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

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. 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. 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. 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

Light hyperweak new gauge bosons from kinetic mixing in string models

String theory is at the moment our best candidate for a unified quantum theory of gravity, aiming to reconcile all the known (and unknown) interactions with gravity as well as provide insights for currently mysterious phenomena that the Standard Model and the modern Cosmology are not able to explain. In fact, it is believed that most of the problems associated to the Standard Model can indeed be resolved in string theory. Supersymmetry is supposed to be an elegant solution to the Hierarchy problem (even though more and more stringent bounds in this direction are being placed by the fact that we have been unable to experimentally find supersymmetry yet), while all the axions that compactifications bring into play can be used to resolve the strong CP problem as well as provide good candidates for Dark Matter. Inflationary models can also be constructed in string theory, providing, then, the most diffused solution to the Horizon problem. This work, in particular, is formulated in type IIB string theory compactified on an orientifolded Calabi-Yau three-fold in LARGE Volume Scenario (LVS) and focuses on the stabilisation of all the moduli in play compatible with the construction of a hidden gauge sector whose gauge boson kinetically mixes to the visible sector U(1), acquiring a mass via a completely stringy process resulting in the St{\"u}ckelberg mechanism. The "compatibility" regards the fact that certain experimental bounds should be respected combined with recent data extrapolated by Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) events at the Spallation Neutron Source at Oak Ridge National Laboratory. We are going to see that in this context we will be able to fix all the moduli as well as present a brane and fluxes set-up reproducing the correct mass and coupling of the hidden gauge boson. We also get a TeV scale supersymmetry, since the gravitino in this model will be of order O(TeV), with an uplifted vacuum to reproduce a de Sitter universe as well