$
3$*$

(The first colums follow the exceptional spinors table.)

The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):

$\stackrel{D}{=}$$p=$123456789
11${\Psi }^{2}{E}^{2}$ on sIso(10,1)${\Psi }^{2}{E}^{5}+{\Psi }^{2}{E}^{2}{C}_{3}$ on m2brane
10${\Psi }^{2}{E}^{1}$ on sIso(9,1)${B}_{2}^{2}+{B}_{2}{\Psi }^{2}+{\Psi }^{2}{E}^{2}$ on StringIIA$\cdots$ on StringIIB${B}_{2}^{3}+{B}_{2}^{2}{\Psi }^{2}+{B}_{2}{\Psi }^{2}{E}^{2}+{\Psi }^{2}{E}^{4}$ on StringIIA${\Psi }^{2}{E}^{5}$ on sIso(9,1)${B}_{2}^{4}+\cdots +{\Psi }^{2}{E}^{6}$ on StringIIA$\cdots$ on StringIIB${B}_{2}^{5}+\cdots +{\Psi }^{2}{E}^{8}$ in StringIIA$\cdots$ on StringIIB
9${\Psi }^{2}{E}^{4}$ on sIso(8,1)
8${\Psi }^{2}{E}^{3}$ on sIso(7,1)
7${\Psi }^{2}{E}^{2}$ on sIso(6,1)
6${\Psi }^{2}{E}^{1}$ on sIso(5,1)${\Psi }^{2}{E}^{3}$ on sIso(5,1)
5${\Psi }^{2}{E}^{2}$ on sIso(4,1)
4${\Psi }^{2}{E}^{1}$ on sIso(3,1)${\Psi }^{2}{E}^{2}$ on sIso(3,1)
3${\Psi }^{2}{E}^{1}$ on sIso(2,1)

Relation to supergravity equations of motion

In the first order formulation of gravity a field configuration on a spacetime manifold $X$ is a Cartan connection

$\nabla :X\to B\mathrm{SuperPoincare}\left(d-1,1{\right)}_{\mathrm{conn}}$\nabla \colon X \to \mathbf{B} SuperPoincare(d-1,1)_{conn}

hence a principal connection for the super Poincaré group such such that at each point $x\in X$ it identifies the tangent space with ${ℝ}^{d;N}=\mathrm{𝔰𝔦𝔰𝔬}\left(d-1,1\right)/𝔬\left(d-1,1\right)$

${T}_{x}X\stackrel{\nabla }{⟶}\mathrm{𝔰𝔦𝔰𝔬}\left(d-1,1\right)⟶{ℝ}^{d;N}\phantom{\rule{thinmathspace}{0ex}}.$T_x X \stackrel{\nabla}{\longrightarrow} \mathfrak{siso}(d-1,1) \longrightarrow \mathbb{R}^{d;N} \,.

Hence given a Lie algebra cocycle

$𝔤⟶ℝ\left[2\right]$\mathfrak{g} \longrightarrow \mathbb{R}[2]

as for the Green-Schwarz superstring we can pull it back along this Cartan connection to a differential 3-form on spacetime.

In general this 3-form is no longer closed. If it is closed, then the Green-Schwarz superstring is again well defined on $\left(X,\nabla \right)$ as a WZW model.

The claim now is that requiring this 3-form still to be closed is, as a condition on the field of gravity $\nabla$, precisely the equations of motion of supergravity (the super-Einstein equations).

This is due to (Nilsson 81).

References

General

The Green-Schwarz action functional (formulated for the superstring) is due to

The observation that this is an example of a WZW-model on super-Minkowski spacetime is due to

• Marc Henneaux, Luca Mezincescu, A Sigma Model Interpretation of Green-Schwarz Covariant Superstring Action, Phys.Lett. B152 (1985) 340 (web)

For more references on this perspective see below.

That the GS-action functionals is consistent on all backgrounds that satisfy the relevant supergravity equations of motion was shown in

A standard textbook reference is appendix 4.A of volume 1 of

and a brief paragraph in Volume II, section 10.2, page 983 of

A more recent and more comprehensive review is

WZW terms, super Lie algebra cohomology and the brane scan

The WZW nature of the second term in the GS action, recognized in (Henneaux-Mezincescu 85) is discussed with its Lie theoretic meaning made fully explicit (in “FDA” language) in chater 8 of

• José de Azcárraga, Izqierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge monographs of mathematical physics, (1995)

The original “brane scan” classification of GS action functionals by WZW terms is due to

For $d=11$ the relevant super Lie algebra cocycles have also been discussed (but not related to the Green-Schwarz action functional) in

A review is in

• Michael Duff, Supermembranes: the first fifteen weeks CERN-TH.4797/87 (1987) (scan)

from which the above table is taken.

A decent systematic account of the principles of super Lie algebra cohomology in the GS-functional, of these cocycles is in the letter

• José de Azcárraga, Paul Townsend, Superspace geometry and the classification of supersymmetric extended objects, Physical Review Letters Volume 62, Number 22 (1989)

and a detailed account building on this, which also discusses the GS/WZW terms for D-branes on the type II supergravity Lie 2-algebra (in its section 6) is in

• C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137)

• I. Bars, C. Deliduman and D. Minic, Phys. Rev D59 (1999) 125004; Phys. Lett. B457 (1999) 275. (arXiv:hep-th/9812161)

More along these lines is in

The Green-Schwarz-type action for the M5-brane was found in

The 7-cocycle on the supergravity Lie 3-algebra which gives the supergravity Lie 6-algebra appears in these articles (somewhat secretly) in equation (BLNPST, equation (9)).

• Igor Bandos, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Superbrane Actions and Geometrical Approach (arXiv:hep-th/9705064)

The 7-cocycle for the M5-brane on the supergravity Lie 3-algebra is equation (8.8) there.

A corresponding refinement of the brane scan to a “brane bouquet” of super L-∞ algebra extensions (hence in infinity-Lie theory via ∞-Wess-Zumino-Witten theory) is discussed in

These cohomologival arguments also appear in what is called the “ectoplasm” method for invariants in super Yang-Mills theory in

• G. Bossard, Paul Howe, K.S. Stelle, A note on the UV behaviour of maximally supersymmetric Yang-Mills theories, Phys. Lett. B682:137-142 (2009) (arXiv:0908.3883)
• Paul Howe, T. G. Pugh, K. S. Stelle, C. Strickland-Constable, Ectoplasm with an Edge, JHEP 1108:081,2011 (arXiv:1104.4387)

• G. Bossard, Paul Howe, U. Lindstrom, K.S. Stelle, L. Wulff, Integral invariants in maximally supersymmetric Yang-Mills theories (arXiv:1012.3142)

The other brane scan, listing consistent asymptotic AdS geometries is due to

• M.P. Blencowea, Mike Duff, Supersingletons, Physics letters B, Volume 203, Issue 3, 31 March 1988, Pages 229–236 .

with further developments discussed in

Supergravity background equations of motion

The consistentcy of the Green-Schwarz action functional for the superstring in a supergravity background should be equivalent to the background satiyfying the supergravity equations of motion.

That the heterotic supergravity equations of motion are sufficient for the 3-form super field strength $H$ to be closed was first argued in

• Bengt Nilsson, Simple 10-dimensional supergravity in superspace, Nuclear Physics B188 (1981) 176-192 (spire)

and the computation there was highlighted and a little simplified in

• Edward Witten, Twistor-like transform in ten dimensions, Nuclear Physics B266 (1986)

Similar arguments for the type II string in type II supergravity appeared in

and for GS sigma-model D-branes in

That the supergravity equations of motion of the background are not just sufficient but also necessary for (and hence equivalent to) the GS-string on that background being consistent was then claimed in

• Joel Shapiro, Cyrus Taylor, Superspace supergravity from the superstring, Physics letter B volume 186, number 1, 1987

That the M2-brane sigma-model is consistent on backgrounds of 11-dimensional supergravity that satisfy their equations of motion is discussed in

These authors amplify the role of closed $\left(p+2\right)$-forms in super $p$-brane backgrounds (p. 3) and clearly state the consistency conditions for the M2-brane in a curved backroundin terms of the Bianchi identities on p. 7-8, amounting to the statment that the 4-form field strenght has to be the pullback of the cocycle $\overline{\psi }\wedge {e}^{a}\wedge {e}^{b}\wedge {\Gamma }^{ab}\psi$ plus the supergravity C-field curvature and has to be closed.

The role of the 4-form here is also amplified around (2.29) in

• Igor Bandos, Carlos Meliveo, Supermembrane interaction with dynamical D=4 N=1 supergravity. Superfield Lagrangian description and spacetime equations of motion (arXiv:arXiv:1205.5885)

and in section 2.2 of

• Igor Bandos, Carlos Meliveo, Three form potential in (special) minimal supergravity superspace and supermembrane supercurrent (arXiv:1107.3232)

All this is actually subsumed by imposing the Bianchi identities of the corresponding supergravity Lie 3-algebra etc. in “rheonomic parameterization”, see at D'Auria-Fré formulation of supergravity.

Symmetries

That higher WZW functionals and hence Green-Schwarz super $p$-brane action functionals should have “higher” extended symmetry algebras in some sense… is observes in

• José de Azcárraga, Jerome P. Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys. Rev. Lett. 63 (1989) 2443 (web)

$\kappa$-Symmetry

The existence of $\kappa$-symmetry was first noticed around

• Warren Siegel, Hidden Local Supersymmetry In The Supersymmetric Particle Action Phys. Lett. B 128, 397 (1983)
• Warren Siegel, Light Cone Analysis Of Covariant Superstring , Nucl. Phys. B 236, 311 (1984).

The meaning of $\kappa$-symmetry in terms of the super-Cartan geometry of super-target space is discussed in

• I.N. McArthur, Kappa-Symmetry of Green-Schwarz Actions in Coset Superspaces (arXiv:hep-th/9908045)

Open branes ending on other branes

Discussion of the Green-Schwarz action for the open M2-brane ending on the M5-brane is in

GS superstrings in various backgrounds

• R. R. Metsaev, Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background (arXiv:hep-th/0112044)

Revised on October 9, 2013 11:17:58 by Urs Schreiber (89.204.139.155)