  This article is an attempt to briefly introduce some of the results from
arXiv:1011.6342 on development of 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 highly frozen triples given by the
data O^r-->F for r>1 where F is a sheaf of pure dimension 1. The moduli space
of such objects does not naturally determine an enumerative theory. Instead, we
build a zero-dimensional virtual fundamental class by truncating a
deformation-obstruction theory coming from the moduli of objects in the derived
category of X. We briefly include the results of calculations in this
enumerative theory for local P^1 using the Graber-Pandharipande virtual
localization technique. We emphasize that in this article we merely include the
statement of our theorems and illustrate the final result of some of the
computations. The proofs and more detailed calculations in arXiv:1011.6342 will
appear elsewhere.
