Supersymmetric theories are highly symmetric and magnificent beautiful. These symmetries exchange fermions with bosons, either in
flat space supersymmetry or in curved space supergravity. Supergravity is a field theory, at present the most promising candidate for quantum gravity that combines the principles of supersymmetry and general relativity. The theory of supergravity suggested a novel approach to unification. In the quest for a unified description of gravity and matter interactions, several higher dimensional theories have been proposed. Supergravity
has emerged as a central ingredient in the search for the unified
theory of nature, and has led to many deep results in theoretical
physics. Supergravity predicts the existence of the graviton as a carrier for the force of gravity, as well as a corresponding particle called the gravitino, neither of which have been observed experimentally. Fields related by supersymmetry transformations form a supermultiplet. The one that contains a graviton is called the supergravity multiplet. The field content of different supergravity theories varies considerably, all supergravity theories contain at least one gravitino and they all contain a single graviton. Thus every supergravity theory contains a single supergravity supermultiplet. The geometry of curved superspace is shown to allow the existence of a large family of supermultiplets that can be used to describe supersymmetric matter, including vector, tensor and hypermultiplets. The first steps in the construction of any supergravity theory are usually based on the observation that local supersymmetry implies the invariance under general coordinate transformation. To construct the full theory usually requires more fields and important restrictions arise on the dimensionality of spacetime. For instance, while minimal supergravity in D = 4 dimensions does not require additional fields, in D = 11 dimensions an additional antisymmetric gauge field is necessary. The need for certain extra fields can be readily deduced from the underlying massless supermultiplets.
The gauged supergravity provides an interesting theoretical framework to the physics beyond the standard model. Gauged supergravities, where the
global isometries of the matter lagrangian are promoted to local
symmetries, have been widely explored and by now almost all allowed
models for diverse spacetime dimensions. Gaugings are the only known supersymmetric deformations of maximal supergravity. They may originate in various ways from fluxes and branes in higher dimensions. When switching on a gauging, the corresponding gauge fields transform according to the adjoint representation of the gauge group. Gaugings of maximal supergravity theories have revealed intriguing insights into the structure of supergravity theories as well as into their higher dimensional origin. Maximal supergravity theories contain a number of vector gauge fields which have an optional coupling to themselves as well as to other supergravity fields. The corresponding gauge groups are nonabelian. To preserve supersymmetry in the presence of these gauge couplings, the Lagrangian must contain masslike terms for the fermions and a potential depending on the scalar fields. The Lagrangian of ungauged maximal supergravity contains the standard Einstein-Hilbert, Rarita-Schwinger and Dirac Lagrangians for the gravitons, the gravitini and the spinor fields. The most systematic approach for a classification and construction of gauged supergravities resorts to exploiting the duality symmetry underlying the ungauged theories. The construction and study of these gauged supergravities is by considering them as deformations of the ungauged theories obtained by simple reduction. The covariant formulation of gauged supergravities provides a universal framework in which the effective theories associated with particular flux compactifications can be conveniently constructed and analyzed. The universal formulation of the five-dimensional gauged supergravity shows a strong similarity with the formulation of the three-dimensional gauged supergravities. In three dimensions, the relevant duality relates scalar and vector fields and the ungauged theory is formulated entirely in terms of scalar fields. The general gauged theory on the other hand combines the complete scalar sector and the dual vector fields. In general, the lower supersymmetric 6D supergravities admit more general couplings than those which can be obtained by truncation of the maximal theory since the quadratic constraints encountered in gauging of the maximal theory are far more stringent than what is required in gauging of the lower supersymmetric theories.
Conformal supergravities are supersymmetric extensions of conformal invariant, higher derivative, power counting renormalizable Weyl theory of gravity. They are gauge theories of the superconformal group and naturaly unify Weyl gravity with gauge and matter fields. The resulting theory is invariant under superconformal transformations which include general coordinate transformations, supersymmetry and special conformal transformations. The construction of off-shell representations of the superconformal algebra, such as the multiplets constructed in the theory, forms only the first step in the formulation of a model of conformal supergravity. These multiplets are controlled by a large set of gauge transformations which contains the superconformal group. Conformal invariance is the highest degree of space-time symmetry that a field theory without dimensional parameters can have. The superconformal transformations are the supersymmetric generalization of this symmetry. Superconformal methods are useful to build invariant actions in supergravity. For the supersymmetric theories, a similar construction allows to get more insight in the structure of supergravity actions. Poincaré supergravity theories are obtained by coupling the Weyl multiplet to additional superconformal multiplets containing Yang-Mills and matter fields. The resulting superconformal theory then becomes gauge equivalent to a theory of Poincaré supergravity. This is conveniently exploited by imposing gauge conditions on certain components of the extra superconformal multiplets. The only phenomenologically acceptable supergravity models that are known at present are based on Poincaré supergravity interacting with a suitable combination of vector and scalar multiplets. By using conformal tensor calculus, conformal supergravities form an elegant way to construct general couplings of Poincaré-supergravities to the matter.
The gauge anomalies manifest themselves as a non-invariance of the effective action under gauge transformations. The anomalous gauge symmetries are more problematic. Since gauge symmetries are needed to decouple the unphysical states of the theory, a violation of these symmetries renders the theory inconsistent. It was studied to what extent a general gauge theory of the above type with gauged axionic shift symmetries, generalized Chern-Simons terms and quantum gauge anomalies can be compatible with N = 1 supersymmetry. Our models constitute the supersymmetric framework for string compactifications with axionic shift symmetries, generalized Chern-Simons terms and quantum anomalies. We review some properties of the field equations of six-dimensional (1,0) supergravity coupled to tensor and vector multiplets, and in particular their relation to covariant, consistent and gravitational anomalies. We construct the complete coupling of (1,0) supergravity to all possible (1,0) multiplets, generalizing the results in order to include hypermultiplets, and extending the results to all orders in the fermi fields, while taking into account the anomalous couplings. The inclusion of charged hypermultiplets gives additional terms in the supersymmetry anomaly. The resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is determined by corresponding Wess-Zumino consistency conditions, aside from a quartic coupling for the gaugini. In formulating the low-energy couplings between tensor and vector multiplets, one has two natural options. The first is related to covariant field equations and to the corresponding covariant anomalies. It has the virtue of respecting gauge covariance and supersymmetry, but the resulting field equations are not integrable. The second is related to consistent, and thus integrable field equations. These may be derived from an action principle that satisfies Wess-Zumino consistency conditions, and as a result embody a supersymmetry anomaly. The subsequent work of some authors has developed the consistent formulation, but one can actually revert to a covariant formulation, at the price of having non-integrable field equations. The relation between the two sets of equations is one more instance of the ink between covariant and consistent anomalies in field theory. This is a remarkable laboratory for current algebra, where one can play explicitly with anomalous symmetries and their consequences. The issue of anomaly cancellation is one crucial importance in field theory, since the presence of anomalies breaks the gauge invariance of the quantum theory. The occurrence of an anomaly for a local supersymmetry makes supergravity inconsistent at the quantum level. Anomaly cancellation has been, for a long time, one of the main guiding principles for the construction of consistent gauge and gravitational theories. Cancellation of gauge anomalies is a dogma nowadays, when one speaks about consistently defining a quantum gauge theory. The anomaly-free models are based on consistency conditions on low-energy supergravity theories, and do not depend upon a specific completion such as superstring theory. The results naturally lead to the question of whether it is possible to place stronger bounds on the set of consistent theories than those understood from anomaly cancellation and other known constraints. We identify a number of models which obey all known low-energy consistency conditions, but which have no known string theory realization. Many of these models contain novel matter representations, suggesting possible new superstring theory constructions. We hope that the variety of new apparently consistent supergravity models identified in the theory will stimulate some further understanding of new string realizations or will help to generate new constraints on quantum theories of gravity. The issue of anomaly freedom in chiral supergravities can be examined not only in the usual context of superstring theories, but also in the context of lower-dimensional supergravities, many of which arise from superstring and M-theory compactifications. Anomaly freedom is generally independent of the existence of superstring theory. The anomaly freedom of our theory strongly suggests the deep significance of such interactions, and may lead to a more fundamental theory of extended objects which are not necessarily superstrings. We hope to have conveyed the idea that anomalies play an important role in supergravity and their cancellation has been and still is a valuable guide for constructing consistent quantum supergravity theories.
Particular 10-dimensional supergravity theories are considered low energy limits of the 10-dimensional superstring theories. More precisely, these arise as the massless, tree-level approximation of string theories. True effective field theories of string theories, rather than truncations, are rarely available. Some of these difficulties could be avoided by moving to a 10-dimensional theory involving superstrings. However, by moving to 10 dimensions one loses the sense of uniqueness of the 11-dimensional theory. The 10D models, that is massless IIA string theory and also IIB when there's a circle around, are equivalent to compactifications of the 11D circle. Thus there can be no experimental evidence for the 10D theory over the 11D theory, and its no harder to find models in the 11D theory than in its 10D description. IIA SUGRA is the dimensional reduction of 11-dimensional supergravity on a circle. This means that 11D supergravity on the spacetime, is equivalent to IIA supergravity on the 10-manifold. In particular the field and brane content of IIA supergravity can be derived via this dimensional reduction procedure. The field however does not arise from the dimensional reduction, massive IIA is not known to be the dimensional reduction of any higher-dimensional theory. For type IIA and IIB supergravities the classical theory is unique, as a classical theory supergravity is consistent with a single supergravity supermultiplet and any number of vector multiplets. It is also consistent without the supergravity supermultiplet, but then it would contain no graviton and so would not be a supergravity theory. While one may add multiple supergravity supermultiplets, it is not known if they may consistently interact.
The dimension D = 11 is the maximal dimension for which one can realize supersymmetry in terms of an ordinary supergravity theory. The conjectured 11-dimensional M-theory is required to have 11-dimensional supergravity as a low energy limit. The low-energy approximation of M-theory is given by the eleven-dimensional supergravity which describes the dynamics of the N = 1 supermultiplet in eleven dimensions. This maximal supergravity is the classical limit of M-theory. Higher-dimensional supergravity is the higher-dimensional, supersymmetric generalization of general relativity. Higher-dimensional SUGRA focuses upon supergravity in greater than four dimensions. A theory of everything must be unique and supergravity is unique in D = 11 dimensions. Among the various supergravity theories, 11-dimensional supergravity plays a special and significant role. The number of supersymmetries imposes a maximum number of dimensions for the spacetime. The symmetry of this supergravity theory is given by the supergroup OSp(1|32) which gives the subgroups O(1) for the bosonic symmetry and Sp(32) for the fermion symmetry. This is because spinors need 32 components in 11 dimensions. 11D supergravity can be compactified down to 4 dimensions which then has OSp(8|4) symmetry. Spinors need 4 components in 4 dimensions. The spacetime has four dimensions, then the theory in 11 dimensions should be 7 extra dimensions compactified in the sense of Kaluza-Klein theories. The 11-dimensional theory generated considerable exaltation as the first potential candidate for the theory of everything. The uniqueness is one reason for interest and development of supergravity theories. Another important reason is that supergravity theories tend to remove some of the problems that are encountered when trying to realize gravity as a quantum field theory. The known supersymmetry transformations of eleven dimensional supergravity lead to symmetries of the theory indicating the consistency of supergravity. The action and boundary conditions provide a supersymmetric theory which is a natural candidate for the a low energy limit of M-theory. Our understanding of M-theory is still very limited, mainly due to the lack of powerful methods to probe it at the quantum level. One approach to encoding information about M-theory is through its low energy effective field theory.
