The 3-fold vertex via stable pairs

TL;DR

This paper develops a new combinatorial approach to compute the equivariant vertex for stable pairs on toric 3-folds, connecting enumerative geometry with box counting and proposing conjectures on rationality and equivalences.

Contribution

It introduces a weighted box counting method for the equivariant vertex of stable pairs and explores its simplification in Calabi-Yau cases, along with conjectures on descendent invariants.

Findings

Equivariant vertex expressed via weighted box counting.

Simplification to pure box counting in Calabi-Yau case.

Conjectures on the rationality of the descendent theory.

Abstract

The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3-folds. We evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting. In the toric Calabi-Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.