Kiefer C. Quantum gravity

QUANTUM SUPERGRAVITY 71
quantum field theory, that is, one can write the spinorial charges as an integral
over a conserved current,
∫
Q i α = d 3 xJ0α(x) i ∂J i
,
mα(x)
=0. (2.141)
∂xm Fermions and bosons are combined into ‘super-multiplets’ by irreducible representations
of this algebra. There would be a fermionic super-partner to each
boson and vice versa. One would thus expect that the partners should have the
same mass. Since this is not observed in Nature, SUSY must be broken. The
presence of SUSY would guarantee that there are an equal number of bosonic
and fermionic degrees of freedom. For this reason, several divergences cancel due
to the presence of opposite signs (e.g. the ‘vacuum energy’). This gave rise to the
hope that SUSY might generally improve the UV behaviour of quantum field
theories.
Performing now an independent SUSY transformation at each space–time
point one arrives at a corresponding gauge symmetry. Because the anticommutator
(2.138) of two SUSY generators closes on the space–time momentum, this
means that space–time translations are performed independently at each space–
time point—these are nothing but general coordinate transformations. The gauge
theory therefore contains GR and is called SUGRA. 33 To each generator one then
finds a corresponding gauge field: P n corresponds to the vierbein field e n µ (see
Section 1.1), J mn to the ‘spin connection’ ωµ mn,andQi
α to the ‘Rarita–Schwinger
fields’ ψµ α,i . The latter are fields with spin 3/2 and describe the fermionic superpartners
to the graviton—the gravitinos. They are a priori massless, but can
acquire a mass by a Higgs mechanism. For N = 1 (simple SUGRA), one has
a single gravitino which sits together with the spin-2 graviton in one multiplet.
The cases N>1 are referred to as ‘extended supergravities’. In the case N =2,
for example, the photon, the graviton, and two gravitinos together form one
multiplet, yielding a ‘unified’ theory of gravity and electromagnetism. One demands
that 0 ≤ N ≤ 8 because otherwise there would be more than one graviton
and also particles with spin higher than two (for which no satisfactory coupling
exists).
For N = 1, the SUGRA action is the sum of the Einstein–Hilbert action and
the Rarita–Schwinger action for the gravitino,
S = 1 ∫
d 4 x (det e n
16πG
µ)R + 1 ∫
d 4 xɛ µνρσ ¯ψµ γ 5 γ ν D ρ ψ σ (2.142)
2
(recall det e n µ = √ −g, and we have γ 5 =iγ 0 γ 1 γ 2 γ 3 ), where we have introduced
here the spinorial covariant derivative
D µ = ∂ µ − 1 2 ωnm µ σ nm ,
33 The gauging of the Poincaré group leads in fact to the Einstein–Cartan theory, which
besides curvature also contains torsion.