Topological recursion of the Weil-Petersson volumes of hyperbolic surfaces with tight boundaries


The Weil-Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani’s recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be tight. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil-Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in $(0,\pi)$ in addition to geodesic boundaries. Moreover, the generating function of Weil-Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani’s formula. We will comment on a possible geometric interpretation of this formula. This work is largely inspired by recent works by Bouttier, Guitter & Miermont on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani’s recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with JT gravity. We show that the multi-boundary correlators of JT gravity with defects (cone points or FZZT branes) are expressible in the tight Weil-Petersson volume generating functions, using a tight generalization of the JT trumpet partition function.