8.7 N = 2, D = 5 Supergravity Before Gauging

Relying on the geometric lore developed in the previous sections it is now easy to state what is the bosonic Lagrangian of a general N = 2 theory in five-dimensions. We just have to choose an n-dimensional very special manifold and some quaternionic manifold QM of quaternionic dimension m. Then recalling (8.6.13) we can specialize it to:

where huv (q) is the quaternionic metric on the quaternionic manifold QM , while gij (φ) is the very special metric on the very special manifold. At the same time the constant tensor dΛΣΓ is that defining the cubic norm (8.6.20) while the kinetic metric N is that defined in (8.6.24). The transformation rule of the gravitino field takes the general form (8.6.16) with the graviphoton defined as in (8.6.17) and the tensor ΦABΛ given by (8.6.31). In this respect it is noteworthy that gravitino supersymmetry transformation rule does depend only on the vector multiplet scalars and it is independent from the hypermultiplets.

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to the field strength of a (d − 1)-form A(d−1) of the higher dimensional theory. Expanding the higher dimensional theory in modes around such a vacuum and keeping only the lightest ones, one obtains a new gravitational theory in d-dimensions including a variety of new fields, whose interactions are dictated by the geometry of MD−d . In particular the geometry of the scalar manifold Mscalar of the lower dimensional theory which, as we know, controls the entire form of the d-dimensional Lagrangian, is related to MD−d in the following general way: Mscalar encodes the moduli-space of the structure-deformations of MD−d . Let us explain this deep and general concept. For instance MD−d is an Einstein manifold. This means that we have a metric gij (y) defined on it, whose Ricci tensor is proportional to the same metric. That metric can be smoothly deformed by means of parameters that we name moduli and fill a subspace of Mscalar. The compact manifold MD−d can have a more refined geometrical structure, a complex structure for instance, a Kähler structure or in any case a restricted holonomy structure. The deformations of all such structures fill moduli space which are included in Mscalar. The special geometry structure of Mscalar follows mathematically from deformation theory.

This scheme, that goes under the name of flux compactification, has been unveiled in supergravity but has a more general validity. The key ingredients are:

(a)The existence of p-forms in higher dimensional theories whose fluxes can drive the compactification.

(b)The choice of restricted holonomy manifolds MD−d for the compact dimensions.

In supersymmetric theories the spectrum of p-forms available in D-dimensions is dictated by the appropriate Free Differential Algebra which, as we learnt in Chap. 6, is ultimately a yield of the super Poincaré Lie algebra cohomology. Without supersymmetry, Free Differential Algebras do exist nonetheless and a more general variety of possibilities is available for the p-form gauge fields.

Similarly, in supersymmetric theories the constraint on the holonomy of the internal manifold MD−d follows from the request that some of the supersymmetries should be preserved by the compactification. This requires the existence of so named Killing spinors, namely of covariantly constant sections of an appropriate spinor bundle on MD−d , whose existence restricts the holonomy. In a more general mathematical set up this is just an instance of the constraints imposed by the existence of some G-structure. Adopting such a language in the context of the more general class of higher dimensional theories postulated above opens a wider spectrum of possibilities.

In such a broader landscape the main mathematical frameworks governing both the construction of the relevant Lagrangians and the search and classification of their solutions will still be the same as in supergravity, namely deformation theory of special geometrical structures, restricted holonomy and G-structures, σ -model reduction of duality symmetric Lagrangians (8.4.1).

The recipe to insert almost all of the most advanced aspects of modern differential geometry into Gravity Theory has been discovered by supergravity but certainly will last as an integral part of it even if our own world should turn out to be non-supersymmetric. The same is probably true of the branes whose existence and duality with the bulk theories is more general and holds true beyond superstrings and supergravity.

For this reason the last chapter of this book is devoted to glances at the classical solutions of supergravity. These form an incredibly rich park with many alleys, islands and ponds. There are vacua solutions, brane-solutions, that were already touched upon, monopole-solutions, instanton solutions, cosmological solutions, black-hole and black-brane solutions and still several other type of geometrical backgrounds. Each of these categories plays a distinctive important role in superstring/supergravity theory and requires appropriate mathematical techniques to be studied and worked out. Even a simple review of the main features of each category would build up a bestiary too long and too much complicated to be presented within the scope of the present book. Hence we necessarily restricted ourselves to an anthology chosen according to the formative criteria that inspire our writing. Indeed we aim at conveying to the reader some general ideas and some paradigms that, according to the writer’s opinion, encapsulate an intrinsically new quality in the understanding of Gravity Theory and introduce new important mathematical structures in its development. Notwithstanding these restrictive conditions, the list of candidate topics and examples came out quite long so that, a little bit arbitrarily, a final short list of three items was drawn, far from being exhaustive, yet providing a very dense conceptual impact.

1.The first addressed topic is that of spherical black solutions in D = 4 supergravity. The interest in this class of solutions, whose classification and construction constitutes an active field of current research, is two-fold. From the technical point of view, the most effective approach to the derivation of these solutions, that depend only on one radial coordinate, is provided by the reduction of the

supergravity field equations to those of an effective σ -model which, in the case

that the scalar manifold Mscalar is equal to a symmetric coset GH , were proved to be completely integrable. The same σ -model reduction can be applied also to the case of other few parameter solutions, like the cosmological ones, yielding the very interesting phenomenon of cosmic billiards, mentioned in Chap. 5.

From the conceptual point of view the main new quality encapsulated in these studies is given by the attraction mechanism1 and by the identification of the extremal black hole entropy with the square root of a quartic symplectic invariant constructed with the electromagnetic charges of the hole. This phenomenon goes beyond supersymmetry and is just related with the symplectic structure of the

1As we illustrate below the attraction mechanism corresponds to the following notable property of supergravity black holes which was discovered by Ferrara and Kallosh in 1995 [1, 2]: independently from their values at spatial infinity, the scalar fields flow to universal fixed values at the event horizon, dictated solely by the electromagnetic charges of the hole.

duality symmetric theories of type (8.4.1). In a wider contest the charges of the hole can be interpreted in terms of branes and brane-wrappings, thus opening an important window on the statistical interpretation of black-holes. In the next pages we just try to introduce the reader to these fundamental ideas, providing some glimpses of this challenging research field.

2.The second topic addressed is that of flux vacuum solutions of M-theory. Supergravity and superstrings impose higher space-time dimensions and D = 11 is the maximal one where supergravity takes its simplest and most elegant form. Yet our world is effectively four-dimensional so that any contact with reality can be established only if seven of the extra dimensions are compactified and made observable only at energy scales of the order of the Planck mass. A challenging mechanism is provided by flux compactifications encoded in (9.1.2). Just be-

cause M-theory contains a three-form and a six-form, giving a vacuum expectation value to their field strengths splits eleven dimensional space-time into 4 + 7. At the beginning of the eighties this raised a lot of expectations that produced a large literature going under the name of Kaluza-Klein Supergravity. Although the hope that the standard model of non-gravitational interactions might be retrieved in this way proved too naive, yet the (flux) compactifications of M-theory provide to the present day a very important theoretical laboratory in connection with the gauge/gravity correspondence and with several other aspects of brane physics. From the conceptual and mathematical point of view, the problem of constructing these vacua and classifying their residual supersymmetry brings in the theory of Killing spinors, G-structures and restricted holonomy. Introducing the reader to these concepts and to their use is the main reason of considering this example. An additional reason resides in the opportunity offered by these examples of deepening our understanding of rheonomy. According to what we explained in Chap. 6, every classical solution of the space-time field equations can be extended to a full superspace solution by integrating the rheonomic conditions. The result is guaranteed but how to do it in practice is a different question. We will show that the integration is immediate and leads to the Maurer Cartan forms of a supercoset manifold in all those θ -directions that correspond to preserved supersymmetries. The θ -integration in the direction of broken supersymmetries is instead more involved and corresponds to some non-trivial fiberings. Our goal is to illustrate this mechanism locating the obstruction both to θ -integration and to supersymmetry preservation, which is the same thing, in a well-defined geometrical datum that is the holonomy tensor.

3.The third addressed topic is similar to the second, dealing with a particular in-

stance of flux vacuum solution of type IIA supergravity, namely that on the product manifold AdS4 × P3. This is done purposely in order to emphasize both the similarities and the essential new features one encounters while solving the same

problem in D = 10 rather than in D = 11. The novelty is provided by a necessary internal flux of the G[2] Ramond form which pairs with the external flux of the G[4] Ramond form and which is possible only due to the Kähler structure of the internal manifold P3. The example of this compactification has an intrinsic value since it corresponds to a situation where we end up with N = 6