Semi perfect obstruction theories and higher rank Donaldson-Thomas type invariants

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---->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 show how to use the technology

of semi perfect obstruction theories and the luxury of infinity stacks in obtaining a well behaved

truncation of an obstruction theory coming from the moduli of objects in the derived category.

After building a suitable zero-dimensional virtual fundamental class by hand, we obtain the
first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau
threefolds. Finally If time permits we explain how to use virtual localization techniques to

compute the invariants using equivariant intersection theory.