Gravity and Strings

Lie groups, symmetric spaces, and Yang–Mills fields 609
A.4.2 Example: round spheres
The n-dimensional sphere S n (see Appendix C) is a homogeneous topological space on
which the orthogonal group SO(n + 1) acts transitively. 18 Any point is invariant under
rotations around the axis that crosses that point: the isotropy group is thus SO(n), and S n
is therefore homeomorphic to SO(n + 1)/SO(n). Ifthe SO(n + 1) generators are { ˆM â ˆb },
â, ˆb = 1,...,n + 1, the generators of SO(n) can be chosen as {Mab ≡ ˆMab} a, b = 1,...,n
and those of the orthogonal complement as Pa ≡ Mn+1 a. Weimmediately see that S n is a
symmetric space:
[Pa, Mbc] = 2δa[c Pb] =−PdƔv(Mbc) d a, [Pa, Pb] = Mab. (A.129)
To construct an SO(n + 1)-symmetric Riemannian metric, 19 we first construct a coset
representative u as above and then the Maurer–Cartan 1-form V :
−V = Pndx n + e −xn Pn Pn−1e xn Pn dx n−1
Here we can use repeatedly the formula
+ e −xn Pn e x n−1 Pn−1 Pn−2e xn Pn−1 e x n Pn dx n−2 ···. (A.130)
[X, Y ] = Z, [Y, Z] = X, [Z, X] = Y, ⇒ e aX Ye −aX = cos(a) Y + sin(a) Z,
(A.131)
for the triplets Pa, Pb, and Mab,or, far better, the definition of the adjoint action:
In both cases, the result is
e −xn Pn Pn−1e xn Pn = 1
2 ˆM â ˆb ƔAdj(e −xn Pn ) â ˆb n−1. (A.132)
−e = Pndx n + Pn−1 cos x n dx n−1 + Pn−2 cos x n cos x n−1 dx n−2 + ···,
−ϑ = sin x n
a=n−1
Mnadx a + cos x n sin x n−1
a=n−2
Mn−1 adx a
a=1
a=1
a=1
+ cos x n cos x n−1 sin x n−2
a=n−3
Mn−2 adx a + ···.
Using the SO(n)-invariant metric 20 δab,weobtain the SO(n + 1)-invariant metric
(A.133)
ds 2 = (dx n ) 2 + cos 2 x n (dx n−1 ) 2 + cos 2 x n−1 (dx n−2 ) 2 + ···. (A.134)
On comparing this with the metric Eq. (C.4) in standard spherical coordinates, we see that
the coset coordinates that we have used are related to them by x n = θn−1 + π/2,...,x 1 =
ϕ.The spin connection is given by
ω a b = 1
2 ϑ cd fcd n+1b n+1a = 1
2 ϑ cd Ɣv(Mcd) a b = ϑ a b, (A.135)
18 SO(n + 1) rotates in the standard form the coordinates of the ambient space Rn+1 , respecting the defining
equation (x1 ) 2 + ···+(x n+1 ) 2 = 1.
19 With
1
2 n(n + 1) isometries, spheres with this metric (round spheres) are also maximally symmetric spaces.
20 On rescaling some entries of δab, one obtains squashed spheres (see Appendix C.2) that have less symmetry.