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

In 1992, Kontsevich introduced certain ribbon graphs [ 47 ] as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten’s conjecture [ 59 ]. 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. 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 expression for the generating series of ciliated maps in terms of 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 also provides concrete tools to explore the conjectural symplectic invariance property of topological recursion for large classes of spectral curves.