A Fock space approach to Severi degrees

Abstract

The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through points in a surface. We express the Severi degrees of CP1 x CP1 as matrix elements of the exponential of a single operator M on Fock space. The formalism puts Severi degrees on a similar footing as the more developed study of Hurwitz numbers of coverings of curves. The pure genus 1 invariants of the product E x CP1 (with E an elliptic curve) are solved via an exact formula for the eigenvalues of M to initial order. The Severi degrees of CP2 are also determined by M via the (-1)^(d-1)/d^2 disk multiple cover formula for Calabi-Yau 3-fold geometries.