Gravity and Strings

392 Unbroken supersymmetry
transformation rules because they transform among themselves. Now, ˆψ + w corresponds to
amatter spinor and therefore M consists basically of matter fields whose truncation sets
M = 0. This is indeed possible because all the terms in M contain two gammas and a combination
of them can always be set to zero. This truncation gives the pure N = 1, d = 5
supergravity action Eq. (11.98) and the supersymmetry transformation rule of the fivedimensional
gravitino [664]:
δˆɛ ˆψâ
= ˆ∇â − 1
8 √ 3 ( ˆγ ˆbĉ
ˆγâ + 2 ˆγ ˆb
ĉ
ˆg â) ˆG ˆbĉ ˆɛ. (13.105)
The relation between the six- and five-dimensional fields is
ˆgww =−1,
ˆB − 1
ˆµw = √ ˆV ˆµ, ˆg ˆµw =
3
1
√ ˆV ˆµ, ˆg ˆµˆν =ˆg ˆµˆν −
3
1
3 ˆV ˆµ ˆVˆν, (13.106)
with the ˆB − µν components determined by anti-self-duality.
The AdS3 × S3 solution can be reduced preserving all the supersymmetry in two directions:
the direction of the Hopf fiber of the S3 when we see it as a fibration over S2 (i.e. the
coordinate ψ in the Euler-angle parametrization Eq. (A.97)) and the analog in AdS 3 when
we see it as a fibration over AdS2, i.e. the coordinate η in
d 2 (3)
1 2
≡ d 4 (2) − (dη + sinh(χ/2)dφ) 2 . (13.107)
Actually it is also possible to perform a dimensional reduction in a combination of the two
directions: on rotating by an angle ξ,
w = R3
(cos ξη+ sin ξψ), y =−sin ξη+ cos ξψ, (13.108)
2
and reducing in the direction w,weobtain the two-parameter family [272]
d ˆs 2 = R 2 2 d2 (2) − R2 2 d2 (2) − R2 2 (dy + cos ξ cos θ dϕ − sin ξ sinh χ dφ)2 ,
ˆG = √ 3R2(cos ξωAdS2 − sin ξω S 2), R2 = R3/2. (13.109)
It is maximally supersymmetric for any ξ and can be obtained as the near-horizon limit
of the d = 5 rotating extreme BH [422] which, as usual, has only half of the maximal
supersymmetry [614]. sin ξ plays the role of the rotation parameter j