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

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 $\mathrm{\infty }$ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral $x(z)=z+1\mathrm{/}z$ 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 ${\mathbb{P}}^1$.