Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers

TL;DR

This paper establishes a combinatorial framework connecting generalized Kontsevich graphs, topological recursion, and r-spin intersection numbers, providing new insights into the combinatorial and geometric structures of moduli spaces.

Contribution

It introduces four types of generalized Kontsevich graphs, proves they satisfy topological recursion, and links them to r-spin intersection numbers via the GKM.

Findings

Ciliated maps satisfy Tutte recursion and are computed by topological recursion.

Established a combinatorial interpretation of loop equations for a broad class of spectral curves.

Connected ciliated maps with r-spin intersection numbers through the GKM and topological recursion.

Abstract

Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich graphs and find combinatorial relations among them. We call the main type ciliated maps and use the auxiliary ones to show they satisfy a Tutte recursion that we turn into a combinatorial interpretation of the loop equations of topological recursion for a large class of spectral curves. It follows that ciliated maps, which are Feynman graphs for the Generalised Kontsevich matrix Model (GKM), are computed by topological recursion. Our particular instance of the GKM relates to the r-KdV integrable hierarchy and since the string solution of the latter encodes intersection numbers with Witten's $r$-spin class, we find an identity between ciliated maps and $r$-spin intersection numbers, implying that they are also governed by topological recursion. In turn, this paves the way towards a combinatorial understanding of Witten's class. This new topological recursion perspective on the GKM provides concrete tools to explore the conjectural symplectic invariance property of topological recursion for large classes of spectral curves.