Gravity and Strings

13.3 N = 1, 2, d = 4 vacuum supersymmetry algebras 383
This term is manifestly electric–magnetic-duality invariant. Since it is not proportional
to g, this agrees with the chiral–dual invariance of the ungauged theory. Now
Fγ[µ Fγν] = 8iF[µ ρ⋆ Fν]ργ5 + 8T (A)[µ|ργ ρ |ν]. (13.56)
The first term here can be shown to be identically zero 10 and T (A)µν is the bosonic part
of the vector-field energy–momentum tensor Eq. (5.79). These two terms are also electric–
magnetic-duality invariant and independent of g.
Putting everything together, we find
[ ˜ˆDµ, ˜ˆDν] =− 1
4 Rµν abγab − 1
2 g2γµν + T (A)[µ|αγ α |ν] − i
2 ∇ Fµν + i ⋆
2
Fµνγ5 iσ
+ i
2 γµνρ∇σ(F σρ + i ⋆F σργ5)iσ 2 − g
8 Fab
ab 3γ γµν + γµνγ abiσ 2 .
(13.57)
We can now use the bosonic part of the Einstein equation rewritten in this form (substituting
the value of R =−12g 2 )
T (A)µν = 1
2 Rµν + 3
2 g2 gµν, (13.58)
plus the bosonic part of the Maxwell equation and the Bianchi identity. We obtain
− 1
Cµν 4
ab γab + 2i ∇ Fµν + i ⋆ 2 g
Fµνγ5 iσ +
2 Fab
ab
3γ γµν + γµνγ ab iσ 2
κ = 0.
(13.59)
This is a homogeneous linear equation for κ. The 8 × 8 matrix is a linear combination
of tensor products of gamma matrices and Pauli matrices, all of them linearly independent.
There are terms with two gammas (and ⊗I2×2), whose coefficients are the components of
the Weyl tensor, there are terms proportional to one gamma and γ5 (⊗σ 2 ), whose coefficients
are the components of the covariant derivative of the electromagnetic tensor and its
dual, and, finally, there are terms with zero, two, and four gammas (⊗σ 2 ), whose coefficients
are the components of the electromagnetic tensor. In order to have maximally supersymmetric
solutions, each of these terms has to vanish. This imposes severe constraints
on the vacuum candidates. Let us now study each case separately, and let us calculate the
symmetry superalgebra using the recipe of Section 13.2.2.
13.3.2 The vacua of N = 1, d = 4 Poincaré supergravity
On setting g = Fµν = 0inEq. (13.59), we find the integrability condition for the Killing
spinors of N = 1, d = 4 Poincaré supergravity. The maximally supersymmetric solutions
are those with vanishing Weyl tensor, which (since the equations of motion are Rµν = 0)
implies a vanishing Riemann curvature tensor. Thus, Minkowski spacetime is the only maximally
supersymmetric vacuum of N = 1, d = 4 Poincaré supergravity.
10 One has to use the self-evident four-dimensional identity
ηa[bɛa1...a4] F a1a2 F a3a4 = 0.