Higher-dimensional supergravity

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 ${\displaystyle {\mathcal {N}}}$ of gravitinos that it has. Sometimes one also includes the choices of supermultiplets in the name of theory. For example, an ${\displaystyle {\mathcal {N}}=2}$, (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.

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.

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 ${\displaystyle {\mathcal {N}}}$, 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.

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 ${\displaystyle 2^{1+\lfloor {\frac {2d-k}{2}}\rfloor }}$ where ${\displaystyle \lfloor x\rfloor }$ is the greatest integer less than or equal to x. When ${\displaystyle -2\leq k\leq 2{\pmod {8}}}$ 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 ${\displaystyle 3\leq k\leq 5}$, 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.

Spinors in n-dimensions are representations (really modules) not only of the n-dimensional Lorentz group, but also of a Lie algebra called the n-dimensional Clifford algebra. The most commonly used basis of the complex ${\displaystyle 2^{\lfloor n\rfloor }}$-dimensional representation of the Clifford algebra, the representation that acts on the Dirac spinors, consists of the gamma matrices.

When n is even the product of all of the gamma matrices, which is often referred to as ${\displaystyle \Gamma _{5}}$ as it was first considered in the case n = 4, is not itself a member of the Clifford algebra. However, being a product of elements of the Clifford algebra, it is in the algebra's universal cover and so has an action on the Dirac spinors.

In particular, the Dirac spinors may be decomposed into eigenspaces of ${\displaystyle \Gamma _{5}}$ with eigenvalues equal to ${\displaystyle \pm (-1)^{-k/2}}$, where k is the number of spatial minus temporal dimensions in the spacetime. The spinors in these two eigenspaces each form projective representations of the Lorentz group, known as Weyl spinors. The eigenvalue under ${\displaystyle \Gamma _{5}}$ is known as the chirality of the spinor, which can be left or right-handed.

A particle that transforms as a single Weyl spinor is said to be chiral. 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.

Recall that the eigenvalues of ${\displaystyle \Gamma _{5}}$, whose eigenspaces are the two chiralities, are ${\displaystyle \pm (-1)^{-k/2}}$. In particular, when k is equal to two modulo four the two eigenvalues are complex conjugate and so the two chiralities of Weyl representations are complex conjugate representations.

Complex conjugation in quantum theories corresponds to time inversion. Therefore, the CPT theorem implies that when the number of Minkowski dimensions is divisible by four (so that k is equal to 2 modulo 4) there be an equal number of left-handed and right-handed supercharges. 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 ${\displaystyle {\mathcal {N}}=({\mathcal {N}}_{L},{\mathcal {N}}_{R})}$ where ${\displaystyle {\mathcal {N}}_{L}}$ and ${\displaystyle {\mathcal {N}}_{R}}$ are the number of left-handed and right-handed supercharges respectively.

All supergravity theories are invariant under transformations in the super-Poincaré algebra, although individual configurations are not in general invariant under every transformation in this group. The super-Poincaré group is generated by the Super-Poincaré algebra, which is a Lie superalgebra. A Lie superalgebra is a ${\displaystyle \mathbf {Z} _{2}}$ graded algebra in which the elements of degree zero are called bosonic and those of degree one are called fermionic. 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.

The fermionic generators are also called supercharges. 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.

The supercharges transform as spinors, and the number of irreducible spinors of these fermionic generators is equal to the number of gravitinos ${\displaystyle {\mathcal {N}}}$ defined above. Often ${\displaystyle {\mathcal {N}}}$ is defined to be the number of fermionic generators, instead of the number of gravitinos, because this definition extends to supersymmetric theories without gravity.

Sometimes it is convenient to characterize theories not by the number ${\displaystyle {\mathcal {N}}}$ of irreducible representations of gravitinos or supercharges, but instead by the total Q of their dimensions. This is because some features of the theory have the same Q-dependence in any number of dimensions. For example, one is often only interested in theories in which all particles have spin less than or equal to two. This requires that Q not exceed 32, except possibly in special cases in which the supersymmetry is realized in an unconventional, nonlinear fashion with products of bosonic generators in the anticommutators of the fermionic generators.

The Lagrangian for 11D supergravity found by brute force by Cremmer, Julia and Scherk[1] is:

${\displaystyle {\begin{array}{rcl}L&=&+{\frac {1}{2\kappa ^{2}}}eR-{\frac {1}{2}}e{\overline {\psi }}_{M}\Gamma ^{MNP}D_{N}[{\frac {1}{2}}(\omega -{\overline {\omega }})]\psi _{P}\\&&+{\frac {1}{48}}eF_{MNPQ}^{2}+{\frac {{\sqrt {2}}\kappa }{384}}e({\overline {\psi }}_{M}\Gamma ^{MNPQRS}\psi _{S}\\&&+12{\overline {\psi }}^{N}\Gamma ^{PQ}\psi ^{R})(F+{\overline {F}})_{NPQR}+{\frac {{\sqrt {2}}\kappa }{3456}}\varepsilon ^{M_{1}\dots M_{11}}F_{M_{1}\dots M_{4}}F_{M_{5}\dots M_{8}}A_{M_{9}M_{10}M_{11}}\end{array}}}$

which contains the three types of field:

${\displaystyle e_{M}^{A},\psi _{M},A_{MNP}}$

The symmetry of this supergravity theory is given by the supergroup OSp(1|32) which gives the subgroups O(1) for the bosonic symmetry and Sp(32) for the fermion symmetry. This is because spinors need 32 components in 11 dimensions. 11D supergravity can be compactified down to 4 dimensions which then has OSp(8|4) symmetry. (We still have 8 × 4 = 32 so there are still the same number of components.) Spinors need 4 components in 4 dimensions. This gives O(8) for the gauge group which is too small to contain the Standard Model gauge group U(1) × SU(2) × SU(3) which would need at least O(10).