246	6 Supergravity: The Principles

supersymmetry algebra. The answer is easily obtained arguing in terms of highest vectors. Consider the highest helicity state of the graviton:

|Ω! = |2, 0, 0, 0!

(6.5.15)

and let us assume that it is the highest state of the entire supermultiplet. This means that it is annihilated by all supercharges that are creation operators, namely:

q( 21 ,± 21 ,± 12 ,± 12 )|2, 0, 0, 0! = 0

(6.5.16)

A non-vanishing result is obtained applying to |2, 0, 0, 0! products of the operators

q(− 21 ,± 12 ,± 12 ,± 21 ), where all factors in the product are different. So we find:
|2, 0, 0, 0!							2 1				q(− 2
8 2 ,			2 ,		2 ,								, 2	, 2 , 2 )				2, 0, 0, 0
8	3		1		1		1		=			1	1	1	1		|							!
8
8										(	1	, 1 ,	1 ,	1 )	q	(		1	, 1	, 1	,	−	1 )		|2, 0, 0, 0!		(6.5.17)
|1, 1, 1, 0! = q											− 2	2	2	2			− 2		2	2			2
· · · = · · ·
· · · = · · ·
| − 2, 0, 0, 0! =												8 sign*choices q− 21									,± 21 ,± 12 ,± 12					|2, 0, 0, 0!

Applying the destruction operators to all other positive weights of the spin two representation we find that the set of weights we can construct is just the union of the weights belonging to the three mentioned irreducible representations. Hence the D = 11 supermultiplet is indeed constituted by the fields mentioned in Table 6.2.

6.6 Summary of Supergravities

Having clarified the construction principles of supergravities and their fundamental algebraic basis, rooted in the Free Differential Algebra structure we pass to a scan of the available theories.

In D = 11 there is just a unique supergravity, M-theory, whose structure we have thoroughly discussed.

In D = 10 we have just five different supergravities, displayed in Fig. 6.12 which are in one-to-one correspondence with the available consistent superstring theories. Actually, from the supergravity viewpoint there are only three possible theories in D = 10. The type II theories that have two Majorana-Weyl supercharges (A and B, according to the choice of their chirality, as we explain below) and the type I theory that has only one Majorana-Weyl supercharge and can be coupled to a vector multiplet. The different choice of the gauge group is the only distinction among type

6.6 Summary of Supergravities

247

Fig. 6.12 The field content of the five D = 10 supergravities. The fields of each of these theories are the massless modes of the corresponding superstring theory. The bosonic fields are organized according to their string origin in the NS-NS or R-R sector, while the fermionic fields are organized according to their chirality

I theories. However a very complex mechanism displayed by these latter resides in the possibility of introducing their coupling to gauge and Lorentz Chern-Simons three-forms: this provides the cancellation of anomalies and selects the three models that complete the list of consistent superstring theories. In this book we do not dwell on D = 10, N = 1 supergravities and on their Lorentz Chern-Simons coupling for which we refer the reader to the book [28]. We rather focus on type II theories of which we give a complete account.

The key point in D = 10 is the existence of Majorana-Weyl spinors satisfying the double condition:

ψL/R = C		L/RT ;	Γ11ψL/R = ±ψL/R	(6.6.1)
	ψ

where C is the charge conjugation matrix and Γ11 is the chirality matrix (see Appendix A.4 for details). The type IIA theory is based on the super-Poincaré Lie algebra containing two Majorana-Weyl supercharges, one QL which is left-handed, the other QR which is right-handed. The type IIB theory is instead based on the superPoincaré Lie algebra that contains two Majorana-Weyl supercharges, QAL (A = 1, 2) of the same chirality, say left-handed. The presence of this doublet of chiral supercharges introduces an SL(2, R) symmetry which is an essential item in the construction of the whole theory.

In dimensions D < 10 the number of available supergravity theories starts growing because we can reduce the number of supercharges and introduce an increasing available choice of matter supermultiplets. As soon as scalar fields appear in the

248	6 Supergravity: The Principles

spectrum they introduce a new quality: the scalars can be regarded as the coordinates of a differentiable target manifold for whose geometry supersymmetry selects a variety of special structures. Chapter 8 presents a bird-eye review of supergravity geometries and couplings.

Chapter 7 is instead devoted to enlighten the vital dualism between the bulk and the brane world-volume perspectives. For that we shall need the explicit structure of the type II theories which we present in the next two sections.

6.7 Type IIA Supergravity in D = 10

The full-fledged rheonomic construction of type IIA supergravity was obtained only recently in [29] which we follow in the present section. The field content of the theory is given in Table 6.3.

This field content corresponds to the basic forms of a specific Free Differential Algebra including the 0-form items entering the rheonomic parameterizations of its curvatures.

The starting point is, as usual, the super-Poincaré algebra. In D = 10 we have two super-Poincaré algebras with 32 supercharges, the type IIA and the type IIB. If we also include the dilaton, as we will do, there are various equivalent definitions of curvatures that are named frames and differ by dilaton pre-factors. The Einstein frame is that which leads to an action where the Einstein kinetic term is canonical without any dilaton pre-factors. The string frame, which has distinguished advantages when writing the string action in its background, corresponds instead to non-canonical Einstein terms in the action. The two frames are just related by a suitable Weyl transformation depending on the dilaton. For type IIA supergravity we use the string frames for two reasons. The first is that in this frame the FDA has a simpler and more elegant form. The second is pedagogical. We want to emphasize the freedom of using different but equivalent frames. For type IIB supergravity we will rather use the Einstein frame in which the SL(2, R) symmetry of that theory is manifest.

The Maurer Cartan description of the type IIA superalgebra is obtained by setting to zero the following curvatures:

Table 6.3 Structure of the graviton multiplet in Type IIA supergravity

SO(1, 9) rep.							# of states	Name
(2, 0, 0, 0, 0)							35	graviton
( 32	, 21 ,		21 ,	21	, 21 )L		56	left gravitino
( 3	,	1 ,	1 ,	1	,	1 )R	56	right gravitino
2		2	2	2		2
(1, 1, 0, 0, 0)							28	NS B-field
(1, 0, 0, 0, 0)							8	RR 1-form
(1, 1, 1, 0, 0)							56	RR 3-form
( 12	, 21 ,		21 ,	21	, 21 )L		8	left dilatino
( 1	,	1 ,	1 ,	1	,	1 )R	8	right dilatino
2		2	2	2		2
(0, 0, 0, 0, 0)							1	dilaton

6.7 Type IIA Supergravity in D = 10

249

Type IIA Super-Poicaré Algebra in the String Frame

	Rab ≡ dωab − ωac ωcb														(6.7.1)
		1
	Ta ≡ D V a − i			ψL Γ a ψL + ψR Γ a ψR											(6.7.2)
		2
					1						ωab ΓabψL,R
	ρL,R ≡ D ψL,R ≡ dψL,R −														(6.7.3)
										4
	G[2] ≡ dC[1] + exp[−ϕ]								R ψL						(6.7.4)
								ψ
	f[1] ≡ dϕ	1													(6.7.5)
	χL/R ≡ dχL,R −				ωab ΓabχL,R										(6.7.6)
		4

where the 0-form dilaton ϕ appearing in (6.7.4) introduces a mobile coupling constant. Furthermore, V a , ωab denote the vielbein and the spin connection 1-forms, respectively, while the two fermionic 1-forms ψL/R are Majorana-Weyl spinors of opposite chirality:

Γ11ψL/R = ±ψL/R

(6.7.7)

The flat metric ηab = diag(+, −, . . . , −) is the mostly minus one and Γ11 is Hermitian and squares to the identity Γ112 = 1.

Setting Rab = T a = G[2] = f[1] = 0 one obtains the Maurer Cartan equations of a superalgebra where the spinor charges, QL,R dual to the spinor 1-forms ψL,R not only anticommute to the translations Pa but also to a central charge Z dual to the (Ramond Ramond) 1-form C[1].

According to Sullivan’s second theorem the FDA extension of the above superalgebra is dictated by its cohomology. In a first step one finds that there exists a cohomology class of degree three which motivates the introduction of a new 2-form generator B[2] which in the superstring interpretation is just the Kalb-Ramond field. Considering then the cohomology of the FDA-extended algebra one finds a degree four cohomology class which motivates the introduction of a 3-form generator C[3]. In the superstring interpretation, this is just the second R-R field, the first being the gauge field C[1]. Altogether the complete type IIA FDA is obtained by adjoining the following curvatures to those already introduced:

The FDA Extension of the Type IIA Superalgebra in the String Frame
H[3] = dB[2] + i(ψ			L Γa ψL −				R Γa ψR ) V a			(6.7.8)
						ψ
G[4] = dC[3] + B[2] dC[1]
1
−		exp[−ϕ](ψL ΓabψR + ψR ΓabψL) V a V b								(6.7.9)
	2

Equations (6.7.1)–(6.7.5) together with (6.7.8)–(6.7.9) provide the complete definition of the type IIA Free Differential Algebra.