Pure $\mathcal {N} = 1$ Supergravity in Four Dimensions

Abstract

In this chapter, we discuss pure supergravity in four dimensions with minimal supersymmetry and show explicitly the construction of its action, also proving the invariance under supersymmetry transformations. We then extend this construction to the case of a non-trivial cosmological constant and finally discuss the concept of mass in spacetimes with negative curvature.

4.A Appendix: Gauging the Poincaré Algebra

When introducing general relativity as well as supergravity, we discussed the possibility of considering the vierbein and the spin connection as independent quantities. Do we have any conceptual reason behind this, in addition to the simplification of some computations? We will now see that an interesting perspective on gravity, which can help when dealing with supergravity, is that of considering gravity itself as a sort of a gauge theory where the gauge group is the Poincaré group. This analogy will work only to a certain extent, but it will be very useful for understanding many specific new features that have to be introduced when one wants to promote supersymmetry to a local symmetry of nature. In fact, supergravity is the gauge theory of supersymmetry, and therefore there must be a way to describe it as a theory where the gauge group is the Poincaré supergroup (or some other supergroup). For the sake of simplicity in this appendix, we set M _P = 1.

Consider an ordinary gauge transformation δ _𝜖 = 𝜖 ^A T _A, where T _A are the gauge generators satisfying

$$\displaystyle \begin{aligned}{}[T_A,T_B] = f_{AB}{}^C T_C. \end{aligned} $$

If this is a global symmetry of an action, it can be made local by introducing vector fields $A_\mu ^A$ for each symmetry so that the algebra

$$\displaystyle \begin{aligned}{}[\delta(\epsilon_1^A),\delta(\epsilon_2^B)] = \delta\left(\epsilon_2^B \epsilon_1^A f_{AB}{}^C\right), {} \end{aligned} $$

(4.90)

where f _AB ^C are the structure constants, has a faithful realization on them,

$$\displaystyle \begin{aligned} \delta_\epsilon A_\mu^A = \partial_\mu \epsilon^A + \epsilon^C\, A^B_\mu\, f_{BC}{}^A, {} \end{aligned} $$

(4.91)

and we can introduce covariant derivatives

$$\displaystyle \begin{aligned} D_\mu = \partial_\mu - A_\mu^A T_A {} \end{aligned} $$

(4.92)

acting non-trivially on fields which transform in non-trivial representations of the gauge group. The curvature, defined as

$$\displaystyle \begin{aligned}{}[D_\mu,D_\nu] = - F_{\mu\nu}^A T^A \qquad \Leftrightarrow \qquad F_{\mu\nu}^A \equiv 2 \partial_{[\mu}A_{\nu]}^A + A_\mu^B A_\nu^C f_{BC}{}^A, {} \end{aligned} $$

(4.93)

transforms covariantly:

$$\displaystyle \begin{aligned} \delta_\epsilon F^A_{\mu\nu} = \epsilon^C F_{\mu\nu}^B f_{BC}{}^A. {} \end{aligned} $$

(4.94)

Let us now imagine that we want to make local the symmetries of the Poincaré group. The usual procedure is to introduce gauge fields in correspondence with the generators of the algebra. For the Poincaré algebra, this means introducing two fields, one $e_\mu ^a$ with a vector gauge index and one $\omega _\mu ^{ab}$ with two, so as to match the gauge generators

$$\displaystyle \begin{aligned} A_\mu^A T_A = e_\mu^a P_a +\frac 12\, \omega_\mu^{ab} M_{ab}. {} \end{aligned} $$

(4.95)

The gauge curvatures of these vectors are precisely T ^a and R ^ab as defined in (3.22) and (3.23), where the spin connection and the vielbein are so far independent fields.

This construction is perfectly legitimate. However, it clearly leads to an ordinary gauge theory and not to a gravity theory as we would like. From the T ^a and R ^ab curvatures, we could construct kinetic terms giving the propagation of independent degrees of freedom and discuss the resulting gauge theory, where the Poincaré group is realized on the vector fields as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta_P \omega^{ab} & =&\displaystyle 0, {} \end{array} \end{aligned} $$

(4.96)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta_P e^{a} & =&\displaystyle D \epsilon^a, {} \end{array} \end{aligned} $$

(4.97)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta_M \omega^{ab} & =&\displaystyle d \varLambda^{ab} - \omega^{a}{}_c \varLambda^{cb} - \varLambda^{ac}\omega_c{}^b, {} \end{array} \end{aligned} $$

(4.98)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta_M e^{a} & =&\displaystyle \varLambda^a{}_c e^c. {} \end{array} \end{aligned} $$

(4.99)

If, on the other hand, we want to get only the metric degrees of freedom, we have to impose a constraint between ω ^ab and e ^a. This is the conventional constraint or torsion constraint

$$\displaystyle \begin{aligned} T^a = 0. {} \end{aligned} $$

(4.100)

This constraint however is not invariant under (4.96)–(4.97):

$$\displaystyle \begin{aligned} \delta_P T^a = \delta_P D e^a = D \delta_P e^a = DD \epsilon^a = -R^a{}_b \epsilon^b \neq 0. {} \end{aligned} $$

(4.101)

This means that if we impose the conventional constraint, translation symmetry is broken. Moreover, it is also clear that now the spin connection ω ^ab cannot be treated as independent of the vielbein anymore and hence the transformation (4.96) will not be valid anymore. Since ω ^ab = ω ^ab(e), the spin connection is also not invariant under translations

$$\displaystyle \begin{aligned} \delta_P \omega^{ab} = \int d^4 x\, \frac{\delta \omega^{ab}}{\delta e^c} \delta_P e^c \neq 0. \end{aligned} $$

The final outcome of this discussion is that, when the conventional constraint is imposed, the Poincaré gauge algebra is deformed and translational symmetry is replaced by a new invariance under diffeomorphisms. This can be seen by considering the commutator of two translation generators on the vielbein:

$$\displaystyle \begin{aligned}{}[\delta_{P2},\delta_{P1}] e^a = - \delta_{P[2}(D \epsilon_{1]}^a) = - \delta_{P[2} \omega^{a}{}_c \epsilon_{1]}^c, {} \end{aligned} $$

(4.102)

and now δ _P ω ^ab ≠ 0. The resulting algebra then has a non-vanishing commutator

$$\displaystyle \begin{aligned}{}[P,P] \neq 0, \end{aligned} $$

as is appropriate for general coordinate transformations, which do not commute. Actually one can check that using the constraint (4.100), the translation generators on the vielbein take the form of general coordinate transformations.

The Lie derivative of a p-form A _p along the flow of a vector field V  is defined as

where $\sigma _{t}^*$ is the pullback of the differential form along the flow generated by the vector field V . When applied to a scalar valued p-form, this reduces to

The action constructed from the curvatures and the vielbein then is invariant with respect to local Lorentz group transformations and diffeomorphisms. The infinitesimal change of a function under a diffeomorphism is given by the Lie derivative , and therefore the action is going to be invariant if

but the first term is a total derivative that can be discarded while the second is zero because $d\mathcal {L}$ has one degree more than the top form. Finally, in the construction of an action, we will not make use of a kinetic term of the form R ^ab ∧ ⋆R _ab because of the conventional constraint which makes it quartic in the derivatives. The appropriate quadratic term is the Einstein–Hilbert action above.

The method we have outlined in this section can be easily extended to generic supergravity theories by extending the Poincaré algebra to the super Poincaré algebra, by including fermionic generators and possibly other bosonic generators for the internal symmetries. The power of this approach lies in the ease of guessing the transformation laws under the various symmetries, including supersymmetry. This means that this approach can be used as a guide to derive and construct the Lagrangian and/or the equations of motion of systems respecting any symmetry group we would like to realize. Once again, we stress that one has to be careful with its application because of the constraints that will be needed to obtain a consistent gravity theory (invariant under diffeomorphisms). Imposing these constraints will break the transformation rules that do not preserve them.

1.1 4.A.1 Gauging the Super Poincaré Algebra

We end this appendix with a few remarks on the gauging of the super Poincaré group. We could gauge this algebra by adding new vector fields ψ _μA for the fermionic generators Q ^A. From the algebra we then have

$$\displaystyle \begin{aligned} A^A_\mu T_A = e_\mu^a P_a + \frac 12\,\omega_\mu^{ab} M_{ab} + \overline \psi_{\mu A} Q^A + \overline \psi_\mu^A Q_A, \end{aligned} $$

(4.103)

and we can read the supersymmetry transformations by applying

$$\displaystyle \begin{aligned} \delta_\epsilon A_\mu^A = \partial_\mu \epsilon^A + \epsilon^C \,A^B_\mu \,f_{BC}{}^A. \end{aligned}$$

For instance, for $\mathcal {N} = 1$ supergravity, we would get that the spin connection is invariant,

$$\displaystyle \begin{aligned} \delta_\epsilon \omega^{ab} = 0, {} \end{aligned} $$

(4.104)

because the Lorentz generator never appears on the right-hand side of any commutator involving the supersymmetry generator. However, just like for the bosonic case, we should impose a torsional constraint in order for the vielbein and spin connection not to be independent. Doing so, we fix the form of the spin connection as ω ^ab = ω ^ab(e, ψ) and check the new realization of the algebra on the fields.

One last interesting remark involves the definition of the gauge curvatures for the super-Poincaré algebra. From the structure constants of the supersymmetry algebra (1.53), one can deduce a new definition for the curvatures, including the one of the translation generators, namely, the torsion T ^a. Since the translation generators P _a appear on the right-hand side of the commutator of two supercharges, the corresponding curvature definition is now

$$\displaystyle \begin{aligned} T^a = D e^a - \frac 14\, \overline \psi^A \gamma^a \psi_A \end{aligned} $$

(4.105)

and involves a fermion bilinear. This means that imposing the conventional constraint T ^a = 0 results in a spin connection depending on the gravitino fields. Hence, supergravity is often referred to as a theory with non-trivial torsion for the spin connection, because T ^a = 0 implies De ^a ≠ 0.

Exercises

4.1

Prove that the supersymmetry variation of the spin connection in the first-order formalism and in the second-order formalism is equivalent upon using the gravitino equation of motion.

4.2

In the first-order formalism, compute the new piece in the variation of the spin connection due to the cosmological constant.

4.3

Find the embedding coordinates that give origin to

$$\displaystyle \begin{aligned} ds^2 = - dt^2 + \mathrm{e}^{2t/\ell}\, d\vec{x}^2, \end{aligned} $$

for de Sitter and to

$$\displaystyle \begin{aligned} ds^2 = - dt^2 + \ell^2\sin^2 ({t}/{\ell})\left(d \psi^2 + \sinh^2 \psi \,d \varOmega^2\right), \end{aligned} $$

for anti-de Sitter spacetime. Discuss the “cosmological” meaning of these metrics.

4.4

Check that the gauge curvatures T ^a and R ^ab coming from (4.95) and (4.93) match (3.22) and (3.23).