Combinatorial classes on the moduli space of curves are tautological

TL;DR

This paper proves that combinatorial classes on the moduli space of curves, described through ribbon graphs, are tautological and provides a recursive method to find their polynomial expressions.

Contribution

It confirms Witten and Kontsevich's conjecture that combinatorial classes are tautological and develops a recursive approach to compute their polynomial forms.

Findings

Confirmed that combinatorial classes are tautological

Derived recursive formulas for polynomial expressions

Validated conjecture of Witten and Kontsevich

Abstract

The combinatorial description via ribbon graphs of the moduli space of Riemann surfaces makes it possible to define combinatorial cycles in a natural way. Witten and Kontsevich first conjectured that these classes are polynomials in the tautological classes. We answer affirmatively to this conjecture and find recursively all the polynomials.