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.
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.
Notably, many four-dimensional models derived from string theory are of this type, with supersymmetry providing crucial control over the compactification procedure.
[6] All these theories were constructed using the iterative Noether method, which does not lend itself towards deriving more general matter coupled actions due to being very tedious.
supergravity action was constructed in 1982 by Eugène Cremmer, Sergio Ferrara, Luciano Girardello, and Antonie Van Proeyen.
[1][2] It was also derived by Jonathan Bagger shortly after using superspace techniques, with this work highlighting important geometric features of the theory.
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.
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.
In supersymmetric theories these manifolds are imprinted with additional geometric constraints arising from the supersymmetry transformations.
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
Since the superpotential transforms by a prefactor, this implies that the scalar manifold must globally admit a consistent line bundle.
must be quantized on any topologically non-trivial two-sphere of the scalar manifold, analogous to the Dirac quantization condition for magnetic monopoles.
For Kähler manifolds, this condition additionally implies that there exists a set of holomorphic functions known as Killing prepotentials
that leaves the Kähler potential invariant, which imposes the condition that the only admissible superpotentials are ones satisfying[25]: 211 Global symmetries involving scalars present in the gauge kinetic matrix still act on the scalar fields as isometry transformations, but now these transformations change the gauge kinetic matrix.
connection on the scalar manifold, with its explicit form given in terms of the Kähler potential described previously.
The three main functions determining the structure of the Lagrangian are the superpotential, the Kähler potential, and the gauge kinetic matrix.
The second line consists of the kinetic terms for the chiral multiplets, with its overall form determined by the scalar manifold metric which itself is fully fixed by the Kähler potential
They provide additional bilinear terms between the gravitino and the other fermions that need to be accounted for when going into the mass basis.
as where the first term is known as the F-term, and is a generalization of the potential arising from the chiral multiplets in global supersymmetry, together with a new negative gravitational contribution proportional to
The second term is called the D-term and is also found in a similar form in global supersymmetry, with it arising from the gauge sector.
The supersymmetry transformation rules, up to three-fermion terms which are unimportant for most applications,[nb 10] are given by[19]: 389 where are known as fermionic shifts.
[19]: 397–401 This setup has a hidden and an observable sector that have no renormalizable couplings between them, meaning that they fully decouple from each other in the global supersymmetry
The general formula is most compactly written in the superspace formalism,[32][33] but in the special case of a vanishing cosmological constant, a trivial gauge kinetic matrix
[19]: 397 No-scale models are models with a vanishing F-term, achieved by picking a Kähler potential and superpotential such that[19]: 401–403 When D-terms for gauge multiplets are ignored, this gives rise to the vanishing of the classical potential, which is said to have flat directions for all values of the scalar field.
Additionally, supersymmetry is formally broken, indicated by a non-vanishing but undetermined mass of the gravitino.
When moving beyond the classical level, quantum corrections come in to break this degeneracy, fixing the mass of the gravitino.
In particular, for supergravity to be consistent as a quantum theory, new constraints come in such as anomaly cancellation conditions and black hole charge quantization.
This is seen in string theory compactifications, which can at most produce field dependent Fayet–Iliopoulos terms associated to Stueckelberg masses for gauged