Geometric and topological recursion and invariants of the moduli space of curves

Abstract

A thread common to many problems of enumeration of surfaces is the idea that complicated cases can be recovered from simpler ones through a recursive procedure. Solving the problem for the simplest topologies and expressing how to glue them together provides an algorithm to solve the enumerative problem of interest. In this dissertation, we consider three distinct but interconnected topics: integration over the moduli space of curves and its combinatorial model, the enumeration of curves and quadratic differentials, and the enumeration of branched covers of the Riemann sphere. The leitmotif that will connect them all is a recursive procedure known as topological recursion. <br /> The moduli space of curves is a key object of study in algebraic geometry. Its combinatorial model has provided powerful tools to compute various invariants of the moduli space, such as the Euler characteristic and Witten's intersection numbers. In this dissertation we further develop the (symplectic) geometry of this combinatorial model, providing a complete parallel with the Weil–Petersson geometry of the hyperbolic model. In particular, we show that certain length and twist coordinates are Darboux, and propose a new geometric approach to Witten's conjecture/Kontsevich's theorem. Namely, it is obtained by integration of a Mirzakhani-type identity on the combinatorial Teichmüller space, which recursively computes the constant function 1 by excision of embedded pairs of pants. <br /> The second topic of interest is the enumeration of multicurves with respect to either the hyperbolic or the combinatorial notion of length. Following ideas of Mirzakhani and Andersen–Borot–Orantin, we show that such problems can again be recursively solved by excision of embedded pairs of pants. As a consequence, the average number of multicurves over the corresponding moduli space can be computed by topological recursion. On the other hand, since the work of Mirzakhani, the average number of multicurves is known to be related to the Masur–Veech volumes of the principal stratum of the moduli space of quadratic differentials. Combining these two results, we find a topological recursion formula to compute Masur–Veech volumes. <br /> To conclude, we turn our attention to spin Hurwitz theory, that is the enumeration of branched covers of the Riemann sphere with respect to their ramification and parity. Thanks to the connection between the fermion formalism and Hurwitz theory, we are able to formulate a precise conjecture to recursively compute spin Hurwitz numbers from the simplest topologies. We also prove that this recursive formula is equivalent to a description of spin Hurwitz numbers as intersection numbers on the moduli space of curves, that is a spin version of the celebrated ELSV formula.