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Loop equations for Gromov-Witten invariants of P1

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Borot,  Gaëtan
Max Planck Institute for Mathematics, Max Planck Society;

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Borot, G., & Norbury, P. (2019). Loop equations for Gromov-Witten invariants of P1. Symmetry, Integrability and Geometry: Methods and Applications, 15: 061. doi:10.3842/SIGMA.2019.061.


Cite as: http://hdl.handle.net/21.11116/0000-0004-9EE4-9
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 $\mathbb{P}^1$.