### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 061, 29 pages      arXiv:1905.01890      https://doi.org/10.3842/SIGMA.2019.061
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

### Loop Equations for Gromov-Witten Invariant of $\mathbb{P}^1$

Gaëtan Borot a and Paul Norbury b
a) Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
b) School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia

Received May 16, 2019, in final form August 14, 2019; Published online August 23, 2019

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 $\mathbb{P}^1$.

Key words: Virasoro constraints; topological recursion; Gromov-Witten theory; mirror symmetry.

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