Intersection numbers of spectral curves

TL;DR

This paper derives a universal formula for symplectic invariants of spectral curves with one branchpoint, linking them to characteristic classes in moduli spaces and unifying several important enumerative geometry formulas.

Contribution

It introduces a general formula connecting spectral curve invariants to characteristic classes, encompassing known results like Kontsevich-Witten, ELSV, and Marino-Vafa formulas.

Findings

Unified formula for spectral curve invariants.

Special cases recover known enumerative geometry results.

Highlights the role of Laplace transform in mirror symmetry.

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.