Gravity and Strings

11.2 KK dimensional reduction on a circle S 1 315
Now we need to identify the d-dimensional field strength. This is going to be related to
ˆGab, which is invariant under δ and δ ˆχ transformations (including the linear ones δm):
ˆGab = ea µ eb ν (2∂[µVν] + 2V ∂[µ Aν]), (11.106)
and we define the gauge-invariant Gµν and the gauge-plus-global-invariant Gab = ˆGab:
On the other hand,
is also invariant under δm,and, therefore,
Gµν = 2∂[µVν], G = G + lF. (11.107)
ˆGaz = k −1 ∂al (11.108)
ˆG 2 = ˆGab ˆG ab − 2 ˆGaz ˆG a z = G 2 − 2k −2 (∂l) 2 , (11.109)
and the full dimensionally reduced Einstein–Maxwell action is
ˆS = 2πℓ
16πG ( ˆd)
N
d ˆd−1
x |g| k R + 1
2k−2 (∂l) 2 − 1
4k2 F 2 − 1
4
2
G . (11.110)
Let us now go back to the ˆd = 5 and let us reduce the Chern–Simons term. First, we
convert the Chern–Simons term into an expression with only Lorentz indices,
and then we use the relation
ˆɛ ˆµ1··· ˆµ5 ˆG ˆµ1 ˆµ2 ˆG ˆµ3 ˆµ4 ˆV ˆµ5 = |ˆg|ˆɛ â1···â5 ˆGâ1â2
ˆGâ3â4
ˆVâ5 , (11.111)
ˆɛ abcdz = ɛ abcd
between the five- and four-dimensional Levi-Cività symbols:
(11.112)
|ˆg|ˆɛ ˆG ˆG ˆV = k |g|ɛ( ˆG ˆG ˆVz − 4 ˆG ˆGz ˆV) = |g|ɛ(GGl − 4G∂lV). (11.113)
On turning back to curved indices and integrating by parts, the action takes the form
S = 2πℓ
16πG (
ˆd)
N
d 4 x |g| k R + 1
2 k−2 (∂l) 2 − 1
4 k2 F 2 (A) − 1
4 G2
+ k−1l 4 √ 3 √
ɛ[G − 2A∂l]2 .
|g|
(11.114)
This theory is a four-dimensional SUGRA theory that is invariant under eight independent
local z-independent supersymmetry transformations. Thus, it is an N = 2, d = 4