12.3 Plumbing Fixture
263
(a)
(b)
(c)
Fig. 12.7 Permutations of punctures while gluing two spheres: they correspond to different
(disconnected) parts of M 0,4 . (a) Degeneration 12 → 34. (b) Degeneration 13 → 24. (c)
Degeneration 14 → 23
are associated to different moduli—Fig. 12.7). Since each puncture is described by
a modulus, choosing different punctures for gluing gives different set of moduli for
g,n , such that each possibility covers a different subspace of M g,n . The part of
the moduli space M g,n covered by the plumbing fixture of all surfaces g 1 ,n 1 and
g 2 ,n 2 (with fixed g 1 , g 2 , n 1 , n 2 ) is denoted by M g 1 ,n 1 #M g 2 ,n 2 :
M g 1 ,n 1 #M g 2 ,n 2 ⊂ M g,n ,
(12.40)
where the operation # includes the plumbing fixture for all values of q and all pairs
of punctures. Similarly, the part covered by the non-separating plumbing fixture is
written as #M g 1 ,n 1 :
#M g 1 ,n 1 ⊂ M g,n .
(12.41)
Importantly, the regions covered by the plumbing fixture depend on the choice
of the local coordinates because (12.30) is written in terms of local coordinates.
The subspaces M g 1 ,n 1 #M g 2 ,n 2 and #M g 1 ,n 1 are not necessarily connected (in the
topological sense).
The moduli space M g,n cannot be completely covered by the plumbing fixture
of lower-dimensional surfaces. We define the propagator and fundamental vertex
263
(a)
(b)
(c)
Fig. 12.7 Permutations of punctures while gluing two spheres: they correspond to different
(disconnected) parts of M 0,4 . (a) Degeneration 12 → 34. (b) Degeneration 13 → 24. (c)
Degeneration 14 → 23
are associated to different moduli—Fig. 12.7). Since each puncture is described by
a modulus, choosing different punctures for gluing gives different set of moduli for
g,n , such that each possibility covers a different subspace of M g,n . The part of
the moduli space M g,n covered by the plumbing fixture of all surfaces g 1 ,n 1 and
g 2 ,n 2 (with fixed g 1 , g 2 , n 1 , n 2 ) is denoted by M g 1 ,n 1 #M g 2 ,n 2 :
M g 1 ,n 1 #M g 2 ,n 2 ⊂ M g,n ,
(12.40)
where the operation # includes the plumbing fixture for all values of q and all pairs
of punctures. Similarly, the part covered by the non-separating plumbing fixture is
written as #M g 1 ,n 1 :
#M g 1 ,n 1 ⊂ M g,n .
(12.41)
Importantly, the regions covered by the plumbing fixture depend on the choice
of the local coordinates because (12.30) is written in terms of local coordinates.
The subspaces M g 1 ,n 1 #M g 2 ,n 2 and #M g 1 ,n 1 are not necessarily connected (in the
topological sense).
The moduli space M g,n cannot be completely covered by the plumbing fixture
of lower-dimensional surfaces. We define the propagator and fundamental vertex