12
Geometry of Moduli Spaces and Riemann
Surfaces
Abstract
In this chapter, we describe how to parametrize the moduli space M g,n and
the local coordinates that together form the fibre bundle P g,n introduced in the
previous chapter. Then, we can characterize the tangent space that we will need in
the next chapter to write the p-forms on P g,n necessary to write the amplitudes.
Finally, we introduce the notion of plumbing fixture, an operation that glues
together punctures located on the same or different surfaces.
12.1 Parametrization of P g,n
The first step is to find a parametrization of the Riemann surfaces. As we have seen
(Chap. 11), the dependence of the surface on the punctures can be described by
local coordinates, that is, transition functions. The patch is defined by cutting a disk
around each puncture, and the transition functions are defined on the circle given
by the intersection of the disk with the rest of the surface. The number of disks is
simply:
#disks = n.
(12.1)
It makes sense to look for a similar description of the other moduli (associated
to the genus) by introducing additional coordinate patches. One can imagine that all
the dependence of the moduli and punctures will reside in the transition functions
between patches if the different patches are isomorphic to a surface without any
moduli: the 3-punctured sphere 0,3 . Hence, one can look for a decomposition of
the surface by cutting disks such that one is left with 3-punctured spheres only, and
transition functions are defined on the circles at the intersections of the spheres.
Next, we need to find the number of spheres with 3 holes (or punctures). We
start first with 0,n : in this case, it is straightforward to find that there will be n − 2
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_12
251
Geometry of Moduli Spaces and Riemann
Surfaces
Abstract
In this chapter, we describe how to parametrize the moduli space M g,n and
the local coordinates that together form the fibre bundle P g,n introduced in the
previous chapter. Then, we can characterize the tangent space that we will need in
the next chapter to write the p-forms on P g,n necessary to write the amplitudes.
Finally, we introduce the notion of plumbing fixture, an operation that glues
together punctures located on the same or different surfaces.
12.1 Parametrization of P g,n
The first step is to find a parametrization of the Riemann surfaces. As we have seen
(Chap. 11), the dependence of the surface on the punctures can be described by
local coordinates, that is, transition functions. The patch is defined by cutting a disk
around each puncture, and the transition functions are defined on the circle given
by the intersection of the disk with the rest of the surface. The number of disks is
simply:
#disks = n.
(12.1)
It makes sense to look for a similar description of the other moduli (associated
to the genus) by introducing additional coordinate patches. One can imagine that all
the dependence of the moduli and punctures will reside in the transition functions
between patches if the different patches are isomorphic to a surface without any
moduli: the 3-punctured sphere 0,3 . Hence, one can look for a decomposition of
the surface by cutting disks such that one is left with 3-punctured spheres only, and
transition functions are defined on the circles at the intersections of the spheres.
Next, we need to find the number of spheres with 3 holes (or punctures). We
start first with 0,n : in this case, it is straightforward to find that there will be n − 2
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_12
251