Higher rank stable pairs and virtual localization

TL;DR

This paper develops a new higher-rank enumerative theory for stable pairs on Calabi-Yau threefolds, constructing a virtual fundamental class and computing it explicitly for local b^1 using localization.

Contribution

It introduces a novel higher-rank stable pairs theory, constructing a virtual fundamental class without a natural perfect obstruction theory, and applies localization for explicit calculations.

Findings

Constructed a zero-dimensional virtual fundamental class for higher-rank stable pairs.

Developed a deformation-theoretic approach for higher-rank enumerative invariants.

Computed explicit invariants for local b^1 using virtual localization.

Abstract

We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold $X$. More precisely, we develop a moduli theory for frozen triples given by the data $O^r(-n)\rightarrow F$ where $F$ is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of $X$. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local $\mathbb{P}^1$ using the Graber-Pandharipande virtual localization technique.