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The Weil-Petersson volumes of moduli spaces of hyperbolic surfaces withgeodesic boundaries are known to be given by polynomials in the boundarylengths. These polynomials satisfy Mirzakhani's recursion formula, which fitsinto the general framework of topological recursion. We generalize therecursion to hyperbolic surfaces with any number of special geodesic boundariesthat are required to be tight. A special boundary is tight if it has minimallength among all curves that separate it from the other special boundaries. TheWeil-Petersson volume of this restricted family of hyperbolic surfaces is shownagain to be polynomial in the boundary lengths. This remains true when we allowconical defects in the surface with cone angles in $(0,\pi)$ in addition togeodesic boundaries. Moreover, the generating function of Weil-Peterssonvolumes with fixed genus and a fixed number of special boundaries is polynomialas well, and satisfies a topological recursion that generalizes Mirzakhani'sformula. We will comment on a possible geometric interpretation of thisformula.  This work is largely inspired by recent works by Bouttier, Guitter & Miermonton the enumeration of planar maps with tight boundaries. Our proof relies onthe equivalence of Mirzakhani's recursion formula to a sequence of partialdifferential equations (known as the Virasoro constraints) on the generatingfunction of intersection numbers.  Finally, we discuss a connection with JT gravity. We show that themulti-boundary correlators of JT gravity with defects (cone points or FZZTbranes) are expressible in the tight Weil-Petersson volume generatingfunctions, using a tight generalization of the JT trumpet partition function.

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