Refined stable pair invariants on local Calabi-Yau threefolds

A refinement of the Pandharipande-Thomas stable pair invariants for local toric Calabi-Yau threefolds is defined by what we call the virtual Bialynicki-Birula decomposition. We propose a product formula for the generating function for the refined stable pair invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local ${\bf P}^1$. I will also describe how the proposed product formula is related to the wall crossing in my first talk. This is joint work with Sheldon Katz and Albrecht Klemm.