42
3 String Theory
where A (φ, c) is a polynomial in vertex operators obtained, according to the
operator-state correspondence, by gluing unit disks with the corresponding vertex
operators at their origin. These states may be off-shell and therefore vertex operators
are not assumed to be primaries. Now, if we let n c and n b be the number of c-ghost
zero modes and the number of b-ghost zero modes, respectively, we have, because
of the ghost number anomaly, that n b − n c = 6g − 6. Consequently, upon expanding
(3.18) in δh we find that F A (h, δh) is a homogeneous function of degree at least
6g − 6 in δh which, due to the anticommuting nature of δh, will be seen to give rise
to a differential form on the space of complex structures J of a Riemann surface
Σ g,n of genus g with n punctures. Furthermore, the exterior derivative of F A is just
the BRST differential
dF A =
Σ
a,b
δh ab
δF A
δh ab
= Q F A .
(3.19)
It is possible to show that F A (h, δh) is the pullback of a differential form on the
moduli space ˆ
P g,n of punctured Riemann surfaces Σ g,n together with a choice
of a coordinate curve around each puncture. Let D p 1 ,··· ,p n denote the group of
orientation-preserving diffeomorphisms that are trivial to all orders near the points
p 1 , · · · , p n . Then ˆ
P g,n = J /D p 1 ,··· ,p n parametrizes not only the punctures
but the coordinate curves homotopic to these punctures as well. By construction
F A (h, δh) is invariant under D p 1 ,··· ,p n . Furthermore, if v is an infinitesimal vector
field representing one of the generators of D p 1 ,··· ,p n , then its contraction with
F A (h, δh) vanishes. Indeed, because
δh ab → δh ab + (v a;b + v b;a ) ,
the contraction of F A (h, δh) with v is given by
δF A (h, δh) =
D[φ, b, c] e
−I −ΔI
A (φ, c)
Σ
(v a;b + v b;a )b
ab ,
which is equivalent to a shift in the c-ghost, δc = v. Since A (φ, c) depends only
on c(p i ), i = 1, · · · , n, and their derivatives at the punctures where v vanishes to
all orders, it follows from the translation invariance of the ghost measure that the
contraction of F A (h, δh) with v vanishes, too. This proves that F A (h, δh) is the
pullback of a form on ˆ
P g,n . In order to integrate F A (h, δh), one would like to
choose a suitable parameterization {m s }, s = 1, · · · dim(ν g,n ), of the subspace ν g,n
of ˆ
P g,n . This change of variables will induce a Jacobian through
δh ab =
dim(ν g,n )
s=1
∂h ab
∂m s
dm s .
3 String Theory
where A (φ, c) is a polynomial in vertex operators obtained, according to the
operator-state correspondence, by gluing unit disks with the corresponding vertex
operators at their origin. These states may be off-shell and therefore vertex operators
are not assumed to be primaries. Now, if we let n c and n b be the number of c-ghost
zero modes and the number of b-ghost zero modes, respectively, we have, because
of the ghost number anomaly, that n b − n c = 6g − 6. Consequently, upon expanding
(3.18) in δh we find that F A (h, δh) is a homogeneous function of degree at least
6g − 6 in δh which, due to the anticommuting nature of δh, will be seen to give rise
to a differential form on the space of complex structures J of a Riemann surface
Σ g,n of genus g with n punctures. Furthermore, the exterior derivative of F A is just
the BRST differential
dF A =
Σ
a,b
δh ab
δF A
δh ab
= Q F A .
(3.19)
It is possible to show that F A (h, δh) is the pullback of a differential form on the
moduli space ˆ
P g,n of punctured Riemann surfaces Σ g,n together with a choice
of a coordinate curve around each puncture. Let D p 1 ,··· ,p n denote the group of
orientation-preserving diffeomorphisms that are trivial to all orders near the points
p 1 , · · · , p n . Then ˆ
P g,n = J /D p 1 ,··· ,p n parametrizes not only the punctures
but the coordinate curves homotopic to these punctures as well. By construction
F A (h, δh) is invariant under D p 1 ,··· ,p n . Furthermore, if v is an infinitesimal vector
field representing one of the generators of D p 1 ,··· ,p n , then its contraction with
F A (h, δh) vanishes. Indeed, because
δh ab → δh ab + (v a;b + v b;a ) ,
the contraction of F A (h, δh) with v is given by
δF A (h, δh) =
D[φ, b, c] e
−I −ΔI
A (φ, c)
Σ
(v a;b + v b;a )b
ab ,
which is equivalent to a shift in the c-ghost, δc = v. Since A (φ, c) depends only
on c(p i ), i = 1, · · · , n, and their derivatives at the punctures where v vanishes to
all orders, it follows from the translation invariance of the ghost measure that the
contraction of F A (h, δh) with v vanishes, too. This proves that F A (h, δh) is the
pullback of a form on ˆ
P g,n . In order to integrate F A (h, δh), one would like to
choose a suitable parameterization {m s }, s = 1, · · · dim(ν g,n ), of the subspace ν g,n
of ˆ
P g,n . This change of variables will induce a Jacobian through
δh ab =
dim(ν g,n )
s=1
∂h ab
∂m s
dm s .