Geodesic
Loop Equations for Gromov–Witten Invariant of $\mathbb{P}^1$
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\mathbb{P}^1$.
Keywords: Virasoro constraints, topological recursion, Gromov–Witten theory, mirror symmetry. 
@article{SIGMA_2019_15_a60,
     author = {Ga\"etan Borot and Paul Norbury},
     title = {Loop {Equations} for {Gromov{\textendash}Witten} {Invariant} of $\mathbb{P}^1$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a60/}
}
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Gaëtan Borot; Paul Norbury. Loop Equations for Gromov–Witten Invariant of $\mathbb{P}^1$. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a60/

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