A generalization of King’s equation via noncommutative geometry

Abstract

We introduce a framework in noncommutative geometry consisting of a * -algebra, a bimodule endowed with a derivation ("1-forms") and a Hermitian structure (a "noncommutative Kähler form"), and a cyclic 1-cochain whose coboundary is determined by the previous structures. This data leads to moment map equations on the space of connections on arbitrary finitely-generated projective Hermitian module. As particular cases, we obtain a large class of equations in algebra (King's equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasov's beautiful proposal for re-interpreting noncommutative instantons on C n R 2n as an infinite-dimensional solution of King's equationwhere H is a Hilbert space completion of a finitely-generated C[T 1 , . . . , Tn]-module (e.g. an ideal of finite codimension).