Supergravity

Like any field theory of gravity, a supergravity theory contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries.

The term "low energy limits" labels some 10-dimensional supergravity theories. These arise as the massless, tree-level approximation of string theories. True effective field theories of string theories, rather than truncations, are rarely available. Due to string dualities, the conjectured 11-dimensional M-theory is required to have 11-dimensional supergravity as a "low energy limit". However, this doesn't necessarily mean that string theory/M-theory is the only possible UV completion of supergravity; supergravity research is useful independent of those relations.

Before we move on to SUGRA proper, let's recapitulate some important details about general relativity. We have a 4D differentiable manifold M with a Spin(3,1) principal bundle over it. This principal bundle represents the local Lorentz symmetry. In addition, we have a vector bundle T over the manifold with the fiber having four real dimensions and transforming as a vector under Spin(3,1). We have an invertible linear map from the tangent bundle TM to T. This map is the vierbein. The local Lorentz symmetry has a gauge connection associated with it, the spin connection.

The following discussion will be in superspace notation, as opposed to the component notation, which isn't manifestly covariant under SUSY. There are actually many different versions of SUGRA out there which are inequivalent in the sense that their actions and constraints upon the torsion tensor are different, but ultimately equivalent in that we can always perform a field redefinition of the supervierbeins and spin connection to get from one version to another.

In 4D N=1 SUGRA, we have a 4|4 real differentiable supermanifold M, i.e. we have 4 real bosonic dimensions and 4 real fermionic dimensions. As in the nonsupersymmetric case, we have a Spin(3,1) principal bundle over M. We have an R^4|4 vector bundle T over M. The fiber of T transforms under the local Lorentz group as follows; the four real bosonic dimensions transform as a vector and the four real fermionic dimensions transform as a Majorana spinor. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handed Weyl spinor (they're not independent of each other). We also have a spin connection as before.

We will use the following conventions; the spatial (both bosonic and fermionic) indices will be indicated by M, N, ... . The bosonic spatial indices will be indicated by μ, ν, ..., the left-handed Weyl spatial indices by α, β,..., and the right-handed Weyl spatial indices by ${\displaystyle {\dot {\alpha }}}$, ${\displaystyle {\dot {\beta }}}$, ... . The indices for the fiber of T will follow a similar notation, except that they will be hatted like this: ${\displaystyle {\hat {M}},{\hat {\alpha }}}$. See van der Waerden notation for more details. ${\displaystyle M=(\mu ,\alpha ,{\dot {\alpha }})}$. The supervierbein is denoted by ${\displaystyle e_{N}^{\hat {M}}}$, and the spin connection by ${\displaystyle \omega _{{\hat {M}}{\hat {N}}P}}$. The inverse supervierbein is denoted by ${\displaystyle E_{\hat {M}}^{N}}$.

The supervierbein and spin connection are real in the sense that they satisfy the reality conditions

${\displaystyle e_{N}^{\hat {M}}(x,{\overline {\theta }},\theta )^{*}=e_{N^{*}}^{{\hat {M}}^{*}}(x,\theta ,{\overline {\theta }})}$ where ${\displaystyle \mu ^{*}=\mu }$, ${\displaystyle \alpha ^{*}={\dot {\alpha }}}$, and ${\displaystyle {\dot {\alpha }}^{*}=\alpha }$ and ${\displaystyle \omega (x,{\overline {\theta }},\theta )^{*}=\omega (x,\theta ,{\overline {\theta }})}$.

The covariant derivative is defined as

${\displaystyle D_{\hat {M}}f=E_{\hat {M}}^{N}\left(\partial _{N}f+\omega _{N}[f]\right)}$.

The covariant exterior derivative as defined over supermanifolds needs to be super graded. This means that every time we interchange two fermionic indices, we pick up a +1 sign factor, instead of -1.

The presence or absence of R symmetries is optional, but if R-symmetry exists, the integrand over the full superspace has to have an R-charge of 0 and the integrand over chiral superspace has to have an R-charge of 2.

A chiral superfield X is a superfield which satisfies ${\displaystyle {\overline {D}}_{\hat {\dot {\alpha }}}X=0}$. In order for this constraint to be consistent, we require the integrability conditions that ${\displaystyle \left\{{\overline {D}}_{\hat {\dot {\alpha }}},{\overline {D}}_{\hat {\dot {\beta }}}\right\}=c_{{\hat {\dot {\alpha }}}{\hat {\dot {\beta }}}}^{\hat {\dot {\gamma }}}{\overline {D}}_{\hat {\dot {\gamma }}}}$ for some coefficients c.

Unlike nonSUSY GR, the torsion has to be nonzero, at least with respect to the fermionic directions. Already, even in flat superspace, ${\displaystyle D_{\hat {\alpha }}e_{\hat {\dot {\alpha }}}+{\overline {D}}_{\hat {\dot {\alpha }}}e_{\hat {\alpha }}\neq 0}$. In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:

${\displaystyle T_{{\hat {\underline {\alpha }}}{\hat {\underline {\beta }}}}^{\hat {\underline {\gamma }}}=0}$

${\displaystyle T_{{\hat {\alpha }}{\hat {\beta }}}^{\hat {\mu }}=0}$

${\displaystyle T_{{\hat {\dot {\alpha }}}{\hat {\dot {\beta }}}}^{\hat {\mu }}=0}$

${\displaystyle T_{{\hat {\alpha }}{\hat {\dot {\beta }}}}^{\hat {\mu }}=2i\sigma _{{\hat {\alpha }}{\hat {\dot {\beta }}}}^{\hat {\mu }}}$

${\displaystyle T_{{\hat {\mu }}{\hat {\underline {\alpha }}}}^{\hat {\nu }}=0}$

${\displaystyle T_{{\hat {\mu }}{\hat {\nu }}}^{\hat {\rho }}=0}$

Here, ${\displaystyle {\underline {\alpha }}}$ is a shorthand notation to mean the index runs over either the left or right Weyl spinors.

The superdeterminant of the supervierbein, ${\displaystyle \left|e\right|}$, gives us the volume factor for M. Equivalently, we have the volume 4|4-superform${\displaystyle e^{{\hat {\mu }}=0}\wedge \cdots \wedge e^{{\hat {\mu }}=3}\wedge e^{{\hat {\alpha }}=1}\wedge e^{{\hat {\alpha }}=2}\wedge e^{{\hat {\dot {\alpha }}}=1}\wedge e^{{\hat {\dot {\alpha }}}=2}}$.

If we complexify the superdiffeomorphisms, there is a gauge where ${\displaystyle E_{\hat {\dot {\alpha }}}^{\mu }=0}$, ${\displaystyle E_{\hat {\dot {\alpha }}}^{\beta }=0}$ and ${\displaystyle E_{\hat {\dot {\alpha }}}^{\dot {\beta }}=\delta _{\dot {\alpha }}^{\dot {\beta }}}$. The resulting chiral superspace has the coordinates x and Θ.

R is a scalar valued chiral superfield derivable from the supervielbeins and spin connection. If f is any superfield, ${\displaystyle \left({\bar {D}}^{2}-8R\right)f}$ is always a chiral superfield.

The action for a SUGRA theory with chiral superfields X, is given by

${\displaystyle S=\int d^{4}xd^{2}\Theta 2{\mathcal {E}}\left[{\frac {3}{8}}\left({\bar {D}}^{2}-8R\right)e^{-K({\bar {X}},X)/3}+W(X)\right]+c.c.}$

where K is the Kähler potential and W is the superpotential, and ${\displaystyle {\mathcal {E}}}$ is the chiral volume factor.

Unlike the case for flat superspace, adding a constant to either the Kähler or superpotential is now physical. A constant shift to the Kähler potential changes the effective Planck constant, while a constant shift to the superpotential changes the effective cosmological constant. As the effective Planck constant now depends upon the value of the chiral superfield X, we need to rescale the supervierbeins (a field redefinition) to get a constant Planck constant. This is called the Einstein frame.

N=8 Supergravity is the most symmetric quantum field theory which involves gravity and a finite number of fields. It can be found from a dimensional reduction of 11D supergravity by making the size of 7 of the dimensions go to zero. It has 8 supersymmetries which is the most any gravitational theory can have since there are 8 half-steps between spin 2 and spin -2. (A graviton has the highest spin in this theory which is a spin 2 particle). More supersymmetries would mean the particles would have superpartners with spins higher than 2. The only theories with spins higher than 2 which are consistent involve an infinite number of particles (such as string theory and higher-spin theories). Stephen Hawking in his A Brief History of Time speculated that this theory could be the Theory of Everything. However, in later years this was abandoned in favour of string theory. There has been renewed interest in the 21st century with the possibility that this theory may be finite.

Higher-dimensional SUGRA is the higher-dimensional, supersymmetric generalization of general relativity. Supergravity can be formulated in any number of dimensions up to eleven. Higher-dimensional SUGRA focuses upon supergravity in greater than four dimensions.

The number of supercharges in a spinor depends on the dimension and the signature of spacetime. The supercharges occur in spinors. Thus the limit on the number of supercharges cannot be satisfied in a spacetime of arbitrary dimension. Some theoretical examples in which this is satisfied are:

• 12-dimensional two-time theory
• 11-dimensional maximal SUGRA
• 10-dimensional SUGRA theories
  • Type IIA SUGRA: N = (1, 1)
  • IIA SUGRA from 11d SUGRA
  • Type IIB SUGRA: N = (2, 0)
  • Type I gauged SUGRA: N = (1, 0)
• 9d SUGRA theories
  • Maximal 9d SUGRA from 10d
  • T-duality
  • N = 1 Gauged SUGRA

The supergravity theories that have attracted the most interest contain no spins higher than two. This means, in particular, that they do not contain any fields that transform as symmetric tensors of rank higher than two under Lorentz transformations. The consistency of interacting higher spin field theories is, however, presently a field of very active interest.