Moduli spaces of geometric structures on surfaces and modular operads
Seminar Room 1, Newton Institute
It is well-known that the moduli space of Riemann surfaces with nonempty boundary is homotopy equivalent to the moduli space of metric ribbon graphs. We generalize this to a large class of geometric structures on surfaces, including unoriented, Spin, r-spin, and principal G-bundles, etc. For any class of geometric structures that can be described in terms of sections a suitable sheaf of spaces on a surface, we define a moduli space and show that it is homotopy equivalent to a moduli space of graphs with appropriate decorations at the vertices. The construction rests on the contractibility of the arc complex and can be interpreted in terms of derived modular envelopes of cyclic operads.