# Loop equations for Gromov-Witten invariants of $\mathbb{P}^1$

## TL;DR

This paper demonstrates how non-stationary Gromov-Witten invariants of bP^1 can be derived from topological recursion correlators, establishing a connection between spectral curve data and Gromov-Witten theory through loop equations.

## Contribution

It introduces a method to extract non-stationary Gromov-Witten invariants of bP^1 from topological recursion, linking local and global loop equations with Virasoro constraints.

## Key findings

- Global loop equations are established for the spectral curve.
- These equations are shown to be equivalent to Virasoro constraints.
- The approach connects topological recursion with Gromov-Witten invariants.

## Abstract

We show that non-stationary Gromov-Witten invariants of $\mathbb{P}^1$ can be extracted from open periods of the Eynard-Orantin topological recursion correlators $\omega_{g,n}$ whose Laurent series expansion at $\infty$ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral $x(z) = z + 1/z$ and $y(z) = \ln z$ from the local loop equations satisfied by the $\omega_{g,n}$, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov-Witten theory of $\mathbb{P}^1$.