Strebel differentials on stable curves and Kontsevich's proof of Witten's conjecture

TL;DR

This paper extends the theory of Strebel differentials to stable curves, establishing their properties and continuity, and applies this to clarify key aspects of Kontsevich's proof of Witten's conjecture.

Contribution

It generalizes Strebel's theorem to stable curves and demonstrates their application in understanding Kontsevich's proof of Witten's conjecture.

Findings

Existence and uniqueness of Strebel differentials on stable curves

Continuity of Strebel differentials over the moduli space

Application to clarify Kontsevich's proof of Witten's conjecture

Abstract

We define Strebel differentials for stable complex curves, prove the existence and uniqueness theorem that generalizes Strebel's theorem for smooth curves, prove that Strebel differentials form a continuous family over the moduli space of stable curves, and show how this construction can be applied to clarify a delicate point in Kontsevich's proof of Witten's conjecture.