# Buryak-Okounkov formula for the $n$-point function and a new proof of the Witten conjecture

## TL;DR

This paper unifies Buryak and Okounkov's formulas for intersection numbers on moduli spaces to provide a novel proof of the Witten conjecture, linking intersection theory with integrable systems through double ramification cycles.

## Contribution

It offers a new proof of the Witten conjecture by connecting Buryak and Okounkov's formulas with the geometry of double ramification cycles.

## Key findings

- Unified formulas for n-point functions of psi-class intersection numbers.
- Established a new proof of the Witten conjecture.
- Linked intersection theory with integrable systems via geometric methods.

## Abstract

We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new proof of the famous Witten conjecture / Kontsevich theorem, where the link between the intersection theory of the moduli spaces and integrable systems is established via the geometry of double ramification cycles.