# Item

ITEM ACTIONSEXPORT

Released

Journal Article

#### Loop equations for Gromov-Witten invariants of P^{1}

##### Locator

https://doi.org/10.3842/SIGMA.2019.061

(Publisher version)

##### Fulltext (public)

Borot-Norbury_Loop equations for Gromov-Witten invariants of P^1_2019.pdf

(Publisher version), 545KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Borot, G., & Norbury, P. (2019). Loop equations for Gromov-Witten invariants of
P^{1}.* 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 $\mathbb{P}^1$.