Higher rank stable pairs and virtual localization

We introduce a higher rank analog of the Pandharipande–Thomas theory of stable pairs [PT09a] on a Calabi–Yau threefold $X$. More precisely, we develop a moduli theory for frozen triples given by the data $O^{\oplus r}_X (-n) \overset{\phi}{\to} F$ 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 deformationobstruction 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 [GP99] virtual localization technique.

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Published 6 June 2016