Fields related by supersymmetry transformations form a supermultiplet; the one that contains a graviton is called the supergravity multiplet.
Gravitinos are fermions, which means that according to the spin-statistics theorem they must have an odd number of spinorial indices.
In fact the gravitino field has one spinor and one vector index, which means that gravitinos transform as a tensor product of a spinorial representation and the vector representation of the Lorentz group.
The canonical example of a spinorial representation is the Dirac spinor, which exists in every number of space-time dimensions.
The available spinor representations depends on k; the maximal compact subgroup of the little group of the Lorentz group that preserves the momentum of a massless particle is Spin(d − 1) × Spin(d − k − 1), where k is equal to the number d of spatial dimensions minus the number d − k of time dimensions.
The CPT theorem, which is required by Lorentz invariance in Minkowski space, implies that when there is a single time direction such particles have antiparticles of the opposite chirality.
On the other hand, if the dimension is equal to 2 modulo 4, there can be different numbers of left and right-handed supercharges, and so often one labels the theory by a doublet
A commutator, that is an antisymmetric bracket satisfying the Jacobi identity is defined between each pair of generators of fixed degree except for pairs of fermionic generators, for which instead one defines a symmetric bracket called an anticommutator.
Any configuration which is invariant under any of the supercharges is said to be BPS, and often nonrenormalization theorems demonstrate that such states are particularly easily treated because they are unaffected by many quantum corrections.
However, some authors have considered nonlinear actions of the supersymmetry in which higher spin fields may not appear.
If instead one considers 10 spatial direction and a second temporal dimension then there is a Majorana–Weyl spinor, which as desired has only 32 components.
Also, the Hamiltonian-based approach to quantum mechanics may have to be modified in the presence of a second Hamiltonian for the other time.
However, in Two-Time Physics it was demonstrated that such potential problems are solved with an appropriate gauge symmetry.
Some other two time theories describe low-energy behavior, such as Cumrun Vafa's F-theory that is also formulated with the help of 12 dimensions.
This maximal supergravity is the classical limit of type IIA string theory.
From the above Bianchi identity we see that a D8-brane is a domain wall between zones of differing G0, thus in the presence of a D8-brane at least part of the spacetime will be described by the Romans theory.
In particular the field and brane content of IIA supergravity can be derived via this dimensional reduction procedure.
This maximal supergravity is the classical limit of type IIB string theory.
This is smaller than the supergravity supermultiplet in type II, it contains only the graviton, a Majorana–Weyl gravitino, a 2-form gauge potential, the dilaton and a dilatino.
There is also a vector supermultiplet, which contains a one-form gauge potential called a gluon and also a Majorana–Weyl gluino.
In addition one is free to make some choices of gravitational couplings in the classical theory.
Generically the quantum versions of these theories suffer from various anomalies, as can be seen already at 1-loop in the hexagon Feynman diagrams.
In 1984 and 1985 Michael Green and John H. Schwarz have shown that if one includes precisely 496 vector supermultiplets and chooses certain couplings of the 2-form and the metric then the gravitational anomalies cancel.
Quantum theories with at least 8 supercharges tend to have continuous moduli spaces of vacua.
In compactifications of these theories, which have 16 supercharges, there exist degenerate vacua with different values of various Wilson loops.
Recall that in 10 dimensions there were two inequivalent maximal supergravity theories, IIA and IIB.
While T-duality in supergravity involves dimensional reduction and so loses information, in the full quantum string theory the extra information is stored in string winding modes and so T-duality is a duality between the two 10-dimensional theories.
The above construction can be used to obtain the relation between the circle bundle's connection and dual Kalb–Ramond field even in the full quantum theory.
These vector multiplets may be coupled so as to admit arbitrary gauge theories, although not all possibilities have quantum completions.
Unlike the 10-dimensional theory, as was described in the previous subsection, the supergravity multiplet itself contains a vector and so there will always be at least a U(1) gauge symmetry, even in the N=2 case.