Gravity and Strings

168 N = 1, 2, d = 4 supergravities
which one obtains by adding supersymmetric matter to the N = 1, d = 4 AdS supergravity
theory.
In any case, the two main characteristics of the theory are the presence of a negative
cosmological constant =−3g 2 and the fact that the gravitinos are minimally coupled to
the vector field with coupling constant g.
We anticipate that there is going to be a third source term in the Maxwell equation,
which is going to break the invariance under chiral–dual transformations of the “ungauged”
(Poincaré) theory.
The gauged N = 2, d = 4“gauged” supergravity action for these fields in the first-order
formalism is, thus,
S =
d4
xe R(e,ω)+ 6g2 + 2e−1ɛ µνρσ
¯ψµγ5γν ˆDρ + igAρσ 2
ψσ
− F 2 + J(m) µν (J(e)µν + J(m)µν) ,
(5.99)
where again ˆD is the SO(2,3) (AdS) gauge covariant derivative
The symmetries of this action are essentially the same as in the ungauged case: GCTs,
local Lorentz transformations, 11 U(1) gauge transformations, which now take the form
A ′ µ = Aµ + ∂µχ, ψ ′ µ
= e−igχσ2ψµ,
(5.100)
and local supersymmetry transformations, which take the same form as in the Poincaré
case, but with the new supercovariant derivative
˜ ˆDµ = ˆDµ + igAµσ 2 + 1
4 ˜Fγµσ 2 . (5.101)
As mentioned before, the chiral–dual invariance of the ungauged theory is broken by the
minimal coupling between gravitinos and vector field, which results in the new Maxwell
equation with a new Noether current,
∂ν(eF νµ ) − ig
2 ɛµνρσ ¯ψνγ5γρσ 2 ψσ . (5.102)
For the sake of completeness, we give the remaining equations of motion
0 = Ga µ − 3g2ea µ − 2T (ψ)a µ − 2 ˜T (A)a µ ,
0 = e−1ɛ µνρσ
γ5γν ˆDρ + igAρσ 2
ψσ − i ˜F µν + i ⋆ ˜F µν
γ5 σ 2ψν, where the equation of motion for ωµ ab has been used and where
T (ψ)a µ =− 1
2e ɛµνρσ
¯ψνγ5γa ˆDρ + igAρσ 2
ψσ − ig
2e ɛµνρσ ¯ψνγ5γρaψσ ,
˜T (A)a µ = ˜Fa ρ ˜F µ ρ − 1
4ea µ ˜F 2 .
11 There is no invariance under the full SO(2,3).
(5.103)
(5.104)