6.2 Algebro-Geometric Structure of Supergravity

tained the same result in a more compact way using a first order formalism for the spin connection ωμab .

In this way 1976 opened a new important season in the theory of gravitation. General Relativity was found to admit a class of natural extensions dictated by a new powerful local symmetry that mixed fermions and bosons. Such supergravity theories, that can be constructed in various dimensions, are nothing else but Einstein gravity coupled to matter fields, both fermionic and bosonic, with very special choices of the spectrum, of the interactions and of the couplings.

In a few year time the complete park of all possible supergravities was constructed showing that they are all codified by a rich set of special geometric structures that can appear in the scalar sector (see Chap. 8).

At the beginning supergravity was developed independently from string theory, but in 1978, on the basis of a fundamental paper [11, 12] also written in Paris at the Ecole Normale by Gliozzi, Olive and Scherck, it became clear that the finite number of supergravities one can construct in D = 10 space-time dimensions, are in association with the corresponding consistent superstring models in that they just describe the low energy interaction of the massless modes of the superstring spectrum.

In 1978 in another fundamental paper [13] written by Cremmer and Julia in the same Paris location, there appeared the Lagrangian of the unique supergravity in D = 11 space-time dimensions which is the highest possible for such theories. Indeed, starting from D = 12, any closed supersymmetry multiplet includes spins higher than two and General Relativity is overcome. To the present moment, no one has been able to construct interacting theories with a finite number of spins higher than two and the unanswered question is whether they exist.

By dimensional reduction, compactification or direct construction a gigantic bestiary of pure and matter coupled supergravity theories has been derived in diverse dimensions and several type of classical solutions thereof have been found that have enormously enriched the landscape of General Relativity and Gravity providing new insights in the relation of gravity with strings, branes and gauge theories. A quick bird-eye survey of these topics will follow in Chaps. 7, 8, 9. In the present chapter we are interested in analyzing in depth the mathematical structure of supergravity theory developing further, in presence of supersymmetry, the principles that yield Einstein Theory as we presented it in Volume 1. This algebro-geometric approach leads us to single out in free differential algebras the appropriate environment for the construction of supergravities by means of the general principle of rheonomy which works as a sort of generalized analiticity.

6.2 Algebro-Geometric Structure of Supergravity

Let us now consider the generic structure of a candidate locally supersymmetric theory. This means that, in some appropriate way to be established, its field equations and eventually its action are invariant against infinitesimal transformations that are elements of the super-Poincaré Lie algebra. Hence the structure of this latter must

224 6 Supergravity: The Principles

be our primary concern. Looking at

that describes the supersymmetry algebra in D = 11, or at (6.1.5) which displays extended supersymmetry in D = 4, we see that the essential new ingredient is provided by the supercharges Qα . These generators transform as Lorentz spinors:

and are fermionic, in the sense that the associated transformation parameter εα is not a real commuting number, rather an anticommuting Grassmann number:

A would-be connection on a would-be supersymmetric principal bundle must be supersymmetry Lie algebra valued and therefore must contain a fermionic one-form ψ which couples to the supercharges Qα . In full analogy with (5.2.27) of the first volume we can introduce a one-form:

which is Poincaré super Lie algebra valued and whose supercurvature takes a form analogous to (5.2.28) of the first volume:

In relation with the above equations we must remember that p-forms have now a double grading with respect to their degree and with respect to their bosonic / fermionic character. Let us convene that the fermion number f is 0 for bosons and 1 for fermions. Then, for a pair of forms, respectively of degrees p1,2 and fermion numbers f1,2, we have the following commutation relations under exterior product:

In particular this implies that the gravitino one-forms ψα commute among themselves:

while they anti-commute with the vielbein and the spin connection:

Why did we name ψα gravitino one-forms? This will become clear in the sequel. Just as the vielbein one-forms Ea encode the spin-two particle named the graviton, in the same way the ψα one-forms encode its supersymmetric partner of spin s = 3/2, whose name has been agreed to be the gravitino, as we already stressed.

Independently from the number D of space-time dimensions, by setting Ta = Rab = ρ = 0 we obtain the dual description of the super Poincaré Lie algebra in terms of Maurer Cartan equations.

In the case of General Relativity the dynamical theory was constructed first by considering a principal Poincaré bundle admitting D-dimensional space-time MD as its base manifold and the D-dimensional Poincaré group as structural group:

secondly by imposing the soldering condition:

which identifies the Lorentz sub-bundle of P(D-Poincaré, MD ) with the tangent bundle T MD . In this way the spin connection could be solved in terms of the vielbein and its derivatives and local translations could be identified with diffeomorphisms of MD , the physical degrees of freedom being represented by the symmetric part of the square matrix Eμa (x). This could work because the Lorentz algebra is a closed subalgebra of the Poincaré Lie algebra, so that, at the end of the day, the spin connection behaves as a true principal connection on a Lorentz bundle. Rather than being an independent dynamical field, such a connection is a composite one in terms of the graviton degrees of freedom, yet mathematically it is a bona-fide principal connection.

In the case of supersymmetry, the generators {Qα , Jab} do not close a subalge-

bra, since the anti-commutator of two Qs produces a translation. Hence we cannot interpret local supersymmetry transformations as gauge-transformations in a principal bundle having the space-time manifold MD as its base-manifold and a group generated by {Qα , Jab} as structural group.

The alternative is that of enlarging the base-manifold by means of as many fermionic coordinates θ α as there are supercharges. This imitates the structure of General Relativity where the space-time manifold MD is a curved deformation of Minkowski space which, on its turn, can be viewed as the coset manifold:

226 6 Supergravity: The Principles

that is a supermanifold with D bosonic coordinates, named xμ, and q fermionic ones named θ α .5 One might conclude that the degrees of freedom of supergravity are those encoded in the supervielbein of superspace:

namely those described by the supermatrix E7AM (x, θ ) defined by the expansion of E7A in differentials of the supercoordinates:

E7A = E7AM (x, θ ) dzM

(6.2.15)

dzM ≡ (dxμ, dθ α )

yet this turns out to be naive and leads to a wrong track. Differently from the pure bosonic case, the number of components contained in the graded symmetric part6 of E7AM (x, θ ) is too big. It does not correspond to the physical off-shell degrees of freedom of a spin 2 and a spin 3/2 particle, as it should. This means that supergravity is not the theory of supermetrics in superspace. A new constructive principle should be added which should be simple, economic, universal and should introduce those appropriate constraints, that reduce the number of components parameterizing superspace geometry to that of the physical degrees of freedom of the relevant fermionic and bosonic particles.

Such a principle was found in the early years of supergravity theory and it is named the rheonomy principle. We explain it in Sect. 6.5. Before addressing this issue we have to dwell on another point of equal fundamental relevance. Not only supergravity theories are characterized by local supersymmetry transformations that are midway between gauge-transformations in a principle bundle and diffeomorphisms requiring the principle of rheonomy to obtain an adequate geometrical interpretation; they also involve, in all higher space-time dimensions, a new type of gauge fields, namely (p + 1)-forms. From the physical view-point this fact is related to the existence of the so named p-branes, since such (p + 1)-forms naturally couple to the world-volumes of p-extended objects, just as standard gauge fields couple to the world-lines of charged particles; from the mathematical side the presence of higher degree gauge forms is a clear indication that the algebraic structure

5In this discussion the index α incorporates both the spinor index running on the dimension of the relevant spinor representation of SO(1, D − 1) and the replica index related with extended supersymmetry.

6By graded symmetric we mean E7AM (x, θ ) = (−)fA fB E7MA(x, θ ).