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12 Geometry of Moduli Spaces and Riemann Surfaces
regions F g,n and V g,n as the subspaces that can and cannot be described by the
plumbing fixture:
F g,n := #M g−1,n+2
n 1 +n 2 =n+2
g 1 +g 2 =g
M g 1 ,n 1 #M g 2 ,n 2
,
(12.42a)
V g,n := M g,n − F g,n .
(12.42b)
In the RHS, it is not necessary to consider multiple non-separating plumbing fixtures
for the first term because #M g−2,n+4 ⊂ M g−1,n+2 , etc. For the same reason, it is
sufficient to consider a single separating plumbing fixture. Note that V g,n and F g,n
are in general not connected subspaces. A simple illustration is given in Fig. 12.8.
The actual decomposition of M 0,4 is given in Fig. 12.9. Importantly, F g,n and V g,n
depend on the choice of the local coordinates for all V g ,n appearing in the RHS.
It is also useful to define the subspaces F 1PR
g,n and V 1PI
g,n of M g,n that can and
cannot be described with the separating plumbing fixture only:
F
1PR
g,n :=
n 1 +n 2 =n+2
g 1 +g 2 =g
M g 1 ,n 1 #M g 2 ,n 2 ,
(12.43a)
V
1PI
g,n := M g,n − F
1PR
g,n .
(12.43b)
Fig. 12.8 Schematic illustration of the covering of M g,n from the plumbing fixture of lowerdimensional spaces. The fundamental region V g,n (usually disconnected) is not covered by the
plumbing fixture