FQFT and cohomology
Types of quantum field thories
For this is the Green-Schwarz superparticle.
The Green-Schwarz action functionals are of the standard sigma-model form for target spaces that are super-homogeneous spaces for a Lie supergroup and a sub-super-group, and for background gauge fields that are super-WZW-circle n-bundles with connection/bundle gerbes on .
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.).
The Chevalley-Eilenberg algebra is generated on
elements and of degree
and elements of degree
with the differential defined by
Removing the terms involving 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.
may be read as saying that 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 which have “all indices contracted”. (See also at torsion constraints in supergravity.)
Notably we have
This remaining operation “” of the differential acting on Loretz scalars is sometimes denoted “”, 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 a Fierz identity implies that the term
vanishes identically, and hence in these dimensions the term
These exist (are closed) only for certain combinations of and . 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 -connections are simply given by globally defined connection form 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 -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 -brane propagating on an -dimensional target spacetimes makes sense only for special combinations of , 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.
Moving down and left the diagonals corresponds to double dimensional reduction.
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
|the real numbers|
|the complex numbers|
|the quaternions||little string|
|the octonions||heterotic/type II string|
What is missing in the “old brane scan” are the D-branes in and the M5-brane in (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.
|10||D0||F1, D1||D2||D3||D4||NS5, D5||D6||D7||D8||D9|
(The first colums follow the exceptional spinors table.)
|11||on sIso(10,1)||on m2brane|
|10||on sIso(9,1)||on StringIIA||on StringIIB||on StringIIA||on sIso(9,1)||on StringIIA||on StringIIB||in StringIIA||on StringIIB|
|6||on sIso(5,1)||on sIso(5,1)|
|4||on sIso(3,1)||on sIso(3,1)|
Hence given a Lie algebra cocycle
In general this 3-form is no longer closed. If it is closed, then the Green-Schwarz superstring is again well defined on as a WZW model.
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 ) 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 , this way the expression is directly analogous to that of definite 3-forms in the theory of G2-manifolds (see this example for details).
In addition the Bianchi identities have to hold:
Discussion with the hetetoric gauge field included is in (Atick-Dhar-Ratra 86).
The Green-Schwarz action functional (formulated for the superstring) 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 original “brane scan” classification of GS action functionals by WZW terms is due to
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
More along these lines is in
The Green-Schwarz-type action for the M5-brane was found in
The 7-cocycle for the M5-brane on the supergravity Lie 3-algebra is equation (8.8) there.
See also division algebras and supersymmetry.
These cohomologival arguments also appear in what is called the “ectoplasm” method for invariants in super Yang-Mills theory in
The connection is made in
The other brane scan, listing consistent asymptotic AdS geometries is due to
with further developments discussed in
These authors amplify the role of closed -forms in super -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 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 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
and for GS sigma-model D-branes in
The role of the 4-form here is also amplified around (2.29) in
and in section 2.2 of
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 -brane action functionals should have “higher” extended symmetry algebras in some sense… is observed in
The existence of -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).
I.N. McArthur, Kappa-Symmetry of Green-Schwarz Actions in Coset Superspaces (arXiv:hep-th/9908045)
Discussion from the point of view of D'Auria-Fré formulation of supergravity is in
Ph. Brax, J. Mourad, Open Supermembranes Coupled to M-Theory Five-Branes, Phys.Lett. B416 (1998) 295-302 (arXiv:hep-th/9707246)