Witten's conjecture and recursions for \(\kappa\) classes (Q2024102)

Witten's conjecture, proved by Kontsevich, says that there exists a countable number of differential operators \(L_{n\ge-1}\) that annihilate \(e^{\mathcal{F}}\), where \(\mathcal{F}\) is a generating function for intersection numbers of \(\psi\) classes in \(\mathcal{\overline{M}}_{g,n}\). In the same fashion, the authors prove that there exists a countable number of differential operators \(\widehat{L}_{n\ge0}\) that annihilate \(e^{\mathcal{K}}\), where \(\mathcal{K}\) is a generating function for intersection numbers of \(\kappa\) classes. The second section of the article is a general overview of \(\psi,\omega\) and \(\kappa\) classes and their behavior under pullback and pushforward of forgetful morphisms. Let \(\mathcal{S}\) be the potential of \(\omega\) classes. The main result of section 3 is an equation involving the (positive genus part of) \(\mathcal{F}\) and \(\mathcal{S}\). The nature of the proof is combinatorial. The result of section 4 is that \(\mathcal{K}\) can be deduced from \(\mathcal{F}\) and \(\mathcal{S}\) using formal substitutions and restrictions. Section 5 is dedicated to the proof of the main result. The idea is to manipulate the operators in Witten's conjecture, and to use the result of Section 4, to construct explicitly the operators that annihilate \(e^{\mathcal{K}}\). In the last section they use the operators \(\widehat{L}_{n\ge0}\) to get complete recursion relations of the top monomials of \(\kappa\) classes of genus \(2\) and \(3\).