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

Symmetry Integrability and Geometry-Methods and Applications

article

Gaëtan Borot¹  Paul Norbury² 

[1] Max Planck Institut für Mathematik;School of Mathematics and Statistics, University of Melbourne

关键词: Virasoro constraints;    topological recursion;    Gromov–Witten theory;    mirror symmetry;   

DOI  :  10.3842/SIGMA.2019.061

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $\mathbb{P}^1$.

【 授权许可】

Unknown