Higher rank stable pairs and 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.