Higher-dimensional supergravity is the supersymmetric generalization of general relativity in higher dimensions. Supergravity can be formulated in any number of dimensions up to eleven. This article focuses upon supergravity (SUGRA) in greater than four dimensions.
Supermultiplets Fields related by supersymmetry transformations form a supermultiplet; the one that contains a graviton is called the supergravity multiplet. The name of a supergravity theory generally includes the number of dimensions of spacetime that it inhabits, and also the number N {\displaystyle {\mathcal {N}}} of gravitinos that it has. Sometimes one also includes the choices of supermultiplets in the name of theory. For example, an N = 2 {\displaystyle {\mathcal {N}}=2} , (9 + 1)-dimensional supergravity enjoys 9 spatial dimensions, one time and 2 gravitinos. While 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. It is still not known whether one can construct theories with multiple gravitons that are not equivalent to multiple decoupled theories with a single graviton in each. In maximal supergravity theories (see below), all fields are related by supersymmetry transformations so that there is only one supermultiplet: the supergravity multiplet.
Gauged supergravity versus Yang–Mills supergravity Often an abuse of nomenclature is used when "gauge supergravity" refers to a supergravity theory in which fields in the theory are charged with respect to vector fields in the theory. However, when the distinction is important, the following is the correct nomenclature. If a global (i.e. rigid) R-symmetry is gauged, the gravitino is charged with respect to some vector fields, and the theory is called gauged supergravity. When other global (rigid) symmetries (e.g., if the theory is a non-linear sigma model) of the theory are gauged such that some (non-gravitino) fields are charged with respect to vectors, it is known as a Yang–Mills–Einstein supergravity theory. Of course, one can imagine having a "gauged Yang–Mills–Einstein" theory using a combination of the above gaugings.
Counting gravitinos 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. This is a Rarita–Schwinger spinor. While there is only one vector representation for each Lorentz group, in general there are several different spinorial representations. Technically these are really representations of the double cover of the Lorentz group called a spin group. The canonical example of a spinorial representation is the Dirac spinor, which exists in every number of space-time dimensions. However the Dirac spinor representation is not always irreducible. When calculating the number N {\displaystyle {\mathcal {N}}} , one always counts the number of real irreducible representations. The spinors with spins less than 3/2 that exist in each number of dimensions will be classified in the following subsection.
A classification of spinors 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. (See helicity (particle physics)) For example, in our world, this is 3 − 1 = 2. Due to the mod 8 Bott periodicity of the homotopy groups of the Lorentz group, really we only need to consider k modulo 8. For any value of k there is a Dirac representation, which is always of real dimension 2 1 + ⌊ 2 d − k 2 ⌋ {\displaystyle 2^{1+\lfloor {\frac {2d-k}{2}}\rfloor }} where ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } is the greatest integer less than or equal to x. When − 2 ≤ k ≤ 2 ( mod 8 ) {\displaystyle -2\leq k\leq 2{\pmod {8}}} there is a real Majorana spinor representation, whose dimension is half that of the Dirac representation. When k is even there is a Weyl spinor representation, whose real dimension is again half that of the Dirac spinor. Finally when k is divisible by eight, that is, when k is zero modulo eight, there is a Majorana–Weyl spinor, whose real dimension is one quarter that of the Dirac spinor. Occasionally one also considers symplectic Majorana spinor which exist when 3 ≤ k ≤ 5 {\displaystyle 3\leq k\leq 5} , which have half has many components as Dirac spinors. When k=4 these may also be Weyl, yielding Weyl symplectic Majorana spinors which have one quarter as many components as Dirac spinors.
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