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. In this article we consider an implementation of superconformal ideas in the context of supergravity in diverse dimensions. The standard conformal symmetries are fused with supersymmetry into so-called superconformal transformations. The gauge theory of these conformal supersymmetries is called conformal supergravity and constitutes the backbone of all supergravity theories. Conformal supergravity has a higher degree of symmetry than any other field theory that is presently known. Its classical action is invariant under conformal transformations, as well as under two different kinds of supersymmetry, called Q and S supersymmetry. In this note we explain the role of conformal invariance in supergravity. We have argued that the complexity of supergravity makes it advantageous to use superconformally invariant formulations. The superconformal symmetries are introduced in order to separate the irreducible part of a supergravity multiplet containing the degree of freedom. Indeed, conformal invariance plays an important role in the formulation of supergravity. It was known how to make supergravity locally scale invariant from the early days of this theory. An important benefit of a gauge invariant reformulation is that it presents a convenient way to relate different field representations to each other. Superconformal invariance and the constraints it imposes on quantum field theories have long been subject of active investigation. 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. We will see that demanding the existence of an action is more restrictive than only considering equations of motion. 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. An equally important ingredient, of course, is the action of the model, which has to be invariant under local superconformal transformations. So it is desirable to have at our disposal the necessary methods for the construction of invariant actions, or more generally, superconformal invariants. We instead concentrate on conformal supergravity theories considered as candidates for a fundamental theory. In general, conformal supergravity, interacting with renormalizable superconformal matter multiplets, is a power counting renormalizable theory.
Conformal Supergravity in D=4
A complete description of N = 1 superconformal multiplet calculus can be found in a number of reviews. Here we shall mainly concentrate on the invariant actions of physical multiplets of conformal supergravity. Our aim is to study the transformation laws and actions for all these multiplets, coupled to supergravity.
Here is the result for the Q and S supersymmetry transformations for D = 4. First for the independent gauge fields
A complete description of N = 1 superconformal multiplet calculus can be found in a number of reviews. Here we shall mainly concentrate on the invariant actions of physical multiplets of conformal supergravity. Our aim is to study the transformation laws and actions for all these multiplets, coupled to supergravity.
Here is the result for the Q and S supersymmetry transformations for D = 4. First for the independent gauge fields
Terms that represent modified structure functions
The supersymmetry (Q and S), and the gauge transformations with parameter of the multiplet in 4 dimensions are
The second term of the transformation of the vector reflects the presence of the new term in the commutator of two supersymmetries, modifying
To complete the superconformal multiplet, one has to add S-transformations, and there are non-linear transformations involving the matter fields of the Weyl multiplet, necessary in order to represent the anticommutators. The full result is
We can immediately give now the transformation rules for the tensor multiplet in the background of conformal supergravity
The idea is to start by constructing an action invariant under superconformal group. The action is determined by the components of the vector multiplet
The complete lagrangian is
In order to get a more useful form of the action, one has to make the conformal covariant derivatives explicit. This leads here to
This leads to the expression for the first term of the action
Deleting total derivatives, the action is at this point
We should now have for the hypermultiplet action
We also want the action to generate the field equations for the scalars. This leads to
and also
The covariant derivatives in the action read
After gauge covariantization and using the values of the conformal gauge fields, the final result of the action is
and the covariant derivatives are
Conformal Supergravity in D=5
The rigid Q- and S-supersymmetry transformation rules are given by
The covariant derivatives and the field strength are now also covariantized the local Q- and S-transformations
The equations of motion for the fields of the vector multiplet are
The action invariant under local superconformal symmetry can be computed by replacing the rigid covariant derivatives by the local covariant derivatives and adding extra terms proportional to gravitinos or matter fields of the Weyl multiplet, dictated by supersymmetry
where the superconformal d’Alembertian is defined as
The tensor supermultiplet constitutes also a representation of the full N = 2 superconformal algebra. In addition to the translations, Lorentz transformations, and R-symmetry transformations, the fields are subject to dilatations. In principle, fields also transform under conformal boosts, but matter multiplets are usually inert under those. On the fermionic side, the conventional Q-supersymmetry is extended with a second, special supersymmetry, called S-supersymmetry. In their presence the Q- and S-supersymmetry transformations of the tensor supermultiplet fields take the following form
To exhibit some of the details we record the expressions for the superconformal derivatives and the superconformal tensor field strength
Here the derivatives are covariant with respect to Lorentz transformations, dilatations and R-symmetry transformations
The superconformally invariant coupling between the two multiplets is
We simply record the corresponding expressions below
We will now generalize the vector action to an action for the vector-tensor multiplets with n vector multiplets and m tensor multiplets. Starting from n vector multiplets we now wish to consider a more general set of fields. The first part of these fields corresponds to the generators in the adjoint representation.
The transformations are
The covariant derivatives are given by
The covariant curvature should be understood as having components and
The superconformal action for the combined system of m = 2k tensor multiplets and n vector multiplets contains the CS-term and the generalization of the vector action to the extended range of indices. Some extra terms are necessary to complete it to an invariant action: we need mass terms and/or Yukawa coupling for the fermions and the scalar potential. We thus find the following action
where the superconformal d’Alembertian is defined as
The terms in the action containing the fields of the tensor multiplets can also be obtained from the field equations. They are now related to the action as
Furthermore, the equation of motion gets corrected by a term proportional to the self-duality equation
Imposing the local superconformal algebra we find the following supersymmetry rules for the hypermultiplet
Requiring closure of the commutator algebra on these transformation rules yields the eoms for the fermions
The scalar eom can be obtained from varying
The superconformal d’Alembertian becomes here
Introducing a metric, the locally conformal supersymmetric action is given by
In particular, the target space is still hypercomplex or, when an action exists, hyperk¨ahler. This action leads to the following dynamical equations
The Lagrangians are the starting point for obtaining matter couplings to Poincar´e supergravity. This involves a gauge fixing of the local scale and SU(2) symmetries.
In the kinetic terms of the action, the derivatives should now be covariantized with respect to the new transformations. We are also forced to include some new terms to the action
Mixing all these ingredients together we reduce the actions and the superconformal transformation rules to their super-Poincar´e versions. The bosonic part of the action characterizing the SUGRA coupling to n non-Abelian vector multiplets and r gauged hypermultiplets has the following form
The explicit form of the total scalar potential can be read off as
The invariant action in the N = 1 superspace is
We have found that the bosonic sector of N = 2, D = 5 supergravity is described by the lagrangian
We now proceed and substitute the above expressions into the supercovariant density formula. In doing this we make use of the conditions for the tensor multiplets and in order to express the Lagrangian in terms of a single function. The complete result of the total tensor multiplet action can be presented as follows
The total action under local superconformal symmetry for N = 2, D = 5 conformal supergravity is
For vector-tensor multiplets, we have written down equations of motion with an odd number of tensor multiplets in the background of an arbitrary number of vector multiplets. This is in contrast with formulations based on an action, where an even number of tensor multiplets is always needed. Even in the case when an action exists, we have found new couplings where vectors and tensors mix non-trivially due to the off-diagonal structure of the representation matrices for the gauge group. For hypermultiplets, it has been known that the geometry of the scalars is hyperk¨ahler for rigid supersymmetry or quaternionic-K¨ahler for supergravity. This was based on an analysis of the requirements imposed by the existence of an invariant action. We believe that this formalism will be useful in systematically including supergravity effects in higher-dimensional theories.
Conformal Supergravity in D=6
We use the superconformal method to construct the full off-shell action of N = (1; 0), D = 6 supergravity, which has apart from the graviton and the gravitino, a 2-form gauge field, a dilaton and a symplectic Majorana spinor. Starting from the linear transformation rules of the superconformal gauge fields, the curvatures and the matter fields we can construct the full nonlinear N = (1; 0), D = 6 Weyl multiplet by applying an iterative procedure. The results are
The relevant modified curvatures have the form
Because of the deformation of the transformation rules and curvatures by the matter fields, the conventional constraints mentioned above must also be adapted. The bosonic superconformal Lagrangian is
We obtain
The fermionic action looks at this point as
The off-shell (1, 0) supergravity action has been constructed by means of a superconformal tensor calculus in which the off-shell so-called dilaton Weyl multiplet with independent fields is coupled to an off-shell linear multiplet.
The coupling of the vector multiplet to supergravity is then achieved by considering the Lagrangian
This formula, up to quartic fermion terms, yields the result
The action corresponding to the Lagrangian is invariant under the supersymmetry transformations supplemented by the supersymmetry transformations of the components of the off-shell vector multiplet.
We present supersymmetric R2-action using the superconformal invariant exact action formula for the Yang-Mills multiplet
The total off-shell action of minimal Poincare supergravity is
The off-shell Poincare supergravity theory already has a local U(1) symmetry but it is gauged by an auxiliary vector field which is not dynamical. Expressing the action in terms of fields of the Weyl multiplet led to an action that contains kinetic terms for the matter fields and has consistent field equations.
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We also obtain off-shell formulation of Poincare supergravity coupled to a tensor multiplet. As a first step in the construction of the N = 2, d = 6 Weyl multiplet we consider the superconformal algebra in six dimensions which is OSp(6, 2/1). To each generator of the superconformal algebra we assign a gauge field in the following way
We find that the full nonlinear Q and S-transformations of N = 2, D = 6 conformal supergravity are given by
We give these explicitly
We now present our results for the N = 2, D = 6 vector muitiplets. We find that the full nonlinear Q and S-transformation rules are given by
Explicitly one finds for the scalar multiplet
We obtain then
The full transformations under Q and S for the tensor multiplet are
The third constraint is
The N = 2, d = 6 linear multiplet consists of a triplet, an SU(2)-Majorana spinor of negative chirality and a constrained vector field. The full Q and S transformation rules are given by
The full Q, S and K transformations of the nonlinear multiplet read
We will obtain actions for matter multiplets coupled to Poincare supergravity. We will start with the action for the scalar multiplets. It will turn out that this action formula applied to a compensating scalar multiplet gives the action for Poincare supergravity coupled to a tensor multiplet.
We then obtain
The action formula we are looking for is contained in the constraint for the linear multiplet which has Weyl weight 6. In fact if we select there the g-dependent terms we obtain the following density
If the linear multiplet is inert under the gauge group, then the second line of previous action is
Inserting the components of the linear multiplet in the action formulas one obtains an action for the abelian vector multiplet
We obtain for linear multiplet action
The action reduces to
The general action which we obtained can be written as
We constructed local supersymmetric matter couplings in six dimensions. Throughout our work we applied superconformal techniques. We first construct a superconformal invariant action. After imposing appropriate gauge conditions these actions lead to matter coupled Poincare supergravity.
Conformal Supergravity in D=10
We present the complete off-shell structure of conformal supergravity in ten dimensions. We present the full nonlinear structure of conformal supergravity, and study its relation with Poincare supergravity. The construction of the Weyl multiplet is based on an analysis of the supermultiplet of currents which describe the coupling of supersymmetric matter to supergravity.
The transformation rules under Q-supersymmetry read
Our starting point is the construction of an invariant is the following quadratic action
The Gauss-Bonnet combination is
The spin connection satisfies
Two relations which are contractions of the Bianchi identity for the Riemann tensor
The action then becomes
The contributions to the action without explicit gravitinos are
The quadratic action for the Riemann tensor with torsion takes on the form
The commutator of two supersymmetry transformations gives, besides the general coordinate transformation, a field-dependent Lorentz transformation
We now present some of the dependent fields, the curvatures and their transformation rules
The Bianchi identity for the gravitino curvature takes on the form
We present the complete result for the D=10 superconformal action
Poincare supergravity in D = 10 is described by the lagrangian density
The equations of motion follow from the following action
After a partial integration the dependence on the six-index gauge field is now contained in its field strength. We give the relevant terms
Considering the commutator of two Yang-Mills transformations with parameters one finds
In the standard approach by which Poincare supergravity is obtained from conformal supergravity one introduces a number of compensating supermultiplets. The resulting field representation is then shown to be gauge equivalent to an irreducible representation of Poincare supergravity. In principle this procedure works in ten dimensions, except that one expects to be left with fields that are still subject to differential constraints.
Superconformal Aspects of D=11 Supergravity
We discuss superconformal aspects of supergravity in eleven dimensions. We suggest that the Poincare theory is obtained by a compensating mechanism using a scalar superfield. Superconformal concepts help to clarify the offshell structure of Poincar6 supergravity. In this letter, we consider the application of this idea to supergravity in eleven dimensions. We construct part of an off-shell multiplet which contains the superconformal gauge fields (the linearized Weyl multiplet). Using a scalar multiplet as a compensating multiplet, we obtain part of the off-shell Poincare theory.
The standart transformation rules are
The covariant curvatures are
The linearized transformation rules of the independent fields are at this stage
The following decomposition rule for the Poincare supersymmetry transformations
After these gauge conditions have been imposed the linearized Q transformations of the B's are given by
The decomposition rule is
We have shown that superconformal methods can be used to elucidate the structure of D = 11 supergravity. In D = 11 there are no matter multiplets, so that the systematic approach through the construction of a supercurrent is not applicable.
We concluded with the presentation of families of examples of such action principles for supergravity-matter systems. We demonstrated that the curved superspace is ideally suited for the construction of various matter couplings as well as a supergravity actions. The superconformal approach to Poincaré supergravity starts from the construction of theories which are gauge invariant under a much larger group, the superconformal group in respective dimensions. The super-Poincaré theories obtained from the superconformal ones are off-shell. Some fields that are required for the superconformal formulation appear as auxiliary fields in the super-Poincaré theory. 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. Subsequently one can eliminate the auxiliary superconformal fields. The additional multiplets are necessary to provide compensating fields and to overcome a deficit in degrees of freedom between the Weyl multiplet and the minimal field representation of Poincaré supergravity. 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 basic idea is that there is a close relation between matter-coupled Poincaré and conformal supergravity theories.