12.3 Plumbing Fixture
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Fig. 12.9 In white are the subspaces of the moduli space M 0,4 covered by the plumbing fixture.
The three different regions correspond to the three different ways to pair the punctures (see
Fig. 12.7). In grey is the fundamental vertex region V 0,4
1PR (1PI) stands for 1-particle (ir)reducible, a terminology that will become clear
later. Note the relation:
V
1PI
g,n = V g,n
g
#M g−g ,n+g
.
(12.44)
The two plumbing fixtures behave as follow:
• separating: increases both n and g (if both surfaces have a non-vanishing g);
• non-separating plumbing: increases g but decreases n.
The construction is obviously recursive: starting from the lowest-dimensional
moduli space, which is M 0,3 (no moduli), one has
V 0,3 = M 0,3 ,
F 0,3 = ∅.
(12.45)