Paul Norbury

String and dilaton equations for counting lattice points in the moduli space of curves

We prove that the Eynard-Orantin symplectic invariants of the curve xy-y^2=1 are the orbifold Euler characteristics of the moduli spaces of genus g curves. We do this by associating to the Eynard-Orantin invariants of xy-y^2=1 a problem of enumerating covers of the two-sphere branched over three points. This viewpoint produces new recursion relations---string and dilaton equations---between the quasi-polynomials that enumerate such covers.

Orbifold Hurwitz numbers and Eynard–Orantin invariants

We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfy the topological recursion of Eynard and Orantin. This generalises the Bouchard-Marino conjecture and places Hurwitz-Hodge integrals, which arise in the Gromov--Witten theory of target curves with orbifold structure, in the context of the Eynard-Orantin topological recursion.

Quantum spectral curve for the Gromov-Witten theory of the complex projective line

We construct the quantum curve for the Gromov-Witten theory of the complex projective line.

Gromov-Witten invariants of

We prove that stationary Gromov-Witten invariants of

\bp^1

\bp^1

and Eynard-Orantin invariants

arise as the Eynard-Orantin invariants of the spectral curve

Vanishing cycles and monodromy of complex polynomials

x=z+1/z

Spectral curves and the mass of hyperbolic monopoles

,

Hyperbolic Monopoles and Holomorphic Spheres

y=\ln{z}

Dubrovin's superpotential as a global spectral curve

. As an application we show that tautological intersection numbers on the moduli space of curves arise in the asymptotics of large degree Gromov-Witten invariants of