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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.
For $p = 0$ this is the Green-Schwarz superparticle.
For $p = 1$ the Green-Schwarz superstring (at the center of attention in string theory).
This model is in contrast to the NSR-string, which instead has manifest worldsheet supersymmetry. See at superstring for more on this.
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).
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 $\mathfrak{siso}(D-1,1)$ by its CE-algebra $CE(\mathfrak{siso}(D-1,1))$ “of left-invariant 1-forms” on its group manifold.
The Chevalley-Eilenberg algebra $CE(\mathfrak{siso}(d-1,1))$ is generated on
elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$
and elements $\{\psi^\alpha\}$ of degree $(1,odd)$
with the differential defined by
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.
The abstract generators in def. 1 are identified with left invariant 1-forms on the super-translation group (= super Minkowski space) as follows.
Let $(x^a, \theta^\alpha)$ be the canonical coordinates on the supermanifold $\mathbb{R}^{d|N}$ 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
The term $\frac{i}{2}\bar \psi \Gamma^a \psi$ is sometimes called the supertorsion of the super-vielbein $e$, because the defining equation
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 $CE(\mathfrak{siso})$ which have “all indices contracted”. (See also at torsion constraints in supergravity.)
Notably we have
This remaining operation “$e \mapsto \Psi^2$” 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 $(D,p)$ a Fierz identity implies that the term
vanishes identically, and hence in these dimensions the term
is a cocycle. See also the brane scan table below.
(…)
(…)
Let $(e^a, \omega^{a b}, \psi^\alpha)$ be the standard generators of the Chevalley-Eilenberg algebra $CE(\mathfrak{siso}(d,1))$ 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
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
The WZW part of the GS action is then
(…)
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). Discussion from the point of view of the D'Auria-Fré formulation of supergravity is in (Fré-Grassi 07, section 2.2).
The Green-Schwarz action functional of a $p$-brane propagating on an $d$-dimensional target spacetimes makes sense only for special combinations of $(p,d)$, 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.
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 dimension | spin group | normed division algebra | brane scan entry |
---|---|---|---|
$3 = 2+1$ | $Spin(2,1) \simeq SL(2,\mathbb{R})$ | $\mathbb{R}$ the real numbers | |
$4 = 3+1$ | $Spin(3,1) \simeq SL(2, \mathbb{C})$ | $\mathbb{C}$ the complex numbers | |
$6 = 5+1$ | $Spin(5,1) \simeq SL(2, \mathbb{H})$ | $\mathbb{H}$ the quaternions | little string |
$10 = 9+1$ | $Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})$ | $\mathbb{O}$ the octonions | heterotic/type II string |
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):
$\stackrel{D}{=}$ | $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
11 | M2 | M5 | ||||||||
10 | D0 | F1, D1 | D2 | D3 | D4 | NS5, D5 | D6 | D7 | D8 | D9 |
9 | $\ast$ | |||||||||
8 | $\ast$ | |||||||||
7 | M2${}_{top}$ | |||||||||
6 | F1${}_{little}$, S1${}_{sd}$ | S3 | ||||||||
5 | $\ast$ | |||||||||
4 | $\ast$ | $\ast$ | ||||||||
3 | $\ast$ |
(The first colums follow the exceptional spinors table.)
The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):
$\stackrel{D}{=}$ | $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
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) |
under construction
In the first order formulation of gravity a field configuration on a spacetime manifold $X$ is a Cartan connection
hence a principal connection for the super Poincaré group such such that at each point $x \in X$ it identifies the tangent space with $\mathbb{R}^{d;N} = \mathfrak{siso}(d-1,1)/\mathfrak{o}(d-1,1)$
Hence given a Lie algebra cocycle
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 $(X,\nabla)$ 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, Bergshoeff-Sezgin-Townsend 86) and others, see the references below.
For the membrane(M2-brane) in a background of 11-dimensional supergravity (Bergshoeff-Sezgin-Townsend 87) find that consistency requires that (in a given coordinate chart with super-vielbein field $(E^A) = (E^a, \Psi^\alpha)$) the 4-form flux is of the form
where the first summand is the super-Lie algebra cocycle that classifies the supergravity Lie 3-algebra and the second is the field strength of the supergravity C-field proper (hence a purely bosonic differential form). In the second line we have rewritten this more manifestly in terms of the super-vielbein $(E^A) = (E^a, E^\alpha) = (E^a, \Psi^\alpha)$, this way the expression is directly analogous to that of definite 3-forms in the theory of G2-manifolds (see this example for details).
Moreover the torsion tensor $T$ is to have its $(T^a)^\alpha{}_\beta$-component equal to $(\Gamma^a)^\alpha{}_\beta$, see at torsion constraints in supergravity.
In addition the Bianchi identities have to hold:
$\nabla T^A = E^B \wedge R_{B}{}^{A}$
$\nabla H = 0$ (covariant constancy).
All this is implied by the equations of motion of 11-dimensional supergravity.
Notice that in view of the above analogy to G2-structure, the covariant constancy condition is precisely the analog of G2-manifold structure.
Discussion of this in the somewhat more streamlined D'Auria-Fré formulation of supergravity is in (AFFFTT 98, section 3.1).
Discussion that for the GS-version of the heterotic string consistency of the background is equivalent to the equations of motion of heterotic supergravity is in (Shapiro-Taylor 87).
Discussion with the hetetoric gauge field included is in (Atick-Dhar-Ratra 86).
Discussion for the GS-version of the type II superstring in type II supergravity-backgrounds is in (GHMNT 85), and for the D-branes in type II in (CGNSW 97).
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
For more references on this WZW perspective see below.
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
Eric D'Hoker, String theory – lecture 10: Supersymmetry and supergravity , in part 3 of
Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
A more recent and more comprehensive review is
For references on curved background see below.
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
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
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
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
See also
More along these lines is in
The Green-Schwarz-type action for the M5-brane was found in
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory (arXiv:hep-th/9701149)
Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, World-Volume Action of the M Theory Five-Brane (arXiv:hep-th/9701166)
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)).
See also
The 7-cocycle for the M5-brane on the supergravity Lie 3-algebra is equation (8.8) there.
See also division algebras and supersymmetry.
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
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 connection is made in
The other brane scan, listing consistent asymptotic AdS geometries is due to
with further developments discussed in
The consistency 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
Eric Bergshoeff, Ergin Sezgin, Paul Townsend, Superstring actions in $D = 3, 4, 6, 10$ curved superspace, Phys.Lett., B169, 191, (1986) (spire)
Eric Bergshoeff, Ergin Sezgin, Paul Townsend, Supermembranes and eleven dimensional supergravity, Phys.Lett. B189 (1987) 75-78, In Mike Duff, (ed.), The world in eleven dimensions 69-72 (pdf, spire)
These authors amplify the role of closed $(p+2)$-forms in super $p$-brane backgrounds (p. 3) and clearly state the consistency conditions for the M2-brane in a curved backround in 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 $\overline{\psi}\wedge e^a \wedge e^b \wedge \Gamma^{a b} \psi$ plus the supergravity C-field curvature and has to be closed.
That the heterotic supergravity equations of motion are sufficient for the 3-form super field strength $H$ to be closed was first argued in
and the computation there was highlighted and a little simplified around p. 17 of
A more comprehensive result arguing that the heterotic supergravity equations of motion of the background are not just sufficient but also necessary for (and hence equivalent to) the heterotic GS-string on that background being consistent was then claimed in
Discussion of this with the heterotic gauge-field included (hence including the Green-Schwarz anomaly cancellation) is in
Similar arguments for the type II string in type II supergravity appeared in
and for GS sigma-model D-branes in
That the M2-brane sigma-model is consistent on backgrounds of 11-dimensional supergravity that satisfy their equations of motion is discussed in (Bergshoeff-Sezgin-Townsend 87).
The role of the 4-form here is also amplified around (2.29) in
and in section 2.2 of
following
All this is actually subsumed by imposing the Bianchi identities of the corresponding supergravity Lie 3-algebra etc. in “rheonomic parameterization”, of the D'Auria-Fré formulation of supergravity, this is discussed in (AFFFTT 98, section 3.1, Fré-Grassi 07).
Discussion including also the RR-field background includes
That higher WZW functionals and hence Green-Schwarz super $p$-brane action functionals should have “higher” extended symmetry algebras in some sense… is observed in
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).
Michael Green, John Schwarz, Covariant Description Of Superstrings , Phys. Lett. B 136, 367 (1984) (web)
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)
Joaquim Gomis, Kiyoshi Kamimura, Peter West, Diffeomorphism, kappa transformations and the theory of non-linear realisations (arXiv:hep-th/0607104)
Discussion from the point of view of D'Auria-Fré formulation of supergravity is in
Gianguido Dall’Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, The $Osp(8|4)$ singleton action from the supermembrane, Nucl.Phys.B542:157-194,1999, (arXiv:hep-th/9807115)
Pietro Fré, Pietro Antonio Grassi, Pure Spinors, Free Differential Algebras, and the Supermembrane, Nucl.Phys.B763:1-34,2007 (arXiv:hep-th/0606171)
AFFFTT98 FreGrassi07
Discussion of the Green-Schwarz action for the open M2-brane ending on the M5-brane is in
C.S. Chu, Ergin Sezgin, M-Fivebrane from the Open Supermembrane, JHEP 9712 (1997) 001 (arXiv:hep-th/9710223)
Ph. Brax, J. Mourad, Open Supermembranes Coupled to M-Theory Five-Branes, Phys.Lett. B416 (1998) 295-302 (arXiv:hep-th/9707246)