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Gromov–Witten invariants of ℙ1 and Eynard–Orantin invariants
We prove that stationary Gromov-Witten invariants of $\bp^1$ arise as the Eynard-Orantin invariants of the spectral curve $x=z+1/z$, $y=\ln{z}$. As an application we show that tautological…
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Weil–Petersson volumes and cone surfaces
The moduli spaces of hyperbolic surfaces of genus g with n geodesic boundary components are naturally symplectic manifolds. Mirzakhani proved that their volumes are polynomials in the lengths of the…
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A new cohomology class on the moduli space of curves
We define a collection of cohomology classes $\Theta_{g,n}\in H^{4g-4+2n}(\overline{\cal M}_{g,n})$ for $2g-2+n>0$ that restrict naturally to boundary divisors. We prove that a generating function…
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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.
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Counting lattice points in the moduli space of curves
We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus $g$ curves with $n$ labeled points. This produces a polynomial with coefficients that include the Euler…
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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…
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Simple geodesics and Markoff quads
The action of the mapping class group of the thrice-punctured projective plane on its $$\mathop {\mathrm{GL}}\nolimits (2,{\mathbb {C}})$$GL(2,C) character variety produces an algorithm for…
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DUBROVIN’S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE
We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some…
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QUANTUM CURVES AND TOPOLOGICAL RECURSION
This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schroperator-like noncommutative analogue of a plane curve which encodes…
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Spectral Curves and the Mass of Hyperbolic Monopoles
The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps,…
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