Gravity and Strings

1.7 Differential forms and integration 23
which would be the action of a dual vector field Ã( ˜p+1) such that
Using
⋆ F( ˜p+2)µ ρ1···ρ ( ˜p+1) ⋆ F( ˜p+2)νρ1···ρ ( ˜p+1)
we obtain
⋆ Fµ1···µ ( ˜p+2) = ( ˜d + 2)∂[µ1 Ã( ˜p+1)µ2···µ ( ˜p+2)].
= (−1)d−1 ( ˜p + 1)!
gµν F
(p + 2)!
2
(p+2) + (−1)d ( ˜p + 1)!
(p + 1)!
T A(p+1)
µν
A useful expression for the energy–momentum tensor is
p (−1)
T A(p+1)
µν
= 1
2
F(p+2)µ σ1···σ(p+1) F(p+2)νσ1···σ(p+1) ,
(1.124)
= T Ã ( ˜p+1)
µν . (1.125)
(p + 1)! F(p+2)µ ρ1···ρ(p+1)
F(p+2)νρ1···ρ(p+1)
+ (−1) ˜p
∗
F( ˜p+2)µ
( ˜p + 1)!
ρ1···ρ
( ˜p+1) ⋆
F( ˜p+2)νρ1···ρ . (1.126)
( ˜p+1)
1.7 Differential forms and integration
As we have said before, a differential form of rank k, ork-form for short, is nothing but a
totally antisymmetric tensor field ωµ1···µk = ω[µ1···µk]. Wewriteallk-forms in this way:
ω = 1
k! ωµ1···µk dxµ1 ∧ ···∧dx µk , (1.127)
so the action of the exterior derivative d on the components is defined by
(dω)µ1···µk+1 = (k + 1)∂[µ1ωµ2···µk+1] = (k + 1)(∂ω)µ1···µk+1 . (1.128)
The Hodge dual is defined by 14
and, as before,
( ⋆ ω)µ1···µn−k
= 1
k! √ |g| ɛµ1···µn−kν1···νk ων1···νk , (1.129)
( ⋆ ) 2 = (−1) k(d−k) sign g = (−1) k(d−k)+d−1 . (1.130)
The adjoint of d with respect to the inner product of k-forms,
(αk|βk) = αk ∧
M
⋆ βk, (1.131)
is defined by
(αk|dβk−1) = (δαk|βk−1), ⇒ δ = (−1) d(k−1)−1 sign g ⋆ d ⋆
14 Observe that we need a metric to do it and that the dual depends explicitly on that metric.
(1.132)