Fock spaces and refined Severi degrees

TL;DR

This paper generalizes the Fock space operator approach to compute Severi degrees for a broader class of toric surfaces, including singular ones, and extends it to refined invariants using deformed algebraic structures.

Contribution

It extends the Fock space operator method to h-transverse lattice polygons and refined Severi degrees, broadening the scope beyond previously known cases.

Findings

Validates the Fock space operator approach for h-transverse polygons

Extends the method to refined Severi degrees with deformed Heisenberg algebra

Connects tropical curve counts with Feynman diagrams in a Fock space framework

Abstract

A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree N^{Delta,delta} is the number of delta-nodal curves in the complete linear system |L| passing through dim|L|-delta general points. Cooper and Pandharipande showed that in the case of PP^1 x PP^1 the Severi degrees can be computed as the matrix elements of an operator on a Fock space. In this note we want to generalize and extend this result in two ways. First we show that it holds more generally for Delta a so called h-transverse lattice polygon. This includes the case of PP^2 and rational ruled surfaces, but also many other, also singular, surfaces. Using a deformed version of the Heisenberg algebra, we extend the result to the refined Severi degrees defined and studied by G\"ottsche and Shende and by Block and G\"ottsche. For Delta an h-transverse lattice polygon, one can, following Brugall\'e and Mikhalkin, replace the count of tropical curves by a count of marked floor diagrams, which are slightly simpler combinatorial objects. We show that these floor diagrams are the Feynman diagrams of certain operators on a Fock space, proving the result.