Editing Supergravity (section)

== 4D ''N'' = 1 SUGRA ==
{{Main|1=4D N = 1 supergravity}} 
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{{which|date=February 2021}} 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'''<sup>4|4</sup> 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 equation|Majorana spinor]]. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handed [[Weyl equation|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 <math>\dot{\alpha}</math>, <math>\dot{\beta}</math>, ... . The indices for the fiber of T will follow a similar notation, except that they will be hatted like this: <math>\hat{M},\hat{\alpha}</math>. See [[van der Waerden notation]] for more details. <math>M = (\mu,\alpha,\dot{\alpha})</math>. The supervierbein is denoted by <math>e^{\hat{M}}_N</math>, and the spin connection by <math>\omega_{\hat{M}\hat{N}P}</math>. The ''inverse'' supervierbein is denoted by <math>E^N_{\hat{M}}</math>.

The supervierbein and spin connection are real in the sense that they satisfy the reality conditions
:<math>e^{\hat{M}}_N (x,\overline{\theta},\theta)^* = e^{\hat{M}^*}_{N^*}(x,\theta,\overline{\theta})</math> where <math>\mu^*=\mu</math>, <math>\alpha^*=\dot{\alpha}</math>, and <math>\dot{\alpha}^*=\alpha</math> and <math>\omega(x,\overline{\theta},\theta)^*=\omega(x,\theta,\overline{\theta})</math>.

The [[covariant derivative]] is defined as
:<math>D_\hat{M}f=E^N_{\hat{M}}\left( \partial_N f + \omega_N[f] \right)</math>.

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-symmetry|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 <math>\overline{D}_{\hat{\dot{\alpha}}}X=0</math>. In order for this constraint to be consistent, we require the integrability conditions that <math>\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}}}</math> for some coefficients ''c''.

Unlike nonSUSY GR, the [[torsion tensor|torsion]] has to be nonzero, at least with respect to the fermionic directions. Already, even in flat superspace, <math>D_{\hat{\alpha}}e_{\hat{\dot{\alpha}}}+\overline{D}_{\hat{\dot{\alpha}}}e_{\hat{\alpha}} \neq 0</math>.
In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:
:<math>T^{\hat{\underline{\gamma}}}_{\hat{\underline{\alpha}}\hat{\underline{\beta}}} = 0</math>
:<math>T^{\hat{\mu}}_{\hat{\alpha}\hat{\beta}} = 0</math>
:<math>T^{\hat{\mu}}_{\hat{\dot{\alpha}}\hat{\dot{\beta}}} = 0</math>
:<math>T^{\hat{\mu}}_{\hat{\alpha}\hat{\dot{\beta}}} = 2i\sigma^{\hat{\mu}}_{\hat{\alpha}\hat{\dot{\beta}}}</math>
:<math>T^{\hat{\nu}}_{\hat{\mu}\hat{\underline{\alpha}}} = 0</math>
:<math>T^{\hat{\rho}}_{\hat{\mu}\hat{\nu}} = 0</math>
Here, <math>\underline{\alpha}</math> is a shorthand notation to mean the index runs over either the left or right Weyl spinors.

The [[superdeterminant]] of the supervierbein, <math>\left| e \right|</math>, gives us the volume factor for M. Equivalently, we have the volume 4|4-superform<math>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}</math>.

If we complexify the superdiffeomorphisms, there is a gauge where <math>E^{\mu}_{\hat{\dot{\alpha}}}=0</math>, <math>E^{\beta}_{\hat{\dot{\alpha}}}=0</math> and <math>E^{\dot{\beta}}_{\hat{\dot{\alpha}}}=\delta^{\dot{\beta}}_{\dot{\alpha}}</math>. 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, <math>\left( \bar{D}^2 - 8R \right) f</math> is always a chiral superfield.

The action for a SUGRA theory with chiral superfields ''X'', is given by
:<math>S = \int d^4x d^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.</math>
where ''K'' is the [[Kähler potential]] and ''W'' is the [[superpotential]], and <math>\mathcal{E}</math> 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'''.