arXiv 2003.031712020-03-27math-phhep-thmath.MP

King方程的非对易几何推广

A Generalization of King's Equation via Noncommutative Geometry

Gourab Bhattacharya, Maxim Kontsevich

备注：: 24 pages, added new section 5.3, corrected typos

摘要：

我们引入了一个框架，在非交换几何组成的$*$ -代数$\mathcal A$，双模$Ω^1$赋予一个导子$\mathcal A\to Ω^1$和一个埃尔米特结构$Ω^1\otimes \barΩ^1\to \mathcal A$（“非交换Kähler形式”），和循环1-上链$\mathcal A\to \mathbb C$，其上边界是由以前的结构。这些数据给出了任意有限生成的投射$\mathcal A$ -模上的联络空间上的矩映射方程。作为特例，我们得到了代数中的一大类方程（表示箭图的King方程，包括ADHM方程），经典规范理论中的一大类方程（Hermitian Yang-Mills方程，Hitchin方程，Bogomolny和Nahm方程等），以及在非对易规范理论的康纳斯，道格拉斯和施瓦茨。我们还讨论了Nekrasov的美丽的建议重新解释非交换瞬子$\mathbb{C}^n\simeq \mathbb{R}^{2n}$作为无穷维的解决方案国王的方程$$\sum_{i=1}^n [T_i^\dagger, T_i]=\hbar\cdot n\cdot\mathrm{Id}_{\mathcal H}$$其中$\mathcal H$是希尔伯特空间完成有限生成的$\mathbb C[T_1,\dots,T_n]$ -模块（例如一个理想的有限余维）。

英文摘要：

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).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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