Gravity and Strings

16.3 Further reduction of N = 2A, d = 10 SUEGRA to nine dimensions 467
where the nine-dimensional RR field strengths are defined as follows,
G (2n+1) = dC (2n) − HC (2n−2) + F (2) C (2n−1) ,
G (2n) = dC (2n−1) − HC (2n−3) + F (1) C (2n−2) ,
and the nine-dimensional gauge transformations that leave them invariant are
δ A (i) = d (i) ,
δB = d − d (1) A (2) − d (2) A (1) ,
δC (2n) = d (2n−1) − H (2n−3) − F (2) (2n−2) ,
δC (2n+1) = d (2n) − H (2n−2) − F (1) (2n−1) .
(16.80)
(16.81)
Using the notation introduced in Section 16.1.3, we can write the nine-dimensional RR
field strengths and gauge transformations in this way:
G = dC − HC + F (2) oddC + F (1) evenC.
δC = d (·) − H (·) − F (2) even (·) − F (1) odd (·) .
(16.82)
The RR kinetic terms in the action reduce as follows:
|ˆg|
−
ˆG
2 · (2n + 2)!
(2n+2) √
2 |g|
=−
2 · (2n + 2)! kG (2n+2) √
2 |g|
+
2 · (2n + 2)! k−1G (2n+1)2 .
(16.83)
The reduction of the Chern–Simons term is straightforward. On putting everything together,
after some integrations by parts, we obtain
S =
g2 A
16πG (9)
NA
d9x √
|g| e−2φ
R − 4(∂φ) 2 + 1
2 · 3! H 2 + (∂ ln k) 2
− 1
4k2 F (1)2 1
− 4k−2F (2)2
− 1
2
n=1,...,4
(−1) n k (−1)n
n!
G (n) 2
− 1
23 · 32 ɛ
(3) (3) (2) (3) (2) √ ∂C ∂C A − 3∂C ∂C
|g|
B + A (1) A (2)
+ 6∂C (3) ∂ A (1) C (2) A (2) + 9∂C (2) ∂ A (1) C (2)B + A (1) A (2)
+ 9∂ A (1) ∂ A (1) C (2) C (2)
A (2) .
16.3.2 Dimensional reduction of fermions and supersymmetry rules
(16.84)
We can use the decomposition of ten- into nine-dimensional gamma matrices explained
in Appendix B.1.5, which corresponds to the decomposition of ten-dimensional