  The ADHM construction establishes a one-to-one correspondence between framed
torsion free sheaves on the projective plane and stable framed representations
of a quiver with relations in the category of complex vector spaces. This paper
studies the geometry of moduli spaces of representations of the same quiver
with relations in the abelian category of coherent sheaves on a smooth complex
projective curve $X$. In particular it is proven that this moduli space is
virtually smooth and related byrelative Beilinson spectral sequence to the
curve counting construction via stable pairs of Pandharipande and Thomas. This
yields a new conjectural construction for the local Donaldson-Thomas theory of
curves as well as a natural higher rank generalization.
