View source for Pure 4D N = 1 supergravity

{{Short description|Minimal supergravity in four dimensions}}
{{DISPLAYTITLE:Pure 4D ''N'' = 1 supergravity}}
In [[supersymmetry]], '''pure 4D <math>\mathcal N=1</math> supergravity''' describes the simplest [[Four-dimensional space|four-dimensional]] [[supergravity]], with a single [[supercharge]] and a [[supermultiplet]] containing a [[graviton]] and [[gravitino]]. The [[action (physics)|action]] consists of the [[Einstein–Hilbert action]] and the [[Rarita–Schwinger equation|Rarita–Schwinger action]]. The theory was first formulated by [[Daniel Z. Freedman]], [[Peter van Nieuwenhuizen]], and [[Sergio Ferrara]], and independently by [[Stanley Deser]] and [[Bruno Zumino]] in 1976.<ref name="paper1">{{cite journal|last1=Freedman|first1=D.Z.|authorlink1=Daniel Z. Freedman|last2=van Nieuwenhuizen|first2=P.|authorlink2=Peter van Nieuwenhuizen|last3=Ferrara|first3=S.|authorlink3=Sergio Ferrara|date=1976|title=Progress toward a theory of supergravity|url=https://link.aps.org/doi/10.1103/PhysRevD.13.3214|journal=Phys. Rev. D|volume=13|issue=12|pages=3214–3218|doi=10.1103/PhysRevD.13.3214|pmid=|arxiv=|bibcode=1976PhRvD..13.3214F |s2cid=|access-date=}}</ref><ref name="paper2">{{cite journal|last1=Deser|first1=S.|authorlink1=Stanley Deser|last2=Zumino|first2=B.|authorlink2=Bruno Zumino|date=1976|title=Consistent supergravity|url=http://cds.cern.ch/record/438874|journal=Physics Letters B|volume=62|issue=3|pages=335–337|doi=10.1016/0370-2693(76)90089-7|pmid=|arxiv=|bibcode=1976PhLB...62..335D |s2cid=|access-date=}}</ref> The only consistent extension to [[spacetime]]s with a [[cosmological constant]] is to [[anti-de Sitter space]], first formulated by [[Paul Townsend]] in 1977.<ref>{{cite journal|last1=Townsend|first1=P.K.|authorlink1=Paul Townsend|date=1977|title=Cosmological constant in supergravity|url=https://link.aps.org/doi/10.1103/PhysRevD.15.2802|journal=Phys. Rev. D|volume=15|issue=10|pages=2802–2804|doi=10.1103/PhysRevD.15.2802|pmid=|arxiv=|bibcode=1977PhRvD..15.2802T |s2cid=|access-date=}}</ref> When additional matter supermultiplets are included in this theory, the result is known as [[4D N = 1 supergravity| matter-coupled 4D <math>\mathcal N = 1</math> supergravity]].

== Flat spacetime ==

To describe the coupling between [[gravity]] and [[particle]]s of arbitrary [[spin (physics)|spin]], it is useful to use the [[tetrad formalism|vielbein formalism]] of [[general relativity]].<ref>{{cite book|last=Nakahara|first=M.|author-link=|date=2003|title=Geometry, Topology and Physics|url=|doi=|edition=2|location=|publisher=CRC Press|chapter=7|page=|isbn=978-0750306065}}</ref> This replaces the [[metric tensor|metric]] by a set of [[vector field]]s <math>e_a = e^\mu_a \partial_\mu</math> indexed by flat indices <math>a</math> such that

:<math>
g_{\mu\nu} = e^a_\mu e^b_\nu \eta_{ab}.
</math>

In a sense the vielbeins are the square root of the metric. This introduces a new [[gauge symmetry (mathematics)|local]] [[Lorentz covariance|Lorentz symmetry]] on the vielbeins <math>e^a_\mu \rightarrow e^b_\mu \Lambda^a{}_b(x)</math>, together with the usual [[diffeomorphism]] invariance associated with the spacetime indices <math>\mu</math>. This has an associated [[connection form|connection]] known as the [[spin connection]] <math>\omega^{ab}_\mu</math> defined through <math>\nabla_\mu e_a = \omega_\mu{}^{b}{}_a e_b</math>, it being a generalization of the [[Christoffel symbol|Christoffel connection]] to arbitrary spin fields. For example, for [[spinor]]s the [[covariant derivative]] is given by

:<math>
D_\mu = \partial_\mu + \frac{1}{4}\omega_\mu^{ab}\gamma_{ab},
</math>

where <math>\gamma_a</math> are [[gamma matrices]] satisfing the [[Dirac algebra]], with <math>\gamma_{ab} = \gamma_{[a}\gamma_{b]}</math>. These are often contracted with vielbeins to construct <math>\gamma_\mu = e^a_\mu \gamma_a</math> which are in general position-dependent fields rather than constants. The spin connection has an explicit expression in terms of the vielbein and an additional [[torsion tensor]] which can arise when there is matter present in the theory. A vanishing torsion is equivalent to the [[Levi-Civita connection]].

The pure <math>\mathcal N=1</math> supergravity action in four dimensions is the combination of the  Einstein–Hilbert action and the Rarita–Schwinger action<ref name="Agata">{{cite book|last1=Dall'Agata|first1=G.|author-link1=|last2=Zagermann|first2=M.|author-link2=|date=2021|title=Supergravity: From First Principles to Modern Applications|url=|doi=|location=|publisher=Springer|chapter=4|pages=43–70|isbn=978-3662639788}}</ref>

{{Equation box 1
|title='''Pure 4D N=1 supergravity action'''
|indent=:
|equation = <math> S =\frac{M_P^2}{2} \int d^4 x \ e R -\frac{M_P^2}{2} \int d^4 x \ e \  \bar \psi_\mu \gamma^{\mu\nu\rho}D_\nu \psi_\rho. </math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

Here <math>M_P</math> is the [[Planck units|Planck mass]], <math>e = \det e^a_\mu=\sqrt{-g}</math>, and <math>\psi_\mu</math> is the [[Majorana equation|Majorana]] gravitino with its spinor index left implicit. Treating this action within the first-order formalism where both the vielbein and spin connection are independent fields allows one to solve for the spin connections equation of motion, showing that it has the torsion <math>T^\mu_{ab} = \tfrac{1}{2}\bar \psi_a\gamma^\mu \psi_b</math>.<ref name="Freedman">{{cite book|last1=Freedman|first1=D.Z.|author-link1=Daniel Z. Freedman|last2=Van Proeyen|first2=A.|author-link2=|date=2012|title=Supergravity|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=9|pages=187–200|isbn=978-0521194013}}</ref> The second-order formalism action is then acquired by substituting this expression for the spin connection back into the action, yielding additional quartic gravitino vertices, with the Einstein–Hilbert and Rarita–Schwinger actions now being written with a torsionless spin connection that explicitly depends on the vielbeins.

The supersymmetry transformation rules that leave the action invariant are

:<math>
\delta e^a_\mu = \tfrac{1}{2}\bar \epsilon \gamma^a \psi_\mu, \ \ \ \ \ \ \ \ \delta \psi_\mu = D_\mu \epsilon,
</math>

where <math>\epsilon(x)</math> is the spinorial gauge parameter. While historically the first order<ref name="paper2"/> and second order<ref name="paper1"/> formalism were the first ones used to show the invariance of the action, the 1.5-order formalism is the easiest for most supergravity calculations. The additional [[symmetry (physics)|symmetries]] of the action are general coordinate transformations and local Lorentz transformations.

== Curved spacetime ==

The four dimensional <math>\mathcal N=1</math> [[super-Poincare algebra]] in [[Minkowski spacetime]] can be generalized to anti-de Sitter spacetime, but not to [[de Sitter space]]time, since the super-[[Jacobi identity]] cannot be satisfied in that case. Its action can be constructed by gauging this [[superalgebra]], yielding the supersymmetry transformation rules for the vielbein and the gravitino.<ref>{{cite book|last=Ortin|first=T.|author-link=|date=2015|title=Gravity and Strings|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=5|pages=175–186|edition=2|isbn=978-0521768139}}</ref>

The action for <math>\mathcal N=1</math> AdS supergravity in four dimensions is<ref name="Freedman"/>

:<math>
S = \frac{M_P^2}{2} \int d^4 x \ e \bigg(R+ \frac{6}{L^2}\bigg) - \frac{M_P^2}{2}\int d^4 x \ e \bigg(\bar \psi_\mu \gamma^{\mu \nu \rho}D_\nu \psi_\rho + \frac{1}{L}\bar \psi_\mu \gamma^{\mu\nu} \psi_\nu\bigg),
</math>

where <math>L</math> is the AdS radius and the second term is the negative cosmological constant <math>\Lambda = -3/L^2</math>. The supersymmetry transformations are

:<math>
\delta e^a_\mu = \tfrac{1}{2}\bar \epsilon \gamma^a \psi_\mu, \ \ \ \ \ \ \delta \psi_\mu =  D_\mu \epsilon - \tfrac{1}{2L} \gamma_\mu \epsilon.
</math>

While the bilinear term in the action appears to be giving a [[mass]] to the gravitino, it still belongs to the [[massless particle|massless]] gravity supermultiplet.<ref name="Agata"/> This is because mass is not well-defined in [[curved space]]times, with <math>P_\mu P^\mu</math> no longer being a [[Casimir element|Casimir operator]] of the AdS super-Poinacre algebra. It is however conventional to define a mass through the [[Laplace–Beltrami operator]], in which case particles within the same supermultiplet have different masses, unlike in flat spacetimes.

== See also ==

* [[N = 8 supergravity|''N'' = 8 supergravity]]

==References==
{{Reflist}}

{{Supersymmetry topics}}
{{Quantum field theories}}

{{DEFAULTSORT:Pure 4D N = 1 supergravity}}
[[Category:Supersymmetric quantum field theory]]
[[Category:Theories of gravity]]