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Register a new userA Generalization of Kings Equation via Noncommutative Geometry
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We introduce a framework in noncommutative geometry consisting of a $*$-algebra $mathcal A$, a bimodule $Omega^1$ endowed with a derivation $mathcal Ato Omega^1$ and with a Hermitian structure $Omega^1otimes bar{Omega}^1to mathcal A$ (a noncommutative Kahler form), and a cyclic 1-cochain $mathcal Ato mathbb C$ 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 (Kings 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 Nekrasovs beautiful proposal for re-interpreting noncommutative instantons on $mathbb{C}^nsimeq mathbb{R}^{2n}$ as infinite-dimensional solutions of Kings equation $$sum_{i=1}^n [T_i^dagger, T_i]=hbarcdot ncdotmathrm{Id}_{mathcal H}$$ 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).
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