Gravity and Strings

11.4 Toroidal (Abelian) dimensional reduction 333
The relation between ˆd- and d-dimensional fields is
ˆgµν = gµν + A m µ A n νGmn,
ˆgµn = A m µGmn = ˆk(n)µ,
ˆgmn = Gmn = ˆk(m) ˆµ ˆk(n) ˆµ.
(11.176)
These fields transform correctly as tensors, vectors, and scalars under ˆd-dimensional
GCTs in the non-compact dimensions (ˆɛ µ ≡ ɛ µ ). Furthermore, under ˆd-dimensional GCTs
in the internal dimensions (ˆɛ m ≡− m ), the vectors undergo standard U(1) transformations,
δ m An µ = δm n ∂µ m . (11.177)
The constant shifts of the internal coordinates have no effect whatsoever on the
d-dimensional fields. Furthermore, under the GL(n, R) transformations only objects with
internal indices transform. Thus, the d-dimensional metric is invariant and, in matrix notation,
the internal metric and vectors transform according to
G ′ = RGR T , A ′ µ = R−1T Aµ. (11.178)
The group GL(n, R) can be decomposed into SL(n, R) × R + × Z2, theR + factor corresponding
to rescalings analogous to those of the n = 1 case, that change the determinant
of the internal metric, and later we will want to redefine the fields so they transform well
under those factors.
To calculate now the components of the spin connection in the above Vielbein basis, we
first calculate the Ricci rotation coefficients ˆ â ˆbĉ and the non-vanishing ones are
They give
ˆabc = abc,
where we have used
and we have defined
ˆabi = 1
2 emi F m ab,
ˆωabc = ωabc, ˆωabi =− 1
2 eimF m ab,
ˆωibc =−ˆωbci, ˆωaij =−e[i| m ∂aem| j],
ˆωibj = 1
2 ei m e j n ∂bGmn,
ˆibj =− 1
2 ei m ∂bemj. (11.179)
(11.180)
e(i| m ∂ae|m| j) = 1
2 ei m e j n ∂aGmn, (11.181)
F m µν ≡ 2∂[µ A m ν], F m ab = ea µ eb ν F m µν. (11.182)
Next, we plug this result into the Ricci scalar term in the action expressed in terms of the
spin-connection coefficients with the help of Palatini’s identity Eq. (D.4) and obtain
d ˆd
ˆx |ˆg| ˆR = d n
z d d x |g| K −ωb ba ωc c a − ωa bc ωbc a + 2ωb ba ∂a ln K
− (∂ ln K ) 2 + 1
4 F 2 − 1
4 ∂aGmn∂ a G mn , (11.183)