Supergravity in Arbitrary Dimensions

Abstract

While in the first nine chapters of this book, we focused on four-dimensional theories, we use this final chapter to give a brief outlook on supergravity theories in a number of dimensions other than four. Many details of their structure differ from the four-dimensional case, but the main terms and relations remain valid.

While in the first nine chapters of this book, we focused on four-dimensional theories, we use this final chapter to give a brief outlook on supergravity theories in a number of dimensions other than four. Many details of their structure differ from the four-dimensional case, but the main terms and relations remain valid. We begin by analyzing the eleven- and ten-dimensional theories, as they are directly related to the low-energy limit of string theory and then discuss the five-dimensional models, which are very useful in the context of the gauge/gravity correspondence.

1 Higher-Dimensional Theories

$\mathcal {N}=1$ supersymmetry in 4D admits a short representation containing only the graviton, g _μν, and the gravitino, ψ _μ. On-shell, both fields have two degrees of freedom, and we can use them to describe a pure supergravity theory. As we have seen in Chap. 4, the presence of the gravitino provides a lot of new interesting features and constraints to the standard gravity theory described by the Einstein–Hilbert action. This model can then be further coupled to matter fields resulting in even richer and more interesting theories. However, the minimal model can contain just the two basic fields g _μν and ψ _μ.

For supergravity models in arbitrary dimensions higher than D = 4, the situation changes. The Lorentz group becomes SO(1,D − 1), and hence massless physical states are classified by SO(D − 2) representations. This means that while in four dimensions we can classify states by their helicities, in higher dimensions one has generically many more states, and the representations of supersymmetry will no longer admit short representations with only the graviton and gravitino fields. For a generic D, the graviton field g _μν describes a total of

$$\displaystyle \begin{aligned} \frac{(D - 2)(D-2+1)}{2}-1 {} \end{aligned} $$

(10.1)

on-shell degrees of freedom (the metric fluctuations are symmetric traceless matrix states). At the same time, the number of degrees of freedom of a spinor representation grows even faster with D, being

$$\displaystyle \begin{aligned} \# \mbox{ dof} = k \, 2^{[{D}/{2}]-1}, \end{aligned} $$

(10.2)

with k = 2 for Dirac spinors, k = 1 for Majorana spinors and for Weyl spinors, and k = 1∕2 for Majorana–Weyl spinors (cf. Table 10.1 and Appendix 10.A). This implies that the gravitino ψ _μ has

$$\displaystyle \begin{aligned} k \, 2^{[{D}/{2}]-1} (D - 2 -1) {} \end{aligned} $$

(10.3)

Table 10.1 The possible values for η and 𝜖 together with the resulting minimal spinor types, the minimal number of real supercharges, and the general form of the R-symmetry groups (M =  Majorana, SM =  Symplectic Majorana, W =  Weyl, MW =  Majorana–Weyl, SMW =  Symplectic Majorana–Weyl)

states, where the (D − 2 − 1) factor comes from the vector index and the fact that the Rarita–Schwinger action is invariant under δψ _μ = ∂ _μ λ and one has to remove the auxiliary spinor γ ^μ ψ _μ. By a simple comparison of (10.1) with (10.3), we immediately see that only in four dimensions, one can simply match bosonic and fermionic degrees of freedom in a multiplet by using only the graviton and gravitino fields. As soon as one moves to higher dimensions, one needs more fields, both bosonic and fermionic ones. For instance, in five dimensions the graviton has five degrees of freedom and the gravitino has eight (one has to take a Dirac spinor; no Weyl or Majorana spinors are allowed.¹) To complete the graviton multiplet, the matching of bosonic and fermionic states thus requires three additional bosonic degrees of freedom represented by a massless vector field, A _μ, the graviphoton. This is just a special example of what is needed to construct a full supergravity multiplet in higher dimensions: antisymmetric tensor fields . These are rank n fields $B_{\mu _1 \ldots \mu _n}$ with complete antisymmetry of their indices and a tensor gauge invariance,

$$\displaystyle \begin{aligned} \delta B_{\mu_1 \ldots \mu_n} = n\,\partial_{[\mu_1} \varLambda_{\mu_2 \ldots \mu_n]}. \end{aligned} $$

(10.4)

The vector field is a special instance where n = 1, and we usually don’t see higher-rank tensor fields in four-dimensional theories because for n = 2 they are equivalent to scalar fields (as long as they are massless) and for n = 3, 4 they have no physical states. However, in D dimensions the number of physical states of a rank-n tensor is

$$\displaystyle \begin{aligned} \frac{(D-2)!}{(D-2-n)!n!} \end{aligned} $$

(10.5)

and they are further reducible into the self- and anti-self-dual parts when n = D∕2. This means they can play a fundamental role to provide the necessary bosonic degrees of freedom needed to complete a supergravity multiplet.

2 Example: D = 11 Supergravity

In four and in any other dimension, the maximal number of real supercharges allowed in constructing a theory with fields of spin ≤ 2 is 32. A Majorana spinor in 11 dimensions has 2⁵ = 32 components, and hence 11D supergravity is the highest-dimensional supergravity model that can be constructed without introducing higher-spin fields. The 11D supersymmetry algebra naturally contains a central charge,

$$\displaystyle \begin{aligned} \{Q_\alpha, Q_\beta\} = (C \varGamma)_{\alpha \beta}^m P_m + (C \varGamma)_{\alpha \beta}^{mn} Z_{mn}, \end{aligned} $$

(10.6)

which is associated with the existence of membrane-like objects in the theory.

The massless 11D graviton has 44 degrees of freedom, while the gravitino has 128 physical states. This implies the need of additional higher-rank tensor fields. It is actually easy to see that a three-form, C _μνρ, with a gauge transformation

$$\displaystyle \begin{aligned} \delta C_{\mu\nu\rho} = 3\,\partial_{[\mu} \varLambda_{\nu \rho]} \end{aligned}$$

has exactly the 84 missing physical states needed to complete a supermultiplet

$$\displaystyle \begin{aligned} \{e_\mu^a, \, \psi_\mu, \, C_{\mu\nu \rho}\}. \end{aligned} $$

(10.7)

This theory clearly has no scalar potentials (there are no scalars at all), and it has a unique parameter given by the 11D gravitational constant, κ ₁₁.

The action of 11-dimensional supergravity was constructed first by Cremmer, Julia, and Scherk [1] and consists of very few terms:

$$\displaystyle \begin{aligned} \begin{array}{rcl} S &=& \displaystyle \frac{1}{2 \kappa_{11}^2} \int d^{11}x\, e \left[ R(\omega) - \overline{\psi}_\mu \varGamma^{\mu\nu \rho} D_\nu\left(\frac{\omega + \widehat{\omega}}{2}\right) \psi_\rho- \frac{1}{24} G_{\mu\nu\rho \sigma}G^{\mu\nu\rho \sigma}\right. \\ {} &\quad -&\displaystyle \frac{2\sqrt{2}}{(144)^2} \,\epsilon^{\mu_1 \ldots \mu_{11}} G_{\mu_1 \ldots \mu_4} G_{\mu_5 \ldots \mu_8} C_{\mu_9 \mu_{10} \mu_{11}} \\ {} & \quad -& \displaystyle \left.\frac{\sqrt{2}}{192}\left(\overline{\psi}_\mu \varGamma^{\mu\nu \rho \sigma \tau \eta} \psi_\nu + 12 \,\overline{\psi}^\rho \varGamma^{\sigma \tau}\psi^\eta\right)\left(2\, G_{\rho \sigma \tau \eta} - \frac 32 \sqrt{2} \kappa\, \overline{\psi}_{[\rho} \varGamma_{\sigma \tau} \psi_{\eta]}\right)\right], \end{array} \end{aligned} $$

(10.8)

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} G_{\mu\nu\rho\sigma} & =&\displaystyle 4\, \partial_{[\mu} C_{\nu \rho \sigma]}, \end{array} \end{aligned} $$

(10.9)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \omega_{\mu ab} & =&\displaystyle \omega_{\mu ab}(e) - \frac{1}{8}\bigg[ \overline \psi_\alpha \varGamma_{\mu ab}{}^{\alpha \beta}\psi_\beta +2 \overline{\psi}_\mu \varGamma_b \psi_a \\ {} & &\displaystyle -2 \overline{\psi}_a \varGamma_\mu \psi_b +2 \overline{\psi}_b \varGamma_a \psi_\mu\bigg], \end{array} \end{aligned} $$

(10.10)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \widehat{\omega}_{\mu ab}& =&\displaystyle \omega_{\mu ab} - \frac{1}{4} \left(\overline{\psi}_\mu \varGamma_b \psi_a - \overline{\psi}_a \varGamma_\mu \psi_b + \overline{\psi}_b \varGamma_a \psi_\mu\right). \end{array} \end{aligned} $$

(10.11)

This action obviously includes the kinetic terms for the graviton, the gravitino, and the three-form field C in the first line. The modified spin connection $\widehat \omega$ has been introduced to take into account the four-Fermi interactions between the gravitino fields. The second line is a special Chern–Simons-like term, which does not depend on the metric. The last line, finally, contains further interaction terms between the three-form and the gravitino. The invariance under supersymmetry follows from the application of

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta e_\mu^a & =&\displaystyle \frac 12\, \overline \epsilon \varGamma^a \psi_\mu, \end{array} \end{aligned} $$

(10.12)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta \psi_\mu & =&\displaystyle D_\mu(\widehat \omega)\epsilon + \frac{\sqrt{2}}{288} \left(\varGamma_\mu{}^{\nu \rho \sigma \tau}-8 \delta_\mu^\nu \varGamma^{\rho \sigma \tau}\right) \bigg( G_{\nu \rho \sigma \tau} \\ {} & &\displaystyle +\frac 32 \sqrt2 \, \overline \psi_{[\nu} \varGamma_{\rho \sigma} \psi_{\tau]} \bigg) \epsilon, \end{array} \end{aligned} $$

(10.13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta C_{\mu \nu \rho} & =&\displaystyle - \frac 34 \sqrt{2}\, \overline \epsilon \varGamma_{[\mu \nu} \psi_{\rho]}. \end{array} \end{aligned} $$

(10.14)

We don’t discuss here the proof of invariance under supersymmetry of this action nor the peculiar new features appearing, thanks to the presence of a three-form field. On the other hand, we are going to use this theory as a starting point for a qualitative discussion of the features of the models that can be obtained by dimensional reduction.

3 Dimensional Reduction and Ten-Dimensional Supergravities

The 11-dimensional supergravity action we just presented should describe the low-energy limit of M-theory, a supposedly consistent quantum theory of membranes in 11 dimensions, which arises as the strong coupling limit of ten-dimensional type IIA string theory. It is therefore conceivable that one could show a detailed relation between supergravity in 11 dimensions and the low-energy limits of ten-dimensional string theory models. Hence, we will now discuss the Kaluza–Klein reduction of 11-dimensional supergravity to ten dimensions and less.

A simple way to dimensionally reduce a theory is to consider a spacetime metric where one of the coordinates runs on a circle of fixed radius. In this way, the fluctuations of the fields over this coordinate will be constrained by the geometry and will result in effective masses and couplings for the fluctuations in the rest of the spacetime. For instance, if we reduce 11D supergravity over a circle and keep only the massless modes in the resulting effective 10D theory, we obtain a much richer spectrum than the one we discussed in the previous section. If we split the 11D coordinates, x ^M, into the non-compact ones, x ^μ, and the circle coordinate, θ ≡ x ¹⁰, we see that the reduction of the 11D metric gives rise to three different fields, with different transformation properties with respect to the residual 10D diffeomorphism invariance:

$$\displaystyle \begin{aligned} g_{MN} \to \left\{\begin{array}{lcl} g_{\mu \nu} && \mbox{ten-dimensional metric,}\\ {} g_{\mu 10} \sim A_\mu && \mbox{ten-dimensional vector,} \\ {} g_{10\, 10} \sim \phi && \mbox{ten-dimensional scalar.} \end{array}\right. \end{aligned} $$

(10.15)

From the 11-dimensional line element,

$$\displaystyle \begin{aligned} ds_{11}^2 = \mathrm{e}^{2 \phi(x)}(d \theta + dx^\mu A_\mu(x))^2 + ds_{10}^2, \end{aligned} $$

(10.16)

we see that the ten-dimensional photon field, A _μ, describes the fibration of the circle on the ten-dimensional base spacetime and the dilaton field, ϕ, is associated with the radius of the circle of the internal direction. In the same fashion, we can reduce the 11-dimensional rank-3 tensor,

$$\displaystyle \begin{aligned} C_{MNP} \to \left\{\begin{array}{lcl} C_{\mu \nu \rho} && \mbox{ten-dimensional three-form,}\\ {} C_{\mu \nu 10} \equiv B_{\mu\nu}&& \mbox{ten-dimensional two-form,}\\ {} {C_{\mu 10 \,10}} = 0&& \mbox{vanishing because of antisymmetry,} \end{array}\right. \end{aligned} $$

(10.17)

and the 11-dimensional gravitino,

$$\displaystyle \begin{aligned} \varPsi_M \to \begin{array}{lcl} \psi_\mu^+,\quad \psi_\mu^- &\qquad & \mbox{Two ten-dimensional gravitini}\\ {} \psi_{10}^+ \equiv \lambda^+, \ \psi_{10}^- \equiv \lambda^-&\qquad & \mbox{Two ten-dimensional spin }{1}/{2}\mbox{ fields}. \end{array} \end{aligned} $$

(10.18)

The plus and minus signs on the fermions refer to the chirality of the resulting ten-dimensional spinor fields. In fact, the 32-component Majorana spinor in 11D reduces to a 32-component Majorana spinor in 10D, which, however, can be further decomposed into two 10D Majorana–Weyl spinors of opposite chiralities, each having 16 independent real components. The same is true for the supercharges, $Q^{11} \to \left \{Q^+ , Q^-\right \}$, which split into two Majorana–Weyl representations of opposite chirality in ten dimensions. The resulting model is therefore a non-chiral ten-dimensional supergravity theory with two supersymmetry generators: type IIA supergravity .

The splitting of the 11-dimensional supercharge implies that one can construct other supergravity models with only one supercharge (type I models) or with both supercharges of the same chirality (type IIB supergravity). It is actually useful to summarize the resulting spectrum of type IIA supergravity as follows (the numbers indicate the independent on-shell degrees of freedom of each field):

$$\displaystyle \begin{aligned} \begin{array}{rcccccccccll} \{&\phi&,& \lambda^-&,& B_{\mu\nu}&,& \psi_\mu^+&,& g_{\mu\nu}&\} & \quad \mbox{Common Sector}\\ {} &1&&8&&28&&56&&35&&= 64_B+64_F\\ {} \{&\lambda^+&,&C_{\mu \nu \rho}&,&A_\mu&,& \psi_\mu^-& \}&&&\quad \mbox{IIA RR sector}\\ {} &8&&56&&8&&56&&&&=64_B+64_F. \end{array} {} \end{aligned} $$

(10.19)

The first line is a consistent 10D supermultiplet by itself when one restricts oneself to a single supersymmetry, while the second line collects the fields that complete the multiplet to the type IIA multiplet. The type IIA supergravity theory based on the above fields arises as the low energy limit of type IIA string theory. In the context of this string theory, the fields in the first line are referred to as the Neveu–Schwarz–Neveu–Schwarz (NSNS) sector , whereas the fields in the second line are called the Ramond–Ramond (RR) sector .

As the fields in the first line of (10.19) form a consistent supermultiplet of the 10D superalgebra with one supersymmetry generator, it is natural to suspect that there is also a consistent supergravity action with this reduced field content. This action indeed exists and is referred to as type I supergravity . Type I supergravity by itself, however, has quantum anomalies, and in order to cancel these anomalies, one has to add suitable super-Yang–Mills sectors based on 10D vector multiplets. We will come back to this point and its relation to string theory at the end of this section.

The Lagrangian of type IIA supergravity is obtained by dimensional reduction of the one of the 11-dimensional theory presented in the previous section [2]. For instance, from the kinetic term of the four-form, G, we obtain the kinetic terms of the ten-dimensional four- and three-forms, F ₄ = dC ₃ + A ∧ H ₃ and H ₃ = dB, respectively:

$$\displaystyle \begin{aligned} \frac{1}{48}\, G_{MNPQ} G^{MNPQ} \to \frac{1}{48}\, F_{\mu\nu\rho\sigma}F^{\mu\nu\rho\sigma} + \mathrm{e}^{-2 \phi}\frac{1}{12} H_{\mu \nu \rho}H^{\mu\nu\rho}. \end{aligned} $$

(10.20)

The 11D Ricci scalar reduces to the 10D Ricci scalar, plus kinetic terms for the additional 10D degrees of freedom obtained from the 11D metric, i.e., for the 10D vector and scalar fields. It is important to point out, however, that the presence of the metric determinant, together with the direct reduction of the curvature term, gives rise to dilaton factors in front of the various 10D terms, so that we cannot view the resulting action as a standard Einstein gravity theory unless we perform a rescaling of the metric. More concretely, the straightforward reduction of the Einstein–Hilbert term gives

$$\displaystyle \begin{aligned} e_{11} R_{11} \to e_{10} \,e^{-2\phi}\left(R_{10} -\frac 14e^{2\phi} F^2 + 4 (\partial \phi)^2\right) . {} \end{aligned} $$

(10.21)

This has an apparent wrong sign for the kinetic term of the dilaton and at the same time a non-trivial scalar factor in front of the ten-dimensional Einstein–Hilbert term. To go to the Einstein frame, one performs a rescaling

$$\displaystyle \begin{aligned} g_{\mu \nu} \to e^{\frac{\phi}{2}} g_{\mu \nu}, \end{aligned} $$

(10.22)

so that (10.21) gets mapped to the more standard form

$$\displaystyle \begin{aligned} e_{10}\,\left( R_{10} - \frac{e^{\frac 32\phi}}{4}\, F^2 - \frac{1}{2} (\partial \phi)^2\right). \end{aligned} $$

(10.23)

Once we reduced the theory to ten dimensions, we see that we now have two parameters: k ₁₁ and the vev of e ^ϕ. The field ϕ is the first modulus we meet, i.e., a massless scalar field whose expectation value is related to some geometrical property of the internal spacetime and which affects the coupling constants of the effective theory. In fact, the scalar potential of type IIA supergravity is trivial,²

$$\displaystyle \begin{aligned} V(\phi) \equiv 0. \end{aligned} $$

(10.24)

The full bosonic sector of the type IIA supergravity action in the Einstein frame reads:

$$\displaystyle \begin{aligned} S_{IIA} = & \frac{1}{2 \kappa_{10}^2} \int d^{10}x\, e\,\left( R - \frac 12\, \partial_\mu \phi \partial^\mu \phi-\frac{1}{12}\, e^{-\phi}\, H_{\mu\nu\rho}H^{\mu\nu\rho} \right) \\ {} &-\frac{1}{2 \kappa_{10}^2} \int d^{10}x \, e\, \left(\frac{1}{48}\, e^{\frac{\phi}{2}} G_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}- \frac 14\, \, e^{\frac 32\phi} \,F_{\mu\nu}F^{\mu\nu} \right)\\ {} & - \frac{1}{4 \kappa_{10}^2} \int B_2 \wedge dC_3 \wedge dC_3\,, \end{aligned} $$

(10.25)

where

$$\displaystyle \begin{aligned} G_4 = d C_3 - A_1 \wedge H_3, \quad H_3 = dB_2, \quad F_2 = dA_1. \end{aligned} $$

(10.26)

In our simple reduction, the expectation value of ϕ is related to the radius of the S ¹ we used to go from eleven to ten dimensions. If we interpret the IIA supergravity as a low energy limit of type IIA string theory, we can identify the vev of the dilaton with the string coupling constant as 〈e ^ϕ〉∼ g _s. It is then clear why taking a strong coupling limit of type IIA strings, g _s →∞, implies to go from type IIA to M-theory. In our supergravity setup, which, however, cannot be fully trusted when one goes beyond the classical regime, this corresponds to send the radius of the S ¹ to infinity and therefore to go back to an 11-dimensional background.

In order to complete this quick survey of ten-dimensional supergravity models, we just mention a couple of things. Minimal supergravity in ten dimensions (Type I) can be coupled to matter. In fact ten-dimensional ($\mathcal {N}=1$) supersymmetry allows for another multiplet: a super-Maxwell multiplet, containing a vector and a spin 1∕2 field:

$$\displaystyle \begin{aligned} \begin{array}{ccccc} \{&A_\mu&,&\lambda^\pm&\}. \\ &8&&8 \end{array} \end{aligned} $$

(10.27)

While at the classical supergravity level we could consider coupling an arbitrary number of such multiplets to the supergravity action, anomaly cancellation restricts the allowed possibilities to well-defined gauge groups and therefore to a well-defined and fixed number of vector multiplets. This is also reflected in the corresponding string theories with 10D, $\mathcal {N}=1$ supersymmetry, namely, the type I string theory and the two heterotic string theories. Their low energy limits consist of type I supergravity coupled to $\mathcal {N}=1$ super-Yang–Mills theory with gauge group SO(32) for the type I string theory and with gauge group SO(32) or E₈ ×E₈ for the two heterotic string theories.³

As we discussed before, we could also consider an $\mathcal {N}=2$ theory with both supercharges of the same chirality. This is called type IIB supergravity [4, 5], and its field content can be obtained by substituting the RR sector in Eq. (10.19) with the following:

$$\displaystyle \begin{aligned} \begin{array}{rcccccccccll} \{&C_0&,&\lambda^-{}^{(2)}&,&C_{\mu \nu}&,&C_{\mu \nu \rho \sigma}&,& \psi_\mu^+{}^{(2)}& \}&\quad \mbox{IIB RR sector}\\ {} &1&&8&&28&&35&&56&&=64_B+64_F . \end{array} \end{aligned} $$

(10.28)

We point out that the rank-4 tensor field appearing in the spectrum has only 35 degrees of freedom on shell, because its field strength is a five-form in ten dimensions, and hence one can impose a self-duality constraint on it,

$$\displaystyle \begin{aligned} F_5 = \star F_5\,, \quad F_{\mu_1\ldots \mu_5} = \frac{1}{5!}\epsilon_{\mu_1\ldots \mu_{10}} F^{\mu_6\ldots\mu_{10}}. \end{aligned} $$

(10.29)

This constraint creates some problems when one wants to construct a Lorentz covariant action because, as we can easily see, the standard kinetic term would be identically zero

$$\displaystyle \begin{aligned} F_5 \wedge \star F_5 = F_5 \wedge F_5 = - F_5 \wedge F_5 = 0, \end{aligned} $$

(10.30)

where we first used the constraint and then swapped the order of the two five-forms. A covariant and supersymmetric Lagrangian with a single scalar auxiliary field, which is a pure gauge of a new symmetry of the action and a singlet under supersymmetry, is provided in [6]. As a mnemonic tool, one can use the following bosonic Lagrangian, where the self-duality equation is imposed on the resulting equations of motion,

$$\displaystyle \begin{aligned} S_{IIA} = & \frac{1}{2 \kappa_{10}^2} \int d^{10}x\, e\,\left( R - \frac 12\, \partial_\mu \phi \partial^\mu \phi -\frac{1}{12}\, e^{-\phi}\, H_{\mu\nu\rho}H^{\mu\nu\rho} \right)\\ {} &-\frac{1}{2 \kappa_{10}^2} \int d^{10}x \, e\, \left(\frac 12\, e^{2 \phi} \partial_\mu C_0 \partial^\mu C_0 + \frac{1}{2\cdot 5!}\, \, \,F_{\mu_1 \ldots \mu_5}F^{\mu_1 \ldots \mu_5} \right)\\ {} & -\frac{1}{2 \kappa_{10}^2} \int d^{10}x \, e\, \frac{1}{12}\, e^{\phi} \left(F_{\mu\nu\rho} - C_0 H_{\mu\nu\rho}\right)\left(F^{\mu\nu\rho}- C_0 H^{\mu\nu\rho}\right) \\ {} & - \frac{1}{4 \kappa_{10}^2} \int C_4 \wedge H_3 \wedge F_3\,, \end{aligned} $$

(10.31)

where

$$\displaystyle \begin{aligned} H = d B_2, \quad F_3 = d C_2\, \quad F_5 = d C_4 - \frac 12\, C_2 \wedge H_3 + \frac 12\, B_2 \wedge F_3. \end{aligned} $$

(10.32)

This theory arises as the low energy limit of type IIB string theory.

4 Dimensional Reduction and the Origin of Gauged Supergravities

In the same way as type IIA supergravity arises by reduction of 11-dimensional supergravity on a circle, one could try to produce many different models in four dimensions by compactifying ten- or eleven-dimensional supergravities on six- or seven-dimensional spaces with various geometries. Obviously, the matter content and the number of supersymmetries will depend on the vacuum expectation value of the metric on the internal space. However, following what happened in the IIA case, these reductions (when they preserve some supersymmetry) will usually generate a number of massless scalar fields associated with the shape and volume of the internal space, just like the IIA dilaton was associated with the volume (the radius) of the circle on which one compactifies M-theory. As discussed in detail in Sect. 7.4.1, these moduli fields are a serious phenomenological problem of the resulting effective theories, e.g., because they produce long range fifth forces, modifying the behavior of Newtonian gravity in an unacceptable way. A simple and efficient way out is given by assuming that also the other higher-rank tensor fields appearing in the ten- and eleven-dimensional theories acquire a non-trivial expectation value (the fluxes). As we will sketch in the following, this results in the deformation of the four-dimensional supergravity models by the gauging process and hence leads to gauged supergravity models, which then do have a scalar potential that can make (at least some of) the moduli sufficiently massive. The remarkable aspect of this type of reduction is that the fluxes can be treated perturbatively and produce closed computable expressions for the lower dimensional couplings and potentials.

We will give here an overview of the main features of flux compactifications leading to gauged supergravities. For this reason we will focus on the simple case of 11-dimensional supergravity reduced on the seven-dimensional torus, ${\mathbb T}^7$, and discuss only its bosonic sector. The standard reduction, where one truncates the spectrum to the massless modes, leads to $\mathcal {N} = 8$ ungauged supergravity in four dimensions. In detail, the reduction of the metric and three-form tensor field produces the following massless spectrum:

g μν	g μI	g IJ
1 graviton		28
C μνρ	C μIJ	C μνI, C IJK
0 dof	7+21 vectors	7   35
		= 63 scalars + 7 tensors

As we already discussed previously, massless tensors can be dualized to massless scalar fields, and hence, after such duality, one gets a total of 70 scalar fields, which is the standard scalar content of the $\mathcal {N} = 8$ supergravity multiplet.

An extremely important aspect of 4D supergravity is the U-duality group emerging as a generalization of the standard electric–magnetic duality (see Chap. 9). For $\mathcal {N} = 8$ supergravity, this is E ₇₍₇₎ ⊂ Sp(56,${\mathbb R}$) and implies that there is an underlying symplectic action on 56 vector fields, 28 of which may appear simultaneously in the action. Using the appropriate language, we can then consider the vectors coming from the ${\mathbb T}^7$ compactification as the 28 electric ones: $A_{\mu }^{\varLambda } = \{g_\mu ^I,C_{\mu IJ}\}$. However, one should also be able to identify the higher dimensional origin of the 28 dual vector fields, A _μΛ. Twenty-one of them can be readily identified by the reduction of the dual form of G ₄ = dC ₃. This dual form is schematically obtained as G ₇ ∼ ⋆G ₄, so that its Bianchi identity reproduces the equation of motion for G ₄: dG ₇ = G ₄ ∧ G ₄. The seven-form field strength then is the curvature of a six-form potential that, upon reduction, produces

$$\displaystyle \begin{aligned} C_{\mu IJKLM} = \epsilon_{IJKLMNP} \,A_\mu^{NP}. \end{aligned} $$

(10.33)

We also note that the seven scalars dual to the tensor fields above are also obtained from the same potential as C _IJKLMN. The remaining seven dual vector fields should have an origin as dual metric fields, $\tilde g_{\mu I}$, but it is not obvious how to achieve this yet, though we expect them to correspond to non-geometric deformations. Anyway, at the level of the 4D action, as we already discussed in Chap. 8, both descriptions are equally valid, and as long as we don’t have gaugings we can (almost) freely dualize the curvatures and the vectors.

For what concerns the scalar fields, we get a σ-model, but no scalar potential, as expected for a standard Einstein–Maxwell extended supergravity.

The duality group is broken, and the 4D model gets deformed to a gauged supergravity by introduction of “fluxes” for the four-form, i.e., an expectation value for this field on the internal space:

$$\displaystyle \begin{aligned} \mathcal{G}_{IJKL}\equiv \langle G_{IJKL} \rangle \neq 0. {} \end{aligned} $$

(10.34)

We now discuss how this is achieved, briefly noting that a similar situation occurs when introducing a flux for the dual seven-form field, or a combination of the two.

A simple sketch of the reduction of the kinetic term of the four-form explains how both the gauging and the scalar potential are generated. Without fluxes, the reduction of the four-form kinetic term produces the expected kinetic terms for the scalars and vectors:

(10.35)

In the presence of a non-trivial flux, a new term emerges at the quadratic level, containing only the scalar fields coming from the metric. This is an obvious scalar potential term, and it is of the second order in the coupling constants dictated by the fluxes $\mathcal {G}$:

$$\displaystyle \begin{aligned} V(g_{IJ}) = g^{IA} g^{JB} g^{KC} g^{LD}\, \mathcal{G}_{IJKL} \mathcal{G}_{ABCD}. \end{aligned} $$

(10.36)

Another important coupling that emerges is linear in the flux,

$$\displaystyle \begin{aligned} \partial_{[\mu}C_{\nu]IJ}\, g^{\mu A} g^{\nu B} g^{IC} g^{JD} \, \mathcal{G}_{ABCD}, \end{aligned} $$

(10.37)

which reconstructs the $\mathcal {O}(\mathcal {G})$ couplings giving origin to non-Abelian covariant field strengths

$$\displaystyle \begin{aligned} \partial_{[\mu}C_{\nu]IJ} + g_\mu^A g_\nu^B \,\mathcal{G}_{ABIJ}. \end{aligned} $$

(10.38)

From this same expression, it is also clear that $\mathcal {G}_{IJKL}$ play the role of structure constants. The gauging of the theory becomes even more evident if we look at the kinetic terms of the scalar fields. Also these kinetic terms get modified to new expressions involving $\mathcal {O}(\mathcal {G})$ couplings reconstructing covariant derivatives:

$$\displaystyle \begin{aligned} \partial_\mu C_{IJK} + \mathcal{G}_{IJKL}\, g_\mu^L = \widehat{\partial}_\mu C_{IJK}. \end{aligned} $$

(10.39)

We therefore see that fluxes not only define the gauge couplings and the scalar potential, but they also tell us the form of the embedding tensor, because they tell us which vectors participate in the gauging and specify the couplings between the scalar and vector fields defining the covariant derivatives.

Since the fluxes now play the role of gauge structure constants, one could expect that they obey standard Jacobi identities. Although it may not be evident from the example above, it can be shown that there is a one- to-one correspondence between the consistency conditions on the gauge structure constants in four dimensions and the Bianchi identities of the form fluxes in ten or 11 dimensions. There is, however, an important subtlety that we want to emphasize here. When dealing with flux compactifications (and especially in the case of non-trivial geometric fluxes, i.e., globally defined torsion terms), the tensor fields appearing in the geometric reduction transform under gauge transformations of the vector fields and vice versa. This implies that generically the standard Jacobi identities coming from the gauge algebra of the vector fields do not close. In fact, the gauge algebra is now really a free differential algebra involving also the tensor fields, and therefore it is only the closure of this structure that imposes all the necessary consistency conditions on the structure constants (also to reproduce the higher dimensional Bianchi identities).

We conclude with some comments on the action of the duality group in relation to the situation described in the previous paragraph. Although the natural setup coming from a straightforward reduction of a flux background may result in a free differential algebra that includes tensor fields, one may always use the duality group to rotate the vector field basis so that no tensor fields are present in the final Lagrangian. Obviously, in order to be consistent, the embedding of the gauge group in the duality group will also be rotated accordingly, and the interpretation of the resulting algebra in terms of the original theory may not be straightforward anymore. Actually, this is a rough explanation of how non-geometric fluxes arise and why we should expect them if we believe that the U-duality group of the effective theory survives as a symmetry of the fully fledged higher dimensional fundamental theory (maybe with some restrictions).

5 Example: D = 5

As a further illustration of the new structures emerging in higher dimensional supergravity, we consider the case of five dimensions, where supergravity theories have numerous applications, e.g., in the context of the AdS/CFT correspondence, the study of black holes and domain walls, or phenomenological scenarios such as the Randall–Sundrum or Hořava–Witten-type models. As we will see, the form of the R-symmetry group provides useful constraints on the scalar manifold geometries, which, for $\mathcal {N}\geq 4$, even fixes the target spaces completely once type and number of multiplets in the theory are specified.

In five dimensions, one cannot impose Majorana or Weyl conditions,⁴ and the possible numbers of real supercharges are 8, 16, 24, 32. One might call this $\mathcal {N}=1,2,3,4$ supersymmetry, but in analogy with the counting in 4D, one usually refers to these possible 5D supersymmetries as $\mathcal {N}=2,4,6,8$, respectively, as we will also do in these lecture notes. As explained in Appendix 10.A, in 5D, the parameter 𝜖 defined in (10.79) is 𝜖 = −1, so that one may impose a symplectic Majorana condition for the supersymmetry generators $(i=1,\ldots ,\mathcal {N})$,

$$\displaystyle \begin{aligned} (Q_{i})^{\ast}=\varOmega_{ij}BQ_j. \end{aligned} $$

(10.40)

This has the advantage of making the action of the R-symmetry group manifest. More concretely, the linear rotations of the Q _i that preserve the above symplectic Majorana condition as well as the 5D supersymmetry algebra,

$$\displaystyle \begin{aligned} \{ Q_{i}, Q_{j}^{T}\}= \varOmega_{ij}(C\varGamma^{a})P_{a}, \end{aligned} $$

(10.41)

form the R-symmetry groups $USp(\mathcal {N})\equiv U({\mathcal {N}})\cap Sp(\mathcal {N},\mathbb {R})$.

5.1 $\mathcal {N}=2$ in 5D

For $\mathcal {N}=2$ ungauged supergravity in 5D, there are three important supermultiplets:

• Supergravity multiplet: $(e_{\mu }^{a},\psi _{\mu }^{i},A_{\mu })$ (i = 1, 2) Apart from the fünfbein, $e_{\mu }^{a}$, it contains two gravitini, $\psi ^{i}_{\mu }$, and a vector field, A _μ, the “graviphoton.”

• Vector multiplet: (A _μ, λ ⁱ, φ)The superpartners of the vector field, A _μ, are two gaugini, λ ⁱ, and one real scalar field, φ.

• Hypermultiplet: (ζ ^{1, 2}, q ^{1, 2, 3, 4})Here, ζ ^{1, 2} are two spin-1/2 fermions (“hyperini”), and the q ^{1, 2, 3, 4} denote four real scalar fields.

In the above, the index i of the gravitini and the gaugini is a doublet index of the R-symmetry group USp(2)_R≅SU(2)_R. The hyperini, by contrast, are inert under SU(2)_R, which is the reason that we do not use i to label these two fermions. The scalar φ is likewise SU(2)_R-inert, whereas the hyperscalars q ^{1, 2, 3, 4} form two doublets under SU(2)_R. All spinors are symplectic Majorana spinors.

In order to write down the general Lagrangian for $\mathcal {N}=2$ supergravity coupled to n _V vector multiplets and n _H hypermultiplets, it is useful to group these fields as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \,& &\displaystyle (e_{\mu}^{a},\psi_{\mu}^{i},A_{\mu})\,\,\oplus\,\, n_{V}\times (A_{\mu},\lambda^{i},\varphi)\,\,\oplus\,\, n_{H}\times (\zeta^{1,2},q^{1,2,3,4})\\ & &\displaystyle \qquad \qquad =(e_{\mu}^{a},\,\, \psi^{i}_{\mu},\,\, A_{\mu}^{I},\,\, \lambda^{i},\,\, \zeta^{A},\,\, \varphi^{x},\,\, q^{u}) \end{array} \end{aligned} $$

(10.42)

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle I=0,1,\ldots,n_V \end{array} \end{aligned} $$

(10.43)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle x=1,\ldots,n_V \end{array} \end{aligned} $$

(10.44)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle u=1,\ldots,4n_H \end{array} \end{aligned} $$

(10.45)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle A=1,\ldots,2n_H. \end{array} \end{aligned} $$

(10.46)

Here we have combined the graviphoton and the n _V vector fields of the vector multiplets into an (n _V + 1)-plet of vectors, $A_{\mu }^{I}$. The indices x and u are curved indices on the scalar manifolds, $\mathcal {M}_{V}$ and $\mathcal {M}_{H}$, of the vector scalars and the hyperscalars, respectively. The gaugini, λ ^ix, transform as tangent vectors of $\mathcal {M}_{V}$, and one may use curved indices, x, to label them, just as we did, or, as one often also finds in the literature, one could use a flat tangent space index, a = 1, …, n _V, instead of the curved tangent space index x. Both notations can be easily converted into one another by contraction with vielbein on $\mathcal {M}_{V}$. As the holonomy group of $\mathcal {M}_V$ has no particular restrictions imposed by the supersymmetry algebra (see below), the use of flat vs. curved indices x and a has no clear advantage. This is different for the hyperini, as we will explain below.

In terms of the above fields, the bosonic Lagrangian of $\mathcal {N}=2$ matter-coupled ungauged supergravity in 5D can be written as follows [7, 8]:

$$\displaystyle \begin{aligned} \begin{array}{rcl} e^{-1}\mathcal{L}_{\mathrm{bos}}& =&\displaystyle \frac{1}{2}R-\frac{1}{4}\tilde{a}_{IJ}(\varphi)F_{\mu\nu}^{I}F^{\mu\nu J}-\frac{1}{2}g_{xy}(\varphi)\partial_{\mu}\varphi^{x}\partial^{\mu}\varphi^{y} \\ & &\displaystyle -\frac{1}{2}h_{uv}(q)\partial_{\mu}q^{u}\partial^{\mu}q^v + \frac{1}{6\sqrt{6}}C_{IJK}\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}.\qquad \end{array} \end{aligned} $$

(10.47)

In this expression, C _IJK is a constant, completely symmetric tensor. As was shown by Sierra [8], the two sigma models associated with the hyperscalars and the vector scalars do not mix, i.e., the scalar manifold metric is block diagonal in these two sectors and the total scalar manifold decomposes into a direct product, $\mathcal {M}_V\times \mathcal {M}_{H}$, just as in four dimensions.

5.1.1 The Geometry of $\mathcal {M}_{V}$

The scalars, φ ^x, of the vector multiplets are inert under the R-symmetry group SU(2)_R. The holonomy group of $\mathcal {M}_{V}$ therefore does not receive any constraints from the R-symmetry group, as we already mentioned above. However, as they are connected by supersymmetry to the vector fields (or rather to n _V of them), the scalar manifold $\mathcal {M}_{V}$ inherits part of the vector field structure. In fact, one finds that $\mathcal {M}_{V}$ is completely determined by the constants C _IJK that define the Chern–Simons term in the action. More precisely, the C _IJK define a cubic polynomial [7],

$$\displaystyle \begin{aligned} N(X^{I})\equiv C_{IJK}X^I X^J X^K \end{aligned} $$

(10.48)

on an auxiliary space, $\mathbb {R}^{n_V+1}$, spanned by real coordinates X ^I (I = 0, 1, …, n _V). On this auxiliary space, N then defines a (not necessarily positive definite) metric,

$$\displaystyle \begin{aligned} a_{IJ}(X)\equiv-\frac{1}{3}\frac{\partial}{\partial X^I}\frac{\partial}{\partial X^J}\log N(X). \end{aligned} $$

(10.49)

The scalar manifold $\mathcal {M}_{V}$ is then given as a cubic hypersurface in the auxiliary $\mathbb {R}^{n_{V}+1}$:

$$\displaystyle \begin{aligned} \mathcal{M}_{V}=\{X^{I}\in \mathbb{R}^{n_{V}+1}|N(X)=1\}. {} \end{aligned} $$

(10.50)

$\mathcal {M}_{V}$ can be parameterized by n _V real coordinates, which are identified with the physical scalar fields, φ ^x. The metric, g _xy, on $\mathcal {M}_{V}$ is given by the pull-back of the auxiliary metric a _IJ, and the “gauge kinetic function” $\tilde {a}_{IJ}$ is the restriction of a _IJ to the hypersurface $\mathcal {M}_{V}$:

$$\displaystyle \begin{aligned} \begin{array}{rcl} g_{xy}(\varphi)& =&\displaystyle \frac{3}{2}\, \frac{\partial X^I}{\partial \varphi^{x}} \frac{\partial X^J}{\partial \varphi^{y}}\, a_{IJ }\Big|{}_{N(X)=1} \end{array} \end{aligned} $$

(10.51)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{a}_{IJ}(\varphi)& =&\displaystyle a_{IJ}\Big|{}_{N(X)=1}. \end{array} \end{aligned} $$

(10.52)

The true scalar manifold is then actually the subspace of (10.50) for which g _xy(φ) and $\tilde {a}_{IJ}(\varphi )$ are positive definite. The above-described geometry of the scalar manifold of the vector multiplet scalars is called very special (real) geometry. Upon dimensional reduction to four dimensions, $\mathcal {M}_{V}$ becomes a special Kähler manifold of restricted type, namely, one for which the holomorphic prepotential is purely cubic. Special Kähler manifolds that arise in this way from 5D are called very special Kähler geometries, and the corresponding map is called the R-map [7].

5.1.2 The Geometry of $\mathcal {M}_{H}$

As the hyperscalars transform non-trivially under the R-symmetry group SU(2)_R, we expect the holonomy group of $\mathcal {M}_{H}$ to respect this structure and hence to contain SU(2) as a factor. The largest group G such that SU(2) × G is still a subgroup of (the maximal holonomy group) O(4n _H) is Usp(2n _H). The holonomy group of $\mathcal {M}_{H}$ should thus be contained in SU(2)× USp(2n _H), with the SU(2) part being non-trivial. Manifolds of this type are called quaternionic Kähler, and we discussed them in detail already in Chap. 8, because the hypermultiplet geometry in 5D is really identical to the scalar field geometry of 4D hypermultiplets. The restricted holonomy group also means that the tangent space group can be restricted to SU(2) × USp(2n _H). Just as in four dimensions, this allows a natural split of the flat tangent space index of $\mathcal {M}_{H}$ into an SU(2) index i = 1, 2 and an USp(2n _H) index A = 1, …, 2n _H. These indices are to be identified with the R-symmetry group index i and the index A of the hyperini ζ ^A.

5.2 $\mathcal {N}=4$ in 5D

For 5D, $\mathcal {N}=4$ supersymmetry, the R-symmetry group is USp(4)_R. USp(4)_R is a double cover of SO(5), which, by abuse of notation, we will also call SO(5)_R. The two relevant multiplets in ungauged supergravity are [9]:

• Supergravity multiplet: This multiplet consists of the graviton, four gravitini, six vector fields, four spin-1/2 fields, and one real scalar field. This scalar is necessarily SO(5)_R-inert and parameterizes the real line:

  $$\displaystyle \begin{aligned} \mathcal{M}_{SG}\cong \mathbb{R}\cong \mathrm{SO}(1,1) \end{aligned} $$

  (10.53)

• Vector multiplet: This multiplet contains one vector, four spin-1/2 fields, and five real scalars in the 5 of SO(5)_R. The holonomy group of the scalar manifold of n _V such vector multiplets should thus contain SO(5) as a factor. The largest remaining group factor that still allows the embedding into the maximal holonomy group O(5n _V) is SO(n _V), i.e., $\mbox{Hol}(\mathcal {M}_{V})\subset$ SO(5) × SO(n _V) with the SO(5) part being non-trivial. According to Berger’s classification [10], the only Riemannian manifold of dimension 5n _V with this property is

  $$\displaystyle \begin{aligned} \mathcal{M}_{V}=\frac{{\mathrm{SO}}(5,n_{V})}{{\mathrm{SO}}(5)\times {\mathrm{SO}}(n_{V})}. \end{aligned} $$

  (10.54)

5.3 $\mathcal {N}=6$ in 5D

In this case the R-symmetry group is USp(6)_R. The only multiplet relevant for supergravity is the supergravity multiplet. The scalars in this multiplet transform non-trivially under the R-symmetry group, and the holonomy group of the scalar manifold should contain USp(6) as a factor, which, together with the dimension fixes it to be

$$\displaystyle \begin{aligned} \mathcal{M}_{SG}\cong \frac{\mathrm{SU}^{\ast}(6)}{\mathrm{USp}(6)}, \end{aligned} $$

(10.55)

where SU^∗(6) is a particular real form of SU(6).

5.4 $\mathcal {N}=8$ in 5D

In the maximally supersymmetric case, the R-symmetry group is USp(8)_R, and the supergravity multiplet contains the graviton, 8 gravitini, 27 vector fields, 48 spin-1/2 fields, and 42 real scalar fields. The latter transform non-trivially under the R-symmetry group, and we expect the holonomy group to contain a USp(8)-factor. The only 42-dimensional space with this property is

$$\displaystyle \begin{aligned} \mathcal{M}_{SG}\cong \frac{\mathrm{E}_{6(6)}}{\mathrm{USp}(8)}, \end{aligned} $$

(10.56)

where E₆₍₆₎ denotes the real form of E₆ for which the difference of compact and non-compact generators is 6.

5.5 Gaugings and Tensor Fields

As we discussed in Sect. 8.1, in four spacetime dimensions, a massless vector field without gauge interactions can be equivalently described by its magnetic dual vector field. In the presence of gauge interactions, on the other hand, this electric–magnetic duality is broken, and one has to make sure that the gauging is performed in a suitable duality frame.⁵

In five dimensions, there is no duality between electric and magnetic vector fields, but if there are no gauge interactions, there is an analogous Poincaré duality between vector fields, A = A _μ dx ^μ, and two-form fields, $B=\frac {1}{2}B_{\mu \nu }dx^\mu \wedge dx^\nu$. In the simplest version, this is just the statement that the 5D source-free Maxwell equations,

$$\displaystyle \begin{aligned} d \star F =0, \qquad dF=0 \end{aligned} $$

(10.57)

for the two-form field strength F = dA read

$$\displaystyle \begin{aligned} dH=0, \qquad d \star H=0, \end{aligned} $$

(10.58)

when expressed in terms the dual three-form field strength, H := ⋆F, which imply H = dB and d ⋆ dB = 0, i.e., the field equation of a massless two-form field, B.

If one now instead considers 5D theories with gauge interactions, the above duality between 5D vector and tensor fields no longer holds, and one would naively expect that a consistent gauging would require working exclusively with vector fields. In many cases, this is also what happens, but there are also important situations where some of the vector fields have to be converted to tensor fields in a specific way in order to perform the gauging.

To understand this, let us assume we start from an ungauged 5D supergravity theory in the standard form, as described in the above subsections, where all potential tensor fields are dualized to vector fields. Suppose further the theory has n vector fields and a global symmetry group, G _global,⁶ such that the n vector fields transform in an n-dimensional representation of that global symmetry group (this representation may be reducible or irreducible, depending on the theory).

If G _global has a subgroup, G, such that this n-dimensional representation of G _global becomes the adjoint representation of G,

$$\displaystyle \begin{aligned} \mathbf{n}(G_{\mathrm{global}})\rightarrow \mbox{adj}(G), \end{aligned} $$

(10.59)

one can replace the Abelian field strengths, F ^I (I = 1, …, n), by the corresponding non-Abelian field strengths, $\mathcal {F}^I$, and the partial derivatives, ∂ _μ, of charged matter fields by gauge covariant derivatives, $\widehat {\partial }_{\mu }$, and gauge the group G. Just as in 4D, this covariantization will break supersymmetry, which, however, can be restored by introducing suitable Yukawa interactions and scalar potentials into the Lagrangian as well as fermionic shifts to the supersymmetry transformation laws.

The above also holds true in the more general situation when the n-dimensional representation of G _global decomposes into the adjoint of G plus singlets of G,

$$\displaystyle \begin{aligned} \mathbf{n}(G_{\mathrm{global}})\rightarrow \mbox{adj}(G)\oplus \mbox{singlets}(G). \end{aligned} $$

(10.60)

If G has no Abelian factor, the singlet fields will just remain Abelian vector fields, and they will have no gauge couplings to the matter fields, i.e., they will be “spectator vector fields” with respect to the gauging. In case G has an Abelian factor, on the other hand, the singlet vector fields will still remain Abelian, but they might have minimal couplings to the matter fields, i.e., they might contribute to the Abelian part of the gauge group G.

As a simple example, consider a theory with four vector fields in the fundamental representation of a global symmetry group SO(4). With respect to the obvious subgroup SO(3), the 4 of SO(4) decomposes into 3 ⊕1. As the 3 is the adjoint of SO(3), one can use the three vector fields in the 3 to gauge SO(3), with the SO(3) singlet vector field remaining a spectator vector field. Gaugings of this standard type were investigated in 5D, $\mathcal {N}=2$ supergravity in [11] and in 5D, $\mathcal {N}=4$ supergravity in [9].

A more problematic situation arises, however, if the decomposition of the n-dimensional representation of G _global with respect to the subgroup G also contains non-singlets of G,

$$\displaystyle \begin{aligned} \mathbf{n}(G_{\mathrm{global}}) \rightarrow \mbox{adj}(G)\oplus \mbox{singlets}(G)\oplus \mbox{non-singlets}(G). \end{aligned} $$

(10.61)

In this case, the gauging of G cannot be performed in the usual way, because vector fields can only couple consistently to other vector fields if they sit in the adjoint of the gauge group.

Historically, the first example of this situation occurred in 5D, $\mathcal {N}=8$ supergravity in the 1980s [12,13,14]. In the ungauged version, this theory has 27 vector fields transforming in the 27-dimensional irreducible representation of the global symmetry group, G _global = E₆₍₆₎. E₆₍₆₎ is the maximally non-compact real form of the exceptional group E₆ and forms the isometry group of the scalar manifold $\mathcal {M}_{\mathrm {scalar}}=\mathrm {E}_{6(6)}/\mathrm {USp}(8)$ of this theory.

A particularly interesting subgroup of the global symmetry group E₆₍₆₎ is the subgroup SO(6), under which the 27 of E₆₍₆₎ transforms as

$$\displaystyle \begin{aligned} \mathbf{27} \rightarrow \mathbf{15}\oplus \mathbf{6}\oplus\mathbf{6} . \end{aligned} $$

(10.62)

Here, the 15 is the adjoint representation of SO(6), whereas the 6 denotes the fundamental representation of SO(6), which is clearly a non-singlet representation. Without the 6 ⊕6, one could gauge SO(6) with the 15 vector fields in the adjoint, but due to the presence of the non-singlets, this is not possible in the standard way.

This by itself would not be a big deal, as not every subgroup of a global symmetry group needs to be gaugeable, but in this particular case, there were very strong arguments in favor of the existence of a gauging with the gauge group SO(6). These arguments have to do with the compactification of type IIB supergravity on the maximally supersymmetric background solution AdS ₅ × S ⁵, which was expected to admit a consistent truncation to the lowest lying Kaluza–Klein modes that should be identical to 5D, $\mathcal {N}=8$ supergravity with gauge group SO(6) (the isometry group of the five-sphere).

The resolution of this problem came from a closer inspection of the Kaluza–Klein spectrum of this compactification [15, 16], which, apart from the 15 vector fields, also revealed the presence of 12 tensor fields, which are not equivalent to vector fields in an AdS ₅ background (they transform in a different representation of the AdS ₅ isometry group as the vector fields).

This suggests that a consistent gauging of SO(6) might require treating the fields in the 6 ⊕6 as antisymmetric tensor fields and not as vector fields. In fact, as tensor fields, they could be treated as a special type of matter fields, so that their derivatives could be covariantized by introducing minimal couplings to the 15 gauge fields using the six-dimensional representation matrices of SO(6) (see Eq. (10.64)). This approach turned out to be correct and led to the successful construction [12,13,14] of 5D, $\mathcal {N}=8$ gauged supergravity with gauge group SO(6), which, many years later, also played a central role in the AdS/CFT correspondence.

The necessity of converting non-singlet vector fields to tensor fields in order to perform certain gaugings also was found in 5D, $\mathcal {N}=4$ [17,18,19] and $\mathcal {N}=2$ theories [20,21,22]. In fact, the $\mathcal {N}=2$ cases allow one to isolate the contribution to the scalar potential that arises due to the presence of charged tensor fields from those contributions that come from the gauging of the R-symmetry group or the presence of non-Abelian gauge fields [20], or from charged hypermultiplets [21]. Interestingly, one finds that this scalar potential contribution is positive semi-definite, i.e., it cannot by itself lead to AdS vacua [20]. In fact, the $\mathcal {N}=2$ theories with charged tensor fields were the first extended supergravity theories in which a meta-stable de Sitter vacuum could be constructed [23, 24]. This implies that the presence of tensor fields in 5D supergravity theories with non-singlet representations outside the adjoint representation of the gauge group is not a consequence of an AdS vacuum structure (although it is consistent with it), but follows from more general considerations.

Another noteworthy feature of the 5D gaugings with charged tensor fields is that the corresponding Lagrangians contain a first-order kinetic term for the tensor fields of the schematic form

$$\displaystyle \begin{aligned} \frac{1}{g}\varOmega_{MN}\varepsilon^{\mu\nu\rho\sigma\kappa}B_{\mu\nu}^M\widehat{\partial}_{\rho}B_{\sigma\kappa}^{N}, \end{aligned} $$

(10.63)

where g denotes the gauge coupling, Ω _MN = −Ω _NM is an antisymmetric constant tensor, and M, N, … = 1, …, m label the m tensor fields. The gauge covariant derivative, $\widehat {\partial }_{\mu }$, describes the minimal coupling of the n gauge fields, $A_{\mu }^{I}$ (I, J, … = 1, …, n), in the adjoint representation of the gauge group to the tensor fields and takes the form

$$\displaystyle \begin{aligned} \widehat{\partial}_{[\rho}B_{\sigma\kappa]}^{N}=\partial_{[\rho}B_{\sigma\kappa]}^{N}+gA_{[\rho}^{I}\varLambda_{IM}^{N}B_{\sigma\kappa]}^{M}, {} \end{aligned} $$

(10.64)

where $\varLambda _{IM}^{N}$ denotes the representation matrices of the tensor fields with respect to the gauge group.

Using this first-order form and taking into account also another, mass-like term for the tensor fields not shown here, it is in principle possible to integrate out half of the tensor fields so as to arrive at a Lagrangian with second-order kinetic term for the remaining (massive) tensor fields (see [25] for a detailed discussion). Doing this explicitly, however, yields fairly complicated expressions in general so that it is usually easier to stick with the above first-order form. For more detailed discussions of 5D gauged supergravity with tensor fields, we refer to the original literature.

10.A Appendix: Clifford Algebras and Spinors in Arbitrary D

As described in Sect. 10.1, the balance between bosonic and fermionic degrees of freedom in supersymmetric field theories in general is implemented differently for different spacetime dimension, D, because the degrees of freedom of the corresponding fields have a different overall scaling with D. This is further complicated by the strong D-dependence of the possible chirality and reality conditions one can impose on Clifford algebra representations so as to generate minimal spinor representations of the Lorentz group. The D-dependence of these chirality and reality conditions also leads to D-dependent R-symmetry groups, which in turn contribute to a rich variety of possible scalar manifold geometries in the respective spacetime dimensions. It is the purpose of this subsection to classify the representations of Clifford algebras, the minimal spinor representations of the corresponding Lorentz groups as well as the resulting R-symmetry groups. This generalizes the discussion of spinors in four dimensions given in Chap. 1.

Starting point is the Clifford algebra, Cliff(1, D − 1), in D Lorentzian dimensions,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \{\varGamma_{a},\varGamma_{b}\}& =&\displaystyle 2\eta_{ab} \qquad (a,b,\ldots = 0,1,\ldots, D-1) {} \end{array} \end{aligned} $$

(10.65)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \eta_{ab}& =&\displaystyle \mbox{diag}(-1,+1,\ldots,+1). \end{array} \end{aligned} $$

(10.66)

Just as in 4D, the relation (10.65) implies that

$$\displaystyle \begin{aligned} \widehat{\rho}(M_{ab})\equiv\varSigma_{ab}\equiv\frac{1}{4} [ \varGamma_{a},\varGamma_{b} ] = \frac{1}{2}\varGamma_{ab} \end{aligned} $$

(10.67)

form a representation of the Lorentz algebra. The exponentials $\exp \left [\frac {\omega ^{ab}\varSigma _{ab}}{2}\right ]$ with ω _ab being finite rotation angles or boost parameters then form a double-valued representation of the Lorentz group SO ₀(1, D − 1).

1.1 10.A.1 Irreducible Representations of Cliff(1, D − 1)

The structure of the irreducible representations of Cliff(1, D − 1) is slightly different for even and odd dimensions:

1.1.1 10.A.1.1 Even Dimensions

Up to equivalence, there is exactly one non-trivial irreducible representation (irrep) of Cliff(1, D − 1) (see, e.g., [26]). It has complex dimension 2^D∕2, i.e., the Γ _a are complex (2^D∕2 × 2^D∕2)-matrices, generalizing the (4 × 4)-matrices in 4D. Explicit forms of these representations can be built up by successive tensor products of the irreps of lower-dimensional Clifford algebras, starting with the case D = 2 (see, e.g., [27]), but we do not need them for this book.

1.1.2 10.A.1.2 Odd Dimensions

If D is odd, irreps of Cliff(1, D − 1) can be obtained from an irrep of Cliff(1, D − 2) by defining the analogue of the γ ₅ matrix in 4D:

$$\displaystyle \begin{aligned} \varGamma_{\ast}\equiv(-i)^{\frac{D+1}{2}}\varGamma_{0}\varGamma_{1}\ldots\varGamma_{D-2} .{} \end{aligned} $$

(10.68)

This matrix satisfies

$$\displaystyle \begin{aligned} \begin{array}{rcl} (\varGamma_{\ast})^{2}& =&\displaystyle {\mathbb 1} \end{array} \end{aligned} $$

(10.69)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \{\varGamma_{\ast},\varGamma_{a}\}& =&\displaystyle 0 \qquad \forall a=0,\ldots,D-2 \end{array} \end{aligned} $$

(10.70)

so that either of

$$\displaystyle \begin{aligned} \varGamma_{D-1}^{(\pm)}\equiv\pm\varGamma_{\ast}{} \end{aligned} $$

(10.71)

can be used as the remaining gamma matrix to promote {Γ ₀, …, Γ _D−2} to a representation of Cliff(1, D − 1). One thus obtains two inequivalent representations of Cliff(1, D − 1) for odd D, one for each sign in (10.71).

1.2 10.A.2 Irreducible Spinor Representations of SO ₀(1, D − 1)

Thus far, we have discussed the irreps of Cliff(1, D − 1) and described how these induce double-valued spinor representations of the corresponding Lorentz groups SO ₀(1, D − 1). Just as in four dimensions, however, the spinor representations of the Lorentz group so-obtained are in general not irreducible, even though they descend from irreducible representations of Cliff(1, D − 1). In order to obtain an irreducible spinor representation of SO ₀(1, D − 1), one in general has to impose additional constraints, which may be of the following type:

• 1. Chirality condition

• 2. Reality condition

• 3. Chirality and a reality condition

The possibilities to impose one of the above are strongly dimension dependent, as we will now describe.

1.2.1 10.A.2.1 Chirality Conditions

For even D, we can always impose the following chirality condition to define a left- or right-handed Weyl spinor:

(10.72)

Note that that this condition is Lorentz covariant because of [Σ _ab, Γ _∗] = 0.

For odd D, on the other hand, there is no non-trivial analogue of Γ _∗, because

(10.73)

Thus a non-trivial chirality condition can only be imposed in even D.

1.2.2 10.A.2.2 Reality Conditions

It is again useful to distinguish between even and odd dimensions:

Even Dimensions

As discussed above, for even D there is only one equivalence class of irreps of Cliff(1, D − 1) generated by matrices Γ _a. Hence, the complex conjugate matrices $\pm \varGamma _{a}^{\ast }$, which also satisfy the Clifford algebra, must be equivalent to the matrices Γ _a, i.e., there has to be a matrix, B, such that

$$\displaystyle \begin{aligned} \varGamma_{a}^{\ast}=\eta B\varGamma_{a}B^{-1} {} \end{aligned} $$

(10.74)

for both signs η = ±1.

Odd Dimensions

If D is odd, we can obviously find a matrix, B, that also satisfies (10.74) for the first (D − 1) gamma matrices with η = ±1. What is non-trivial, however, is to extend (10.74) also to the remaining gamma matrix $\varGamma _{D-1}^{(\pm )}=\pm \varGamma _{\ast }$ (cf. Eq. (10.68)), i.e., to have

$$\displaystyle \begin{aligned} (\varGamma_{\ast})^{\ast}=\eta B\varGamma_{\ast}B^{-1}. {} \end{aligned} $$

(10.75)

Indeed, using the definition (10.68) and (10.74) for Γ ₀, …, Γ _D−2, one easily shows

$$\displaystyle \begin{aligned} (\varGamma_{\ast})^{\ast} = (-1)^{\frac{D+1}{2}}B\varGamma_{\ast}B^{-1}, \end{aligned} $$

(10.76)

which is consistent with (10.75) only for one sign:

$$\displaystyle \begin{aligned} \eta=(-1)^{\frac{D+1}{2}}=\left\{\begin{array}{l} -1 \mbox{ for D=5 mod 4}\\ +1 \mbox{ for D=3 mod 4} \end{array}\right. \end{aligned} $$

(10.77)

Obviously, the defining Eqs. (10.74) and (10.75) define B only up to an arbitrary rescaling. We may thus choose the overall scaling such that

$$\displaystyle \begin{aligned} |\det B|=1 \qquad \mbox{(choice)} {} \end{aligned} $$

(10.78)

With this normalization, one has (cf. Exercise 10.1)

$$\displaystyle \begin{aligned} \begin{array}{rcl} B^{\ast}B& =&\displaystyle \epsilon {\mathbb 1}{} \end{array} \end{aligned} $$

(10.79)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \epsilon& =&\displaystyle \pm 1. {} \end{array} \end{aligned} $$

(10.80)

The important point now is that this parameter 𝜖 is not arbitrary, but is instead fixed by the values of η and D. Concretely, for D = 2n or D = 2n + 1, one finds⁷

$$\displaystyle \begin{aligned} \epsilon = -\eta \sqrt{2}\cos \Big[ \frac{\pi}{4}(1+\eta 2n) \Big], \end{aligned} $$

(10.83)

and one arrives at the possible values for 𝜖 and η shown in Table 10.2.

Table 10.2 The possible values for η and 𝜖 together with the resulting minimal spinor types, the minimal number of real supercharges, and the general form of the R-symmetry groups (M =  Majorana, SM =  Symplectic Majorana, W =  Weyl, MW =  Majorana–Weyl, SMW =  Symplectic Majorana–Weyl)

The Majorana Condition

What makes the possible values of 𝜖 so important is that it determines whether one can impose a Majorana condition on a spinor, which, in terms of B, reads

$$\displaystyle \begin{aligned} \psi^{\ast}=\alpha B\psi \qquad \mbox{(Majorana condition)}, {} \end{aligned} $$

(10.84)

where α is an arbitrary phase. This condition is consistent with Lorentz invariance, because $\varGamma _{ab}^{\ast }=B\varGamma _{ab}B^{-1}$. A Majorana spinor thus furnishes a complete representation of the Lorentz algebra and has only half as many degrees of freedom as an unconstrained complex Dirac spinor. The consistency of (10.84) with ψ ^∗∗ = ψ, however, imposes the consistency condition

$$\displaystyle \begin{aligned} \epsilon=+1 \qquad \mbox{(for Majorana condition)}, \end{aligned} $$

(10.85)

limiting the possibility of Majorana spinors to certain dimensions, as indicated in Table 10.2.

Symplectic Majorana Spinors

If 𝜖 = −1, one can impose a symplectic Majorana condition. To this end, one needs an even number of Dirac spinors ψ _i, (i, j, … = 1, …, 2N) and an antisymmetric real matrix Ω _ij with $\varOmega ^{2}=-{\mathbb 1}_{2N}$ and imposes

$$\displaystyle \begin{aligned} (\psi_{i})^{\ast}=\varOmega_{ij}B\psi_{j}. \end{aligned} $$

(10.86)

As one needs at least two Dirac spinors to impose the symplectic Majorana condition, it does not lead to a reduction of the minimal number of degrees of freedom relative to a single Dirac spinor. The symplectic Majorana condition is, however, convenient, because it makes the action of the R-symmetry group (which in these dimensions involve symplectic groups; see Table 10.2) manifest.

1.3 10.A.3 Majorana and Weyl Condition

In some dimensions, the Majorana and the Weyl condition can be imposed simultaneously. This reduces the number of independent degrees of freedom to one quarter relative to an unconstrained Dirac spinor. Imposing (we set the phase α = 1 for simplicity)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \psi^{\ast}& =&\displaystyle B\psi \end{array} \end{aligned} $$

(10.87)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varGamma_{\ast}\psi & =&\displaystyle \pm \psi, \end{array} \end{aligned} $$

(10.88)

at the same time, obviously requires the consistency condition

$$\displaystyle \begin{aligned} (\varGamma_{\ast})^{\ast}=B\varGamma_{\ast}B^{-1} \end{aligned} $$

(10.89)

which is possible only if D = 4n − 2. But as there are no Majorana spinors in D = 6, 14, …, Majorana–Weyl spinors can only exist for

$$\displaystyle \begin{aligned} D=2 \mbox{ mod } 8 \qquad \mbox{(Condition for Majorana--Weyl spinors)}. \end{aligned} $$

(10.90)

Note, in particular, that in 4D, one can have Majorana spinors or Weyl spinors, but not Majorana–Weyl spinors.

Analogously, in dimensions in which 𝜖 = −1 allows a symplectic Majorana condition, one can sometimes also simultaneously impose a Weyl condition, and the corresponding spinors are then called symplectic Majorana–Weyl spinors. These are the dimensions D = 6 mod 8

The minimal amount of supersymmetry in each spacetime dimension is generated by a spinor operator that corresponds to the minimal spinor representation of the Lorentz group in the respective spacetime dimension. Extended supersymmetries then correspond to multiples of such minimal spinors. The R-symmetry group of the corresponding supersymmetry algebra has to respect these reality and chirality conditions and thus depends on the minimal spinor type as shown in Table 10.2. If the scalar fields of a given type of multiplet transform non-trivially under the R-symmetry group (or a factor thereof), the holonomy group of the scalar manifold typically contains this group (factor) as a factor. Especially for large amounts of supersymmetry, this already strongly constrains the possible scalar manifolds, as we described in detail for the theories in 4D and 5D.

For more than 32 real supercharges, one always has states with helicity |h| > 2 in the supermultiplets, which, for Lorentzian signature, limits supersymmetric field theories to D ≤ 11.

We finally note that in spacetimes with non-Lorentzian signature, the possible reality and chirality conditions for a given D are in general different. This is in particular true for the Euclidean signature of the compactification manifolds in string compactifications, so that the possible spinor type on these manifolds cannot be read off from Table 10.2.

Exercises

10.1

Using (10.74) and Schur’s Lemma, show that (10.79) holds for some $\epsilon \in \mathbb {C}$. Using the complex conjugate of (10.79) and the choice (10.78), show that this implies (10.80).