A Generalization of King's Equation via Noncommutative Geometry

Abstract

We introduce a framework in noncommutative geometry consisting of a $*$-algebra $\mathcal A$, a bimodule $\Omega^1$ endowed with a derivation $\mathcal A\to \Omega^1$ and with a Hermitian structure $\Omega^1\otimes \bar{\Omega}^1\to \mathcal A$ (a "noncommutative K\"ahler form"), and a cyclic 1-cochain $\mathcal A\to \mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $\mathcal A$-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 infinite-dimensional solutions of King's equation $$\sum_{i=1}^n [T_i^\dagger, T_i]=\hbar\cdot n\cdot\mathrm{Id}_{\mathcal H}$$ 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).