Essentially irreducible maps and Weil-Petersson volumes


I will discuss the enumeration of certain maps on a genus-g surface, $g \geq 0$, with n faces of prescribed even degrees with a girth constraint. More precisely I consider maps that have no vertices of degree one and that are essentially 2b-irreducible for $b \geq 0$ meaning that simple contractible cycles of length 2b or less are disallowed unless they bound a single face of degree 2b. Using results of Bouttier & Guitter I will show that such maps are counted by polynomials in b and the face degrees and that the polynomials satisfy recursion relations (string and dilaton equations) in n. This generalizes a result by Norbury in the absence of the irreducibility constraint. Next, I will show that the leading orders in b of these polynomials are related to the Weil-Petersson volumes of hyperbolic surfaces with geodesic boundary components (at least when g=0,1). If time permits I will comment on a special case where this relation can be understood bijectively.