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

Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory