On the Formality of Configuration Spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$

Abstract

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.