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.

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).