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

Abstract

This paper presents a complete classification of the formality of configuration spaces of ${\mathbb{R}}^{n^{\prime }}\times {\mathbb{C}}^n$. 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 $(n^{\prime }=0,n\ge 2)$ or $(n^{\prime }=1,n\ge 1)$, the CDGAs are non-formal. For $n^{\prime }\ge 2,n\ge 1$, 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 ${\mathbb{R}}^{n^{\prime }}\times {\mathbb{C}}^n$ ($n^{\prime }\ge 2,n\ge 1$) is homotopically equivalent to a higher dimensional analog of vertex algebras.

57 pages. Comments are welcome