Descendents on local curves: Rationality

TL;DR

This paper proves the rationality of the partition function for descendent invariants in the stable pairs theory of local curves, including various conditions and higher genus cases, using geometric and pole analysis techniques.

Contribution

It establishes the rationality of the descendent partition function for local curves in 3-folds, extending previous results to include relative conditions and higher genus cases.

Findings

Rationality of the descendent partition function is proven.

The capped 1-leg descendent vertex is shown to be rational.

The approach combines geometric constraints with pole analysis.

Abstract

We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the scaling 2-torus) including relative conditions and odd degree insertions for higher genus curves. The capped 1-leg descendent vertex (equivariant with respect to the 3-torus) is also proven to be rational. The results are obtained by combining geometric constraints with a detailed analysis of the poles of the descendent vertex.