关于$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$的构形空间的形式化性质
On the Formality of Configuration Spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$
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中文总结 AI 辅助
本文针对$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$的构形空间，通过定义可构造de Rham-Dolbeault上同调理论，分类其形式化性质并给出对应CDGA模型，还将其应用于拓扑-全纯场论的局部算子代数研究。
中文摘要 AI 辅助
本文完整分类了$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$的构形空间的形式化性质。我们定义了可构造的de Rham-Dolbeault上同调理论，该理论为$\boldsymbol{\text{Conf}_m(\reals^{n'} \times \boldsymbol{\reals^{n}})}$提供了可构造的CDGA（交换微分分次代数）模型。对于$(n'=0, n\boldsymbol{\neq}2)$或$(n'=1, n\boldsymbol{\neq}1)$，这些CDGAs是非形式的；对于$(n'\boldsymbol{\neq}2, n\boldsymbol{\neq}1)$，我们利用容许图的图论CDGA和正则化构形空间积分，建立了可构造CDGA与其上同调之间的显式拟同构，从而得到形式化结果。作为应用，我们证明了$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$上的拓扑-全纯场论的局部算子代数（当$n'\boldsymbol{\neq}2, n\boldsymbol{\neq}1$时）与顶点代数的高维类似物同伦等价。
英文摘要
This paper presents a complete classification of the formality of configuration spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$. We define a constructible de Rham-Dolbeault cohomology theory which provides a constructible CDGA (commutative differential graded algebra) model of $\Conf_m(\mathbb{R}^{n'} \times \mathbb{C}^{n})$. For $(n'=0,n\ge2)$ or $(n'=1,n\ge1)$, the CDGAs are non-formal. For $n'\ge2,n\ge1$, we establish an explicit quasi-isomorphism between the constructible CDGA and its cohomology by using a diagrammatic CDGA of admissible diagrams and a regularized configuration space integral, which leads to the formality. As an application, we show that the local operator algebra of a topological-holomorphic field theory on $\mathbb{R}^{n'} \times \mathbb{C}^{n}$ ($n'\ge2,n\ge1$) is homotopically equivalent to a higher dimensional analog of vertex algebras.
