论文标题
通过非共同几何形状对国王方程的概括
A Generalization of King's Equation via Noncommutative Geometry
论文作者
论文摘要
我们介绍了一个非交易性几何形状框架，该框架由$*$ - 代数$ \ Mathcal a $，bimodule $ω^1 $赋予了衍生$ \ Mathcal a \ toω^1 $，并带有Hermitian结构$ω^1 \ otimes \ otimes \ otimimes \ ot \ ot \ ot \ bar^1 $ \ \ \ \ \ \ \ to $（形式”），一个环状1-Cochain $ \ Mathcal a \ to \ Mathbb c $，其串联由先前的结构确定。这些数据给出了在任意有限生成的投射$ \ Mathcal a $ -module上连接空间上的力矩映射方程。作为特殊情况，我们在代数（包括ADHM方程在内的国王方程式）中获得了一大型方程式，在古典仪表理论（Hermitian Yang-Mills方程，Hitchin方程，Bogomolny和Nahm方程等）中，以及在Connes，Douglas，Douglas，Douglas和Scharz和Schars和Schars的非强制学计算理论中。 我们还讨论了Nekrasov在$ \ Mathbb {C}^n \ simeq \ Mathbb {r}^{2n} $上为国王方程$ uni-dimentional解决方案$ \ sum_ \ sum_ \ sum_ \ sum_ = 1 = 1}^n [t_^_i^d_t_i^d _] n \ cdot \ mathrm {id} _ {\ mathcal h} $$其中$ \ mathcal h $是Hilbert Space完成有限生成的$ \ Mathbb c [t_1，\ dots，t_n] $ - $ - 模块（例如，有限的编码）。
We introduce a framework in noncommutative geometry consisting of a $*$-algebra $\mathcal A$, a bimodule $Ω^1$ endowed with a derivation $\mathcal A\to Ω^1$ and with a Hermitian structure $Ω^1\otimes \barΩ^1\to \mathcal A$ (a "noncommutative Kähler form"), and a cyclic 1-cochain $\mathcal A\to \mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $\mathcal A$-module. As particular cases, we obtain a large class of equations in algebra (King's equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasov's beautiful proposal for re-interpreting noncommutative instantons on $\mathbb{C}^n\simeq \mathbb{R}^{2n}$ as infinite-dimensional solutions of King's equation $$\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).
