In supersymmetry, 4D supergravity is the theory of supergravity in four dimensions with a single supercharge. It contains exactly one supergravity multiplet, consisting of a graviton and a gravitino, but can also have an arbitrary number of chiral and vector supermultiplets, with supersymmetry imposing stringent constraints on how these can interact. The theory is primarily determined by three functions, those being the Kähler potential, the superpotential, and the gauge kinetic matrix. Many of its properties are strongly linked to the geometry associated to the scalar fields in the chiral multiplets. After the simplest form of this supergravity was first discovered, a theory involving only the supergravity multiplet, the following years saw an effort to incorporate different matter multiplets, with the general action being derived in 1982 by Eugène Cremmer, Sergio Ferrara, Luciano Girardello, and Antonie Van Proeyen.
This theory plays an important role in many Beyond the Standard Model scenarios. Notably, many four-dimensional models derived from string theory are of this type, with supersymmetry providing crucial control over the compactification procedure. The absence of low-energy supersymmetry in our universe requires that supersymmetry is broken at some scale. Supergravity provides new mechanisms for supersymmetry breaking that are absent in global supersymmetry, such as gravity mediation. Another useful feature is the presence of no-scale models, which have numerous applications in cosmology.
History
Supergravity was first discovered in 1976 in the form of pure 4D supergravity. This was a theory of only the graviton and its superpartner, the gravitino. The first extension to also couple matter fields to the theory was acquired by adding Maxwell and Yang–Mills fields, as well as a Fayet–Iliopoulos term. Chiral multiplets were also incorporated, starting first by coupling a single massless chiral multiplet. The primary construction technique in these early years was using the iterative Noether method, which does not lend itself towards deriving more general matter coupled actions due to being very tedious.
The development of tensor calculus techniques in 1978 allowed for the construction of supergravity actions more efficiently, with them quickly used to derive the general action coupling a single chiral multiplet to supergravity. Using the same technique, the general four-dimensional matter-coupled supergravity action was constructed in 1982 by Eugène Cremmer, Sergio Ferrara, Luciano Girardello, and Antonie Van Proeyen. It was also derived by Jonathan Bagger shortly after using superspace techniques, with this work highlighting important geometric features of the theory. Around this time two other features of the models were identified. These are the Kähler–Hodge structure present in theory and the presence and importance of no-scale models.
Overview
The particle content of a general four-dimensional supergravity consists of a single supergravity multiplet and an arbitrary number of chiral multiplets and gauge multiplets. The supergravity multiplet contains the spin-2 graviton describing fluctuations in the spacetime metric , along with a spin-3/2 Majorana gravitino , where the spinor index is often left implicit. The chiral multiplets , indexed by lower-case Latin indices , each consist of a scalar and its Majorana superpartner . Similarly, the gauge multiplets consist of a Yang–Mills gauge field and its Majorana superpartner the gaugino , with these multiplets indexed by capital Latin letters .
One of the most important structures of the theory is the scalar manifold, which is the field space manifold whose coordinates are the scalars. Global supersymmetry implies that this manifold must be a special type of complex manifold known as a Kähler manifold. Local supersymmetry of supergravity further restricts its form to be that of a Kähler–Hodge manifold.
The theory is primarily described by three arbitrary functions of the scalar fields, the first being the Kähler potential which fixes the metric on the scalar manifold. The second is the superpotential, which is an arbitrary holomorphic function that fixes a number of aspects of the action such as the scalar field F-term potential along with the fermion mass terms and Yukawa couplings. Lastly, there is the gauge kinetic matrix whose components are holomorphic functions determining, among other aspects, the gauge kinetic term, the theta term, and the D-term potential.
Additionally, the supergravity may be gauged or ungauged. In ungauged supergravity, any gauge transformations present can only act on abelian gauge fields. Meanwhile, a gauged supergravity can be acquired from an ungauged one by gauging some of its global symmetries, which can cause the scalars or fermions to also transform under gauge transformations and result in non-abelian gauge fields. Besides local supersymmetry transformations, local Lorentz transformations, and gauge transformations, the action must also be invariant under Kähler transformations , where is an arbitrary holomorphic function of the scalar fields.
Construction
Historically, the first approach to constructing supergravity theories was the iterative Noether formalism which uses a globally supersymmetric theory as a starting point. Its Lagrangian is then coupled to pure supergravity through the term which couples the gravitino to the supercurrent of the original theory, with everything also Lorentz covariantized to make it valid in curved spacetime. This candidate theory is then varied with respect to local supersymmetry transformations yielding some nonvanishing part. The Lagrangian is then modified by adding to it new terms that cancel this variation, at the expense of introducing new nonvanishing variations. More terms are the introduced to cancel these, and the procedure is repeated until the Lagrangian is fully invariant.
Since the Noether formalism proved to be very tedious and inefficient, more efficient construction techniques were developed. Various sets of auxiliary fields were found which allow one to construct off-shell supersymmetric multiplets, meaning that they satisfy the supersymmetry algebra without needing to also impose equations of motion. The discovery of these led to the development of the tensor calculus formalism, where one can take products and sums of the multiplets to construct supersymmetrically invariant densities from which a supergravity action can be acquired. This method, using the old minimal set of auxiliary fields, was the one first used to construct the general matter-coupled 4D supergravity theory.
The superspace approach was being developed at the same time, with this generalizing the notion of superspace to a curved superspace whose tangent space at each point behaves like the traditional flat superspace from global supersymmetry. The general invariant action can then be constructed in terms of the superfields, which can then be expanded in terms of the component fields to give the component form of the supergravity action.
Another approach which unifies various tensor calculus methods is the superconformal tensor calculus approach which uses conformal symmetry as a tool to construct supergravity actions that do not themselves have any conformal symmetry. This is done by first constructing a gauge theory using the superconformal algebra. This theory contains extra fields and symmetries, but they can be eliminated using constraints or through gauge fixing to yield Poincaré supergravity without conformal symmetry.
The superconformal and superspace ideas have also been combined into a number of different supergravity conformal superspace formulations. The direct generalization of the original on-shell superspace approach is the Grimm–Wess–Zumino formalism. There is also the superspace formalism proposed by Paul Howe. Lastly, the conformal superspace approach has the convenient property that any other formulation of conformal supergravity is either equivalent to it or can otherwise be obtained from a partial gauge fixing. Other approaches also exist, such as the group manifold method which treats fields as the components of one-forms on a supergroup manifold.
Symmetries
Scalar manifold and Kähler transformations
Supergravity often uses Majorana spinor notation over that of Weyl spinors since four-component notation is easier to use in curved spacetime. Weyl spinors can be acquired as projections of a Majorana spinor , with the left and right handed Weyl spinors denoted by .
Complex scalars in the chiral multiplets act as coordinates on a complex manifold in the sense of the non-linear sigma model, known as the scalar manifold. In supersymmetric theories these manifolds are imprinted with additional geometric constraints arising from the supersymmetry transformations. In supergravity this manifold may be compact or noncompact, while for supergravities it is necessarily noncompact.
Global supersymmetry already restricts the manifold to be a Kähler manifolds. These are a type of complex manifold, which roughly speaking are manifolds that look locally like and whose transition maps are holomorphic functions. Complex manifolds are also Hermitian manifolds if they admit a well-defined metric whose only nonvanishing components are the components, where the bar over the index denotes the conjugate coordinate . More generally, a bar over scalars denotes complex conjugation while for spinors it denotes an adjoint spinor. Kähler manifolds are Hermitian manifolds that admit a two-form called a Kähler form
that is closed . A property of these manifolds is that their metric can be written in terms of the derivatives of a scalar function , where the is known as the Kähler potential. Here denotes a derivative with respect to . This potential corresponding to a particular metric is not unique and can be changed by the addition of the real part of a holomorphic function in what are known as Kähler transformations
Since this does not change the scalar manifold, supersymmetric actions must be invariant under such transformations.
While in global supersymmetry, fields and the superpotential transform trivially under Kähler transformations, in supergravity they are charged under the Kähler transformations as
where is the Majorana spinor supersymmetry transformation parameter. These transformation rules impose further restrictions on the geometry of the scalar manifold. Since the superpotential transforms by a prefactor, this implies that the scalar manifold must globally admit a consistent line bundle. The fermions meanwhile transform by a complex phase, which implies that the scalar manifold must also admit an associated principal bundle. The nondynamical connection corresponding to this principal bundle is given by
with this satisfying , where is the Kähler form. Here are holomorphic functions associated to the gauge sector, described below. This condition means that the scalar manifold in four-dimensional supergravity must be of a type which can admit a connection whose field strength is equal to the Kähler form. Such manifolds are known as Kähler–Hodge manifolds. In terms of characteristic classes, this condition translates to the requirement that where is the first Chern class of the line bundle, while is the cohomology class of the Kähler form.
An implication of the presence of an associated principal bundle on the Kähler–Hodge manifold is that its field strength must be quantized on any topologically non-trivial two-sphere of the scalar manifold, analogous to the Dirac quantization condition for magnetic monopoles. This arises due to the cocycle condition, which is the consistency of the connection across different coordinate patches. This can have various implications for the resulting physics, such as on an scalar manifold, it results in the quantization of Newton's constant.
Global symmetries of ungauged supergravity