Journal article

# Loop Equations for Gromov-Witten Invariant of P-1

Gaetan Borot, Paul Norbury

SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS | NATL ACAD SCI UKRAINE, INST MATH | Published : 2019

#### Abstract

We show that non-stationary Gromov–Witten invariants of P1 can be extracted from open periods of the Eynard–Orantin topological recursion correlators ωg,n whose Laurent series expansion at ∞ 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 ωg,n, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov–Witten theory of P1.

#### Related Projects (2)

#### Grants

*Awarded by Australian Research Council*

#### Funding Acknowledgements

This work was initiated during a visit of G.B. at the University of Melbourne supported by P. Zinn-Justin, which he thanks for hospitality. G.B. also thanks Hiroshi Iritani for discussions on mirror symmetry, and acknowledges the support of the Max-Planck-Gesellschaft. Part of this work was carried out during a visit of P.N. to Ludwig-Maximilians-Universitat which he thanks for its hospitality. P.N. is supported by the Australian Research Council grants DP170102028 and DP180103891.