252
12 Geometry of Moduli Spaces and Riemann Surfaces
spheres. Indeed, for each additional puncture beyond n = 3, an additional sphere is
created by cutting a circle. For g ≥ 1, natural places to split the surface are handles:
two circles can be cut for each of them. By inspection, one finds that it leads to 2
spheres for each handle (one on the right and the other on the left). 1 This shows that
the number of spheres is
#spheres = 2g − 2 + n.
(12.2)
The number of circles corresponds to the number of boundaries divided by two
since the boundaries are glued pairwise: each disk has one boundary and each sphere
has 3, which leads to
#circles =
n + 3(2g − 2 + n)
2
= 3g − 3 + 2n.
(12.3)
The idea of the construction is to split the surface into elementary objects
(spheres and disks) such that the full surface is seen as the union of all of them
(gluing along circles), and no information is left in the individual geometries. This
parametrization is particularly useful because there are simple coordinate systems
on spheres and disks and these surfaces are easy to visualize and to work with. For
example, they can be easily mapped to the complex plane.
To conclude, a genus-g Riemann surface g,n with n punctures can be seen as
the collection of:
• 2g − 2 + n three-punctured spheres {S a } with coordinates z a ,
• n disks {D i } with coordinates w i around each puncture,
• 3g − 3 + 2n circles {C α } at the intersections of the spheres and disks.
Examples for 0,4 and 2,2 are given in Figs. 12.1 and 12.2, respectively.
There are two types of circles: respectively, the ones at the overlap between two
spheres, and between a disk and a 3-sphere:
C := S a ∩ S b ,
C i(a) := S a ∩ D i ,
{C α } = {C , C i },
(12.4)
where counts in fact the number of moduli:
= 1, . . . , M
c
g,n ,
M
c
g,n = 3g − 3 + n.
(12.5)
On the overlap circles, the coordinate systems are related by transition functions:
on C :
z a = F ab (z b ),
on C i(a) :
z a = f ai (w i ).
(12.6)
1 The simplest way to find this result is to consider g,2 and to write one puncture at each side of the
surface (as in Fig. 12.2). To generalize further, one can consider a generic n and put all punctures
but one on one side of the surfaces.
12 Geometry of Moduli Spaces and Riemann Surfaces
spheres. Indeed, for each additional puncture beyond n = 3, an additional sphere is
created by cutting a circle. For g ≥ 1, natural places to split the surface are handles:
two circles can be cut for each of them. By inspection, one finds that it leads to 2
spheres for each handle (one on the right and the other on the left). 1 This shows that
the number of spheres is
#spheres = 2g − 2 + n.
(12.2)
The number of circles corresponds to the number of boundaries divided by two
since the boundaries are glued pairwise: each disk has one boundary and each sphere
has 3, which leads to
#circles =
n + 3(2g − 2 + n)
2
= 3g − 3 + 2n.
(12.3)
The idea of the construction is to split the surface into elementary objects
(spheres and disks) such that the full surface is seen as the union of all of them
(gluing along circles), and no information is left in the individual geometries. This
parametrization is particularly useful because there are simple coordinate systems
on spheres and disks and these surfaces are easy to visualize and to work with. For
example, they can be easily mapped to the complex plane.
To conclude, a genus-g Riemann surface g,n with n punctures can be seen as
the collection of:
• 2g − 2 + n three-punctured spheres {S a } with coordinates z a ,
• n disks {D i } with coordinates w i around each puncture,
• 3g − 3 + 2n circles {C α } at the intersections of the spheres and disks.
Examples for 0,4 and 2,2 are given in Figs. 12.1 and 12.2, respectively.
There are two types of circles: respectively, the ones at the overlap between two
spheres, and between a disk and a 3-sphere:
C := S a ∩ S b ,
C i(a) := S a ∩ D i ,
{C α } = {C , C i },
(12.4)
where counts in fact the number of moduli:
= 1, . . . , M
c
g,n ,
M
c
g,n = 3g − 3 + n.
(12.5)
On the overlap circles, the coordinate systems are related by transition functions:
on C :
z a = F ab (z b ),
on C i(a) :
z a = f ai (w i ).
(12.6)
1 The simplest way to find this result is to consider g,2 and to write one puncture at each side of the
surface (as in Fig. 12.2). To generalize further, one can consider a generic n and put all punctures
but one on one side of the surfaces.