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.