Nekrasov's formula and refined sheaf counting

We revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on R4, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. Our main example will be the case of the resolved conifold, corresponding to the gauge group U(1). We will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to an isomorphism of vector spaces.