Stable pairs on local curves and Bethe roots

TL;DR

This paper provides explicit formulas for stable pair invariants of local curves using Bethe roots, revealing new properties and offering a novel approach to quantum multiplication spectra.

Contribution

It introduces explicit formulas linking stable pair invariants to Bethe roots and derives new descriptions and properties of these roots, advancing understanding in enumerative geometry and representation theory.

Findings

Explicit formulas for stable pair invariants in terms of Bethe roots

New descriptions and properties of Bethe roots

Conjectured rationality, functional equations, and pole characterizations

Abstract

We give an explicit formula for the descendent stable pair invariants of all (absolute) local curves in terms of certain power series called Bethe roots, which also appear in the physics/representation theory literature. We derive new explicit descriptions for the Bethe roots which are of independent interest. From this we derive rationality, functional equation and a characterization of poles for the full descendent stable pair theory of local curves as conjectured by Pandharipande and Pixton. We also sketch how our methods give a new approach to the spectrum of quantum multiplication on $\mathsf{Hilb}^n(\mathbf{C}^2)$.