Genus one free energy contribution to the quartic Kontsevich model

Abstract

We prove a formula for the genus one free energy ${\mathcal{F}}^{(1)}$ 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 ${\mathcal{F}}^{(1)}$ compared with its generic representation for ordinary topological recursion. In particular, we clarify the role of the Bergman $\tau$-function in blobbed topological recursion. As a by-product, we show that considering the holomorphic additions contributing to ${\omega }_{g,1}$ or not gives a distinction between the enumeration of bipartite and non-bipartite quadrangulations of a genus-$g$ surface.

arXiv comment33 pages, 4 figures. Final step of the proof of Th. 4.6: Readers with expertise in TR are encouraged to interpret the compensation term

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