6.4The Super FDA of M Theory and Its Cohomological Structure

We discussed Sullivan’s theorems for Lie algebras and their corresponding FDA extensions but, as we stressed, they hold true, with obvious modifications, also for superalgebras Gs and for their FDA extensions. Actually, in view of superstring and supergravity, it is precisely in the supersymmetric context that FDAs have found their most relevant applications. As an illustration of the general set up and also for its intrinsic interest, by recalling the results of [15] and [16], we present here the structure of the M-theory FDA, namely the algebraic basis of maximal supergravity in eleven space-time dimensions. Indeed M-theory is the name frequently given in contemporary literature to D = 11 supergravity. This results from the so called second string revolution that showed that all consistent ten-dimensional string theories can be related to each other by non-perturbative dualities and can be regarded as special limits of a unified non-perturbative mother-theory (this is the meaning of M) whose exact microscopic definition has not yet been given, yet is somehow defined by its own low energy limit that is precisely D = 11 supergravity.

Within this context we are also able to illustrate the bearing of a quite relevant question: is an FDA M always equivalent to a normal Lie algebra G7 G larger than the Lie algebra of which M is a cohomological extension? How we can mathematically formulate and answer such a question we will show below by recalling results of [15] and also more recent literature [18–20].

Let us begin by writing the complete set of curvatures, plus their Bianchi identities. This will define the complete FDA:

?

A = M C (6.4.1)

the curvatures being the contractible generators C. By setting them to zero we retrieve, according to Sullivan’s first theorem, the minimal algebra M. This latter, according to Sullivan’s second theorem, has to be explained in terms of cohomology of the normal subalgebra G M, spanned by the 1-forms. In this case G is just the D = 11 superalgebra spanned by the following 1-forms:

1.the vielbein V a ,

2.the spin connection ωab ,

3.the gravitino ψ .

The higher degree generators of the minimal FDA M are:

1.the bosonic 3-form A[3],

2.the bosonic 6-form A[6].

The complete set of curvatures is given below [15, 16]:

Ta = D V a − i 1 ψ Γ a ψ

2

Rab = dωab − ωac ωcb

F[4] = dA[3] − 1 ψ Γabψ V a V b

2

F[7] = dA[6] − 15F[4] A[3] − 15 V a V b ψ Γabψ A[3]

2

−i 1 ψ Γa1...a5 ψ V a1 · · · V a5

2

From their very definition, by taking a further exterior derivative one obtains the Bianchi identities which play an even more fundamental role in constructing supergravity theories then they played in constructing General Relativity:

The dynamical theory is defined, according to a general constructive scheme of supersymmetric theories which we explain in Sect. 6.5, by the principle of rheonomy, implemented into Bianchi identities. Indeed, as we discuss in that section, there is a unique rheonomic parameterization of the curvatures (6.4.2) which solves the Bianchi identities and it is the following one:

Ta = 0

F[4] = Fa1...a4 V a1 · · · V a4

The expressions (6.4.8) satisfy the Bianchi provided the space-time components of the curvatures satisfy the following constraints

which are the space-time field equations. We will come back to these equations and study some of their solutions. We postpone this to the sequel. Here we concentrate on the structure of the FDA.

6.4.1 The Minimal FDA of M-Theory and Cohomology

Setting Ta = Rab = ρ = F[4] = F[7] = 0 in (6.4.2) we obtain the Maurer Cartan equations of the minimal algebra M. In particular we have:

≡15 V a V b ψ Γabψ A[3]

2

+i 1 ψ Γa1...a5 ψ V a1 · · · V a5

2

The reason why the three-form generator A[3] does exist and also why the six-form generator A[6] can be included is, in this set up, a direct consequence of the cohomology of the super Poncaré algebra in D = 11, via Sullivan’s second theorem. Indeed the 4-form Γ [4](V , ψ) defined in the first line of (6.4.10) is a cohomology class of the super Poincaré Lie algebra whose Maurer Cartan equations are the first three of (6.4.2) upon setting Ta = Rab = ρ = 0. We have:

and there is no Φ[3](V , ψ) such that Γ [4](V , ψ) = dΦ[3](V , ψ).

An important issue to be stressed at this point is the following one. As we will emphasize in Chap. 7, the main new idea underlying superstring/supergravity theories is the interplay between gravitational-like bulk-theories and world-volume gauge-like brane-theories. In bulk-theories there are p-forms. These latter couple to the degrees of freedom corresponding to (p − 1)-extended objects spanning a p- dimensional world-volume in the same way as the electromagnetic field (a 1-form)

algebra G7? Indeed by adding new 1-form generators φp which, together with the generators eI of G satisfy the Maurer Cartan equations of the larger algebra G7 G, it may happen that we are able to construct a polynomial Φ[p−1](e, φ) such that:

In this case the generator A[p−1] of the FDA associated with the cohomology class

(e) can be simply deleted by the list of independent generators and simply identified with the polynomial Φ[p−1](e, φ).

In these terms the question was already posed twenty three years ago by D’Auria and the author of this book in [15] obtaining a positive answer [15] which was revisited in [18, 19].

The enlarged algebra G7 contains, besides the generators of G a bosonic 1-form which is in the rank two antisymmetric representation of the Lorentz group, a bosonic 1-form Ba1a2,...a5 which is in the rank five antisymmetric representation

and finally a fermionic 1-form η which is in the spinor representation just as the generator ψ .

The Maurer Cartan equations of G7 are:

These Maurer Cartan equations are consistent, namely closed, provided the following equation is satisfied by the coefficients:

Using all the generators of G7 one can construct a cubic polynomial

Φ[3] V , ψ, B(2), B(5)

=λBa1a2 V a1 V a2 + α1Ba1a2 Ba2a3 Ba3a1

+α2Bb1a1...a4 Bb1b2 Bb2a1...a4 + α3εa1...a5b1...b4mBa1...a5 Bb1...b5 V m

+α4εm1...m6n1...n5 Bm1m2m3p1p2 Bm4m5m6p1p2 Bn1...n5 iβ1ψ Γ a η V a

The coefficients appearing in (6.4.22) are completely fixed by (6.4.23) for any of the 1-parameter family of algebras described by (6.4.15)–(6.4.20). Indeed the closure condition (6.4.21) is one equation on three parameters which are therefore reduced to two. One of them, say γ1 can be reabsorbed into the normalization of the extra fermionic generator η, but the other remains essential and its value selects one algebra within a family of non-isomorphic ones. Following [18] it is convenient to set:

and s is the parameter which parameterizes the inequivalent algebras G7s . For each of them we have a solution of (6.4.23) realized by

In the original paper [15] only two of this infinite class of solutions were found, namely those corresponding to the values:

which are the roots of the equation λ = 1. Indeed in [15] the additional condition λ = 1 was imposed, which is unnecessary as it has been shown in [18, 19] where the more general solution (6.4.24), (6.4.25) was found.

It remains to be seen whether the equivalence between the minimal FDA M and the Lie algebra G7 can be promoted to a dynamical equivalence between their gaugings. In other words whether one can consistently parameterize the curvatures of G7 in such a way that identifying the three form A[3] with the polynomial Φ[3], the rheonomic parameterizations (6.4.8) are automatically reproduced? This is a rather formidable algebraic problem and to the present time no one has been able to answer it in the positive way. Actually although this has not been established like a theorem it appears from all undertaken attempts that the correct answer is the negative one. Hence the dynamical theory of supergravity is nicely and necessarily based