Loop Equations for Gromov–Witten Invariant 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 $\infty$ 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 $y(z) = \mathrm{ln}\, z$ 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 $\mathbb{P}^1$.