Algebraic deformations of toric varieties II: noncommutative instantons

We continue our study of the noncommutative algebraic and differentialgeometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction andenumeration of noncommutative instantons on these varieties. Wedevelop a noncommutative version of twistor theory, which introducesa new example of a noncommutative four-sphere. Wedevelop a braided version of the ADHM construction and show that itparameterizes a certain moduli space of framed torsion freesheaves on a noncommutative projective plane. We use theseconstructions to explicitly build instanton gauge bundles withcanonical connections on the noncommutative four-sphere that satisfy appropriateanti-selfduality equations. We construct projective moduli spaces forthe torsion free sheaves and demonstrate that they are smooth. Wedefine equivariant partition functions of thesemoduli spaces, finding that they coincide with the usualinstanton partition functions for supersymmetric gauge theories on~$\complex^2$.

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Published 24 October 2012