Loop Equations for Gromov-Witten Invariant of P¹

Abstract

We show that non-stationary Gromov-Witten invariants of P¹ can be extracted from open periods of the Eynard-Orantin topological recursion correlators ωg,ₙ whose Laurent series expansion at ∞ 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)=lnz from the local loop equations satisfied by the ωg,ₙ, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov-Witten theory of P¹.