254
12 Geometry of Moduli Spaces and Riemann Surfaces
are related as
f i (w i ) = ˜
f i (e
iα i w i ).
(12.9)
Hence, this motivates to consider the smaller space
ˆ
P g,n = P g,n /U(1)
n ,
(12.10)
where the action of each U(1) is defined by the equivalence (12.9). The necessity
to consider this subspace will be strengthened further later and will correspond to
the level-matching condition. Below, global phase rotations are also interpreted in
terms of the plumbing fixture, see (12.57).
The different spaces that we need are parametrized by the transition functions up
to the following identifications:
• P g,n = {F ab , f ai } modulo reparametrizations of z a ,
• ˆ
P g,n = {F ab , f ai } modulo reparametrizations of z a and phase rotations of w i ,
• M g,n = {F ab , f ai } modulo reparametrizations of z a and of w i , keeping the
points w i = 0 fixed,
• M g = {F ab , f ai } modulo reparametrizations of z a and w i .
At each step, the dimension of the space is reduced because one divides by
bigger and bigger groups. The highest reduction occurs when dividing by the
reparametrizations of w i that form an infinite-dimensional group (phase rotations
form a finite-dimensional subgroup of them).
For concreteness, it is useful to introduce explicit coordinates x s on P g,n (s ∈ N
since the space is infinite dimensional). The transition functions on the Riemann
surface depend on the x s , which explains why they can be used to parametrize the
moduli spaces. Describing the spheres S a by complex planes with punctures located
at z a,1 , z a,2 and z a,3 , the transition functions on the M g,n circles C = S a ∩ S b
for all a < b can be taken to be
on C :
z a − z a,m =
q
z b − z b,n
,
(12.11)
where z a,m and z b,n denote the punctures of S a and S b lying in C . Then, the
complex parameters q with = 1, . . . , M c
g,n are coordinates on the moduli space
M g,n . On the remaining n circles C i(a) = S a ∩ D i , the transition functions can be
expanded in series
on C i(a) :
z a − z a,m = w i +
∞
N =1
p i,N w
N
i ,
(12.12)
12 Geometry of Moduli Spaces and Riemann Surfaces
are related as
f i (w i ) = ˜
f i (e
iα i w i ).
(12.9)
Hence, this motivates to consider the smaller space
ˆ
P g,n = P g,n /U(1)
n ,
(12.10)
where the action of each U(1) is defined by the equivalence (12.9). The necessity
to consider this subspace will be strengthened further later and will correspond to
the level-matching condition. Below, global phase rotations are also interpreted in
terms of the plumbing fixture, see (12.57).
The different spaces that we need are parametrized by the transition functions up
to the following identifications:
• P g,n = {F ab , f ai } modulo reparametrizations of z a ,
• ˆ
P g,n = {F ab , f ai } modulo reparametrizations of z a and phase rotations of w i ,
• M g,n = {F ab , f ai } modulo reparametrizations of z a and of w i , keeping the
points w i = 0 fixed,
• M g = {F ab , f ai } modulo reparametrizations of z a and w i .
At each step, the dimension of the space is reduced because one divides by
bigger and bigger groups. The highest reduction occurs when dividing by the
reparametrizations of w i that form an infinite-dimensional group (phase rotations
form a finite-dimensional subgroup of them).
For concreteness, it is useful to introduce explicit coordinates x s on P g,n (s ∈ N
since the space is infinite dimensional). The transition functions on the Riemann
surface depend on the x s , which explains why they can be used to parametrize the
moduli spaces. Describing the spheres S a by complex planes with punctures located
at z a,1 , z a,2 and z a,3 , the transition functions on the M g,n circles C = S a ∩ S b
for all a < b can be taken to be
on C :
z a − z a,m =
q
z b − z b,n
,
(12.11)
where z a,m and z b,n denote the punctures of S a and S b lying in C . Then, the
complex parameters q with = 1, . . . , M c
g,n are coordinates on the moduli space
M g,n . On the remaining n circles C i(a) = S a ∩ D i , the transition functions can be
expanded in series
on C i(a) :
z a − z a,m = w i +
∞
N =1
p i,N w
N
i ,
(12.12)