Intersection numbers of spectral curves

Abstract

We compute the symplectic invariants of an arbitrary spectral curve with only 1 branchpoint in terms of integrals of characteristic classes in the moduli space of curves. Our formula associates to any spectral curve, a characteristic class, which is determined by the laplace transform of the spectral curve. This is a hint to the key role of Laplace transform in mirror symmetry. When the spectral curve is y=\sqrt{x}, the formula gives Kontsevich--Witten intersection numbers, when the spectral curve is chosen to be the Lambert function \exp{x}=y\exp{-y}, the formula gives the ELSV formula for Hurwitz numbers, and when one chooses the mirror of C^3 with framing f, i.e. \exp{-x}=\exp{-yf}(1-\exp{-y}), the formula gives the Marino-Vafa formula, i.e. the generating function of Gromov-Witten invariants of C^3. In some sense this formula generalizes ELSV, Marino-Vafa formula, and Mumford formula.

53 pages, 1 fig, Latex, minor modifications

References in corpus (12)

• Open string amplitudes and large order behavior in topological string theory
• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Holomorphic anomaly and matrix models
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Symplectic invariants, Virasoro constraints and Givental decomposition
• Local Mirror Symmetry for One-Legged Topological Vertex
• CFT and topological recursion
• Invariants of algebraic curves and topological expansion
• Bouchard-Klemm-Marino-Pasquetti Conjecture for ${\mathbb{C}}^3$
• On a Proof of a Conjecture of Marino-Vafa on Hodge Integrals
• Hodge Integrals and Integrable Hierarchies
• Evaluating tautological classes using only Hurwitz numbers

Cited by in corpus (29)

• The Virasoro Minimal String
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula
• A generalized topological recursion for arbitrary ramification
• A short overview of the "Topological recursion"
• The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line
• Topological recursion for irregular spectral curves
• On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds
• All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds
• Loop equations and a proof of Zvonkine's $qr$-ELSV formula
• Higher Airy structures and topological recursion for singular spectral curves
• Invariants of spectral curves and intersection theory of moduli spaces of complex curves
• A Unified Approach to Holomorphic Anomaly Equations and Quantum Spectral Curves
• Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion
• Emergent Geometry of Matrix Models with Even Couplings
• On a proof of the Bouchard-Sulkowski conjecture
• Monotone orbifold Hurwitz numbers
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Large Strebel graphs and $(3,2)$ Liouville CFT
• Large genus behavior of topological recursion
• The SYZ mirror symmetry and the BKMP remodeling conjecture
• Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part II : For the Confluent Family of Hypergeometric Equations
• $\mathcal{N}=1$ Super Topological Recursion
• Painlevé equations, topological type property and reconstruction by the topological recursion
• A formalism of abstract quantum field theory of summation of fat graphs
• Topological recursion for the conifold transition of a torus knot
• Les Houches lecture notes on moduli spaces of Riemann surfaces
• Seiberg-Witten Theory and Topological Recursion
• The Laplace transform, mirror symmetry, and the topological recursion of Eynard-Orantin