Configuration models for moduli spaces of Riemann surfaces with boundary
The author studies the moduli space \(\mathfrak{M}^{\bullet}_g(m,n)\) of Riemann surfaces of genus \(g\geq 0\) with \(n\geq 1\) incoming and \(m\geq 1\) outgoing boundary curves and with one marked point on each incoming curve. Such a surface is seen as a bordism between non-empty compact 1-manifolds. The author establishes a homeomorphism between \(\mathfrak{M}^{\bullet}_g(m,n)\) and a space \(\mathfrak{Rad}_h(m,n)\) consisting of configurations of \(h=2g-2+m+n\) pairs of radial slits distributed over \(n\) annuli, a slit being a radial cut from the outer boundary to some interior point. (Certain combinatorial conditions are required to insure genus \(g\) and \(m\) outgoing circles for the corresponding surface.) The association of a configuration to a surface of the type considered is called Hilbert uniformization, since the idea is already contained in David Hilbert's work. The author of the paper under review already used this method in earlier works. The space \(\mathfrak{Rad}_h(m,n)\) has a topology which makes it a noncompact manifold of dimension \(6g-6+3m+4n\), and the author gives a description of this space as the complement of a subcomplex in a finite cell complex. The author also studies the boundary structure obtained by degenerations of configurations. He describes a compactification of the moduli space where degenerate surfaces are added. This compactification is different from the Deligne-Mumford compactification of the usual moduli space of Riemann surfaces, since the degeneration here does not only consist of pinching curves, but also of degenerating handles to intervals. The author, in joint work with J. Abhau and E. Ehrenfried, used this compactification to compute the homology of the space \(\mathfrak{M}^{\bullet}_g(m,n)\). The author also mentions connections with operad structures conformal field theories.
scientific article; zbMATH DE number 5184538