Genus one free energy contribution to the quartic Kontsevich model

Branahl Johannes, Hock Alexander

Abstract

We prove a formula for the genus one free energy F^(1) of the quartic Kontsevich model for arbitrary ramification by working out a boundary creation operator for blobbed topological recursion. We thus investigate the differences in F^(1) compared with its generic representation for ordinary topological recursion. In particular, we clarify the role of the Bergman τ-function in blobbed topological recursion. As a by-product, we show that considering the holomorphic additions contributing to ω_g,1 or not gives a distinction between the enumeration of bipartite and non-bipartite quadrangulations of a genus-g surface.