A generalization of King’s equation via noncommutative geometry

Abstract

We introduce a framework in noncommutative geometry consisting of a $\ast$-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 ${\mathbb{C}}^n\simeq {\mathbb{R}}^{2n}$ as an infinite-dimensional solution of King’s equation

where $\mathcal{H}$ is a Hilbert space completion of a finitely-generated $\mathbb{C}[T_1,\dots ,T_n]$-module (e.g., an ideal of finite codimension).