262
12 Geometry of Moduli Spaces and Riemann Surfaces
Again, the two missing parameters are provided by varying q, and we obtain a M g,n -
dimensional subspace of M g,n .
Example 12.2
Here are some examples of surfaces obtained by gluing:
• 0,4 = 0,3 # 0,3
• 0,5 = 0,3 # 0,3 # 0,3 , , 0,3 # 0,4
• 1,1 = # 0,3
• 1,2 = # 0,4 , , 1,1 # 0,3
Note that the moduli on the LHS and RHS are fixed (we will see later that not all
surfaces can be obtained by gluing).
12.3.3 Decomposition of Moduli Spaces and Degeneration Limit
We have seen that the separating and non-separating plumbing fixtures yield a
family of surfaces in M g,n described in terms of lower-dimensional moduli spaces.
The question is whether all points in M g,n can be obtained in this way by looking
at all the possible gluing (varying g 1 , n 1 , g 2 and n 2 ). It turns out that this is not
possible, which is at the core of the difficulties to construct a string field theory.
Which surfaces are obtained from this construction? In order to interpret the
regions of M g,n covered by the plumbing fixture, the parametrization (12.31) is the
most useful. Previously, we explained that s gives the size of the tube connecting
the two surfaces. Since the latter is like a sphere with two punctures, it corresponds
to a cylinder (interpreted as a propagating intermediate closed string). The angle θ
in (12.31) is the twist of the cylinder connecting both components. This amounts to
start with θ = 0, then to cut the cylinder, to twist it by an angle θ and to glue again.
The limit s → ∞ (|q| → 0) is called the degeneration limit: the degenerate
surface g,n reduces to g 1 ,n 1 and g 2 ,n 2 connected by a very long tube attached
to two punctures (separating case), or to g−1,n+2 with a very long handle (nonseparating case). So it means that the family of surfaces described by the plumbing
fixture are “close” to degeneration. Another characterization (for the separating
case) is that the punctures on g 1 ,n 1 are closer (according to some distance, possibly
after a conformal transformation) to each other than to the punctures on g 2 ,n 2 .
Conversely, there are surfaces that cannot be described in this way: the plumbing
fixture does not cover all the possible values of the moduli. For a given M g,n , we
denote the surfaces that cannot be obtained by the plumbing fixture by V g,n . This
space does not contain any surface arbitrarily close to degeneration (i.e. with long
handles or tubes). In terms of punctures, it also means that there is no conformal
frame where the punctures split into two sets.
In the previous subsection, we considered two specific punctures, but any other
punctures could be chosen. Hence, there are many ways to split g,n into two
surfaces g 1 ,n 1 and g 2 ,n 2 (with fixed g 1 , g 2 , n 1 and n 2 ): every partition of the
punctures and holes in two sets leads to different degeneration limits (because they
12 Geometry of Moduli Spaces and Riemann Surfaces
Again, the two missing parameters are provided by varying q, and we obtain a M g,n -
dimensional subspace of M g,n .
Example 12.2
Here are some examples of surfaces obtained by gluing:
• 0,4 = 0,3 # 0,3
• 0,5 = 0,3 # 0,3 # 0,3 , , 0,3 # 0,4
• 1,1 = # 0,3
• 1,2 = # 0,4 , , 1,1 # 0,3
Note that the moduli on the LHS and RHS are fixed (we will see later that not all
surfaces can be obtained by gluing).
12.3.3 Decomposition of Moduli Spaces and Degeneration Limit
We have seen that the separating and non-separating plumbing fixtures yield a
family of surfaces in M g,n described in terms of lower-dimensional moduli spaces.
The question is whether all points in M g,n can be obtained in this way by looking
at all the possible gluing (varying g 1 , n 1 , g 2 and n 2 ). It turns out that this is not
possible, which is at the core of the difficulties to construct a string field theory.
Which surfaces are obtained from this construction? In order to interpret the
regions of M g,n covered by the plumbing fixture, the parametrization (12.31) is the
most useful. Previously, we explained that s gives the size of the tube connecting
the two surfaces. Since the latter is like a sphere with two punctures, it corresponds
to a cylinder (interpreted as a propagating intermediate closed string). The angle θ
in (12.31) is the twist of the cylinder connecting both components. This amounts to
start with θ = 0, then to cut the cylinder, to twist it by an angle θ and to glue again.
The limit s → ∞ (|q| → 0) is called the degeneration limit: the degenerate
surface g,n reduces to g 1 ,n 1 and g 2 ,n 2 connected by a very long tube attached
to two punctures (separating case), or to g−1,n+2 with a very long handle (nonseparating case). So it means that the family of surfaces described by the plumbing
fixture are “close” to degeneration. Another characterization (for the separating
case) is that the punctures on g 1 ,n 1 are closer (according to some distance, possibly
after a conformal transformation) to each other than to the punctures on g 2 ,n 2 .
Conversely, there are surfaces that cannot be described in this way: the plumbing
fixture does not cover all the possible values of the moduli. For a given M g,n , we
denote the surfaces that cannot be obtained by the plumbing fixture by V g,n . This
space does not contain any surface arbitrarily close to degeneration (i.e. with long
handles or tubes). In terms of punctures, it also means that there is no conformal
frame where the punctures split into two sets.
In the previous subsection, we considered two specific punctures, but any other
punctures could be chosen. Hence, there are many ways to split g,n into two
surfaces g 1 ,n 1 and g 2 ,n 2 (with fixed g 1 , g 2 , n 1 and n 2 ): every partition of the
punctures and holes in two sets leads to different degeneration limits (because they