Eleven-dimensional supergravity

Theory

In supersymmetry, the maximum number of real supercharges that give supermultiplets containing particles of spin less than or equal to two, is 32. Supercharges with more components result in supermultiplets that necessarily include higher spin states, making such theories unphysical. Since supercharges are spinors, supersymmetry can only be realized in dimensions that admit spinoral representations with no more than 32 components, which only occurs in eleven or fewer dimensions.

Eleven-dimensional supergravity is uniquely fixed by supersymmetry, with its structure being relatively simple compared to supergravity theories in other dimensions. The only free parameter is the Planck mass, setting the scale of the theory. It has a single multiplet consisting of the graviton, a Majorana gravitino, and a 3-form gauge field. The necessity of the 3-form field is seen by noting that it provides the missing 84 bosonic degrees of freedom needed to complete the multiplet since the graviton has 44 degrees of freedom while the gravitino has 128.

Superalgebra

The maximally-extended algebra for supersymmetry in eleven dimensions is given by

${\displaystyle \{Q_{\alpha },Q_{\beta }\}=(C\gamma {)}_{\alpha \beta }^{\mu }{P}_{\mu }+(C\gamma {)}_{\alpha \beta }^{\mu \nu }{Z}_{\mu \nu }+(C\gamma {)}_{\alpha \beta }^{\mu \nu \rho \sigma \gamma }{Z}_{\mu \nu \rho \sigma \gamma },}$

where ${\displaystyle C}$ is the charge conjugation operator which ensures that the combination ${\displaystyle C{\gamma }^{{\mu }_1\cdots {\mu }_n}}$ is either symmetric or antisymmetric. Since the anticommutator is symmetric, the only admissible entries on the right-hand side are those which are symmetric on their spinor indices, which in eleven dimensions only occurs for one, two, and five spacetime indices, with the rest being equivalent up to Poincaré duality. The corresponding coefficients ${\displaystyle Z_{\mu \nu }}$ and ${\displaystyle Z_{\mu \nu \rho \sigma \gamma }}$ are known as quasi-central charges. They aren't regular central charges in the group theoretic sense since they are not Lorentz scalars and so do not commute with the Lorentz generators, but their interpretation is the same. They indicate that there are extended objects that preserve some amount of supersymmetry, these being the M2-brane and the M5-brane. Additionally, there is no R-symmetry group.

Supergravity action

The action for eleven-dimensional supergravity is given by

${\displaystyle S=\frac{1}{2{\kappa }_{11}^2}\int d^{11}xe[R(\omega )-{\overline{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{D}_{\nu }(\frac{1}{2}(\omega +\hat{\omega })){\psi }_{\rho }-\frac{1}{24}{F}_{\mu \nu \rho \sigma }{F}^{\mu \nu \rho \sigma }}$

${\displaystyle -\frac{\sqrt{2}}{192}({\overline{\psi }}_{\nu }{\gamma }^{\alpha \beta \gamma \delta \nu \rho }{\psi }_{\rho }+12{\overline{\psi }}^{\gamma }{\gamma }^{\alpha \beta }{\psi }^{\delta })(F_{\alpha \beta \gamma \delta }+{\hat{F}}_{\alpha \beta \gamma \delta })}$

${\displaystyle -\frac{2\sqrt{2}}{(144{)}^2}{ϵ}^{\alpha \beta \gamma \delta {\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }{\delta }^{\prime }\mu \nu \rho }{F}_{\alpha \beta \gamma \delta }{F}_{{\alpha }^{\prime }{\beta }^{\prime }{\gamma }^{\prime }{\delta }^{\prime }}{A}_{\mu \nu \rho }].}$

Here gravity is described using the vielbein formalism ${\displaystyle e_{\mu }^a}$ with an eleven-dimensional gravitational coupling constant ${\displaystyle {\kappa }_{11}}$ and

${\displaystyle {\omega }_{\mu ab}={\omega }_{\mu ab}(e)+K_{\mu ab},}$

${\displaystyle {\hat{\omega }}_{\mu ab}={\omega }_{\mu ab}-\frac{1}{8}{\overline{\psi }}_{\nu }{\gamma }^{\nu \rho }{}_{\mu ab}{\psi }_{\rho },}$

${\displaystyle K_{\mu ab}=-\frac{1}{4}({\overline{\psi }}_{\mu }{\gamma }_b{\psi }_a-{\overline{\psi }}_a{\gamma }_{\mu }{\psi }_b+{\overline{\psi }}_b{\gamma }_a{\psi }_{\mu })+\frac{1}{8}{\overline{\psi }}_{\nu }{\gamma }^{\nu \rho }{}_{\mu ab}{\psi }_{\rho },}$

${\displaystyle {\hat{F}}_{\mu \nu \rho \sigma }=F_{\mu \nu \rho \sigma }+\frac{3}{2}\sqrt{2}{\overline{\psi }}_{[\mu }{\gamma }_{\nu \rho }{\psi }_{\sigma ]}.}$

The torsion-free connection is given by ${\displaystyle {\omega }_{\mu ab}(e)}$, while ${\displaystyle K_{\mu ab}}$ is the contorsion tensor. Meanwhile, ${\displaystyle D_{\nu }(\omega )}$ is the covariant derivative with a spin connection ${\displaystyle \omega }$, which acting on spinors takes the form

${\displaystyle D_{\mu }(\omega ){\psi }_{\nu }={\mathrm{\partial }}_{\mu }{\psi }_{\nu }+\frac{1}{4}{\omega }_{\mu }^{ab}{\gamma }_{ab}{\psi }_{\nu },}$

where ${\displaystyle {\gamma }_{ab}={\gamma }_{[a}{\gamma }_{b]}}$. The regular gamma matrices satisfying the Dirac algebra are denote by ${\displaystyle {\gamma }_a}$, while ${\displaystyle {\gamma }_{\mu }=e_{\mu }^a{\gamma }_a}$ are position-dependent fields. The first line in the action contains the covariantized kinetic terms given by the Einstein–Hilbert action, the Rarita–Schwinger equation, and the gauge kinetic action. The second line corresponds to cubic graviton-gauge field terms along with some quartic gravitino terms. The last line in the Lagrangian is a Chern–Simons term.

The supersymmetry transformation rules are given by

${\displaystyle {\delta }_s{e}_{\mu }^a=\frac{1}{2}\overline{ϵ}{\gamma }^a{\psi }_{\mu },}$

${\displaystyle {\delta }_s{\psi }_{\mu }=D_{\mu }(\hat{\omega })ϵ+\frac{\sqrt{2}}{288}({\gamma }^{\alpha \beta \gamma \delta }{}_{\mu }-8{\gamma }^{\beta \gamma \delta }{\delta }_{\mu }^{\alpha }){\hat{F}}_{\alpha \beta \gamma \delta }ϵ,}$

${\displaystyle {\delta }_s{A}_{\mu \nu \rho }=-\frac{3\sqrt{2}}{4}\overline{ϵ}{\gamma }_{[\mu \nu }{\psi }_{\rho ]},}$

where ${\displaystyle ϵ}$ is the supersymmetry Majorana gauge parameter. All hatted variables are supercovariant in the sense that they do not depend on the derivative of the supersymmetry parameter ${\displaystyle {\mathrm{\partial }}_{\mu }ϵ}$. The action is additionally invariant under parity, with the gauge field transforming as a pseudotensor ${\displaystyle A\to -A}$. The equations of motion for this supergravity also have a rigid symmetry known as the trombone symmetry under which ${\displaystyle g_{\mu \nu }\to {\alpha }^2{g}_{\mu \nu }}$ and ${\displaystyle A_{\mu \nu \rho }\to {\alpha }^3{A}_{\mu \nu \rho }}$.

Special solutions

There are a number of special solutions in 11D supergravity, with the most notable ones being the pp-wave, M2-branes, M5-branes, KK-monopoles, and the M9-brane. Brane solutions are solitonic objects within supergravity that are the low-energy limit of the corresponding M-theory branes. The 3-form gauge field couples electrically to M2-branes and magnetically to M5-branes. Explicit supergravity solitonic solutions for the M2-branes and M5-branes are known.

M2-branes and M5-branes have a regular non-degenerate event horizon whose constant time cross-sections are topologically 7-spheres and 4-spheres, respectively. The near-horizon limit of the extreme M2-brane is given by an ${\displaystyle Ad{S}_4\times S^7}$ geometry while for the extreme M5-brane it is given by ${\displaystyle Ad{S}_7\times S^4}$. These extreme-limit solutions preserve half of the supersymmetry of the vacuum solution, meaning that both the extreme M2-branes and the M5-branes can be seen as solitons interpolating between two maximally supersymmetric Minkowski vacua at infinity, with an ${\displaystyle Ad{S}_4\times S^7}$ or ${\displaystyle Ad{S}_6\times S^4}$ horizon, respectively.

Compactification

The Freund–Rubin compactification of 11D supergravity shows that it preferentially compactifies to seven and four dimensions, the latter of which led to it being extensively studied throughout the 1980s. This compactification is most easily achieved by demanding that the compact and noncompact manifolds have a Ricci tensor that is proportional to the metric, meaning that they are Einstein manifolds. One additionally demands that the solution is stable against fluctuations, which in anti-de Sitter spacetimes requires that the Bretenlohner–Freedman bound is satisfied. Stability is guaranteed if there is some unbroken supersymmetry, although there also exist classically stable solutions that fully break supersymmetry.

One of the main compactification manifolds studied was the 7-sphere. The manifold has 8 Killing spinors, meaning that the resulting four dimensional theory has ${\displaystyle \mathcal{N}=8}$ supersymmetry. Additionally, it also results in an ${\displaystyle \text{SO}(8)}$ gauge group, corresponding to the isometry group of the sphere. A similar widely studied compactification was using a squashed 7-sphere, which can be acquired by embedding the 7-sphere in a quaternionic projective space, with this giving a gauge group of ${\displaystyle \text{SO}(5)\times \text{SU}(2)}$.

A key property of 7-sphere Kaluza-Klein compactifications is that their truncation is consistent, which is not necessarily the case for other Einstein manifolds besides the 7-torus. An inconsistent truncation means that the resulting four dimensional theory is not consistent with the higher dimensional field equations. Physically this needs not be a problem in compactifications to Minkowski spacetimes as the inconsistent truncation merely results in additional irrelevant operators in the action. However, most Einstein manifold compactifications are to anti-de Sitter spacetimes which have a relatively large cosmological constant. In this case irrelevant operators can be converted to relevant ones through the equation of motion.

Related theories

While eleven-dimensional supergravity is the unique supergravity in eleven dimensions at the level of an action, a related theory can be acquired at the level of the equations of motion, known as modified 11D supergravity. This is done by replacing the spin connection by one that is conformally related to the original. Such a theory is inequivalent to standard 11D supergravity only in spaces that are not simply connected. An action for a massive 11D theory can also be acquired by introducing an auxiliary nondynamical Killing vector field, with this theory reducing to massive type IIA supergravity upon dimensional reduction. This is not a proper eleven-dimensional theory since the fields explicitly do not depend on one of the coordinates, but it is nonetheless useful for studying massive branes.

Dimensionally reducing 11D supergravity to ten dimensions gives rise to type IIA supergravity, while dimensionally reducing it to four dimensions can give ${\displaystyle \mathcal{N}=8}$ supergravity, which was one of the original motivations for constructing the theory. While eleven-dimensional supergravity is not UV finite, it is the low energy limit of M-theory. The supergravity also receives corrections at the quantum level, where these corrections sometimes playing an important role in various compactification mechanisms.

Unlike for supergravity in other dimensions, an extension to eleven dimensional anti-de Sitter spacetime does not exist. While the theory is the supersymmetric theory in the highest number of dimensions, the caveat is that this only holds for spacetime signatures with one temporal dimension. If arbitrary spacetime signatures are allowed, then there also exists a supergravity in twelve dimensions with two temporal dimensions.