Gravity and Strings

16.5 Toroidal compactification of the heterotic string 479
where F I µν = 2∂[µ A I ν],whereas the KR field strength decomposes as
ˆHaij = ei me j n∂a Bmn − 1
2a I m∂aa I n + 1
2a I n∂aa I m
ˆHabi = ei m F (2) mab − CmnF (1) n ab + a I m F I ab
ˆHabc = Habc,
where the d-dimensional KR field strength and the scalars Cmn are given by
,
,
(16.144)
Hµνρ = 3∂[µ B νρ] − 3
2 LA [µ F νρ] , Cmn = Bmn − 1
2 a I ma I n, (16.145)
we have defined the (2n + p)-dimensional vector
A µ =
⎛
⎝ A(1)m µ
A (2) m µ
AI µ
⎞
⎠, F µν = 2∂[µ A ν], = 1,...,2n + p, (16.146)
and L is the O(p, p + n) metric in a non-diagonal basis:
⎛
0 Ip×p
⎜
(L) = ⎝ Ip×p 0
0
0
⎞
⎟
⎠. (16.147)
0 0 −In×n
On putting everything together, we obtain an action of the form Eq. (16.123) but with the
fields and L defined above and the matrix M now of dimension (2n + p) × (2n + p)
parametrizing an O(n, n + p)/(O(n) × O(n + p)) coset space and given by
⎛
−G
⎜
M = ⎝
−1 G−1C G−1aT C TG −1 −G − C TG −1C + aTa −C TG −1aT + aT ⎞
⎟
⎠. (16.148)
aG −1 −aG −1 C + a Ip×p − aG −1 a T
It can be constructed with the Vielbein
V ≡ 1 ⎛
−E + (E
⎜
√ ⎝
2
−1 ) TC −(E −1 ) T −(E −1 ) TaT −E − (E −1 ) TC (E −1 ) T (E −1 ) TaT ⎞
⎟
⎠, V
√ √
2 a 0 2 Ip×p
T V = M −1 . (16.149)
All the properties enjoyed by the old M as an O(p, p) matrix are now enjoyed by the new
M as an O(p, p + n) matrix with the new metric L. This metric can also be diagonalized
to η = diag(In×n, −I(n+p)×(n+p)) by the same matrix R given in Eq. (16.127) acting only