# From Kontsevich-Witten to linear Hodge integrals via Virasoro operators

## TL;DR

This paper proves Alexandrov's conjecture linking Kontsevich-Witten and Hodge tau-functions using Virasoro operators, confirming the formula's constant factor as one through series expansion analysis.

## Contribution

It provides a rigorous proof of the conjectured formula connecting two important tau-functions via Virasoro operators.

## Key findings

- Confirmed the constant factor in the formula is one
- Established a new proof using series expansions of Lambert W function
- Connected Kontsevich-Witten and Hodge tau-functions through Virasoro operators

## Abstract

We give a proof of Alexandrov's conjecture on a formula connecting the Kontsevich-Witten and Hodge tau-functions using only the Virasoro operators. This formula has been confirmed up to an unknown constant factor. In this paper, we show that this factor is indeed equal to one by investigating series expansions for the Lambert W function on different points.