标题： 分数自旋、展开和全息：I. 双场高自旋引力约化的基本场方程 显示英文标题

标题： Fractional Spins, Unfolding, and Holography: I. Parent field equations for dual higher-spin gravity reductions

摘要： 在这项工作和配套论文 arXiv:2403.02301 中，我们基于 AKSZ 形式主义发起了一种全息方法。 作为第一步，我们在 arXiv:1203.5554 中改进了 Vasilev 的全息提案，通过获得 4D 高自旋引力（HSG）和 3D 彩色共形高自旋引力（CCHSG） ——即耦合到共形高自旋规范场和颜色规范场的彩色共形物质场 —— 作为单一母理论的两种不同且经典的相容约化。 后者在物理上由取值于 Vasiliev 高自旋代数分数自旋扩展的平坦超联络组成。 HSG 和 CCHSG 约化由对偶结构群和二形式上同调元素表征，并嵌入共同母模型中，为从具有对偶边界条件的圆柱多维 AKSZ 分区函数推导全息关系提供了理由，这些内容将在后续分别展示。 在这项工作中，我们 i) 构造了基础非交换几何作为一个表示在一对共形粒子的厄米模组上的金属量子算符代数；ii) 识别了一个离散的模群，它来自第一量子化系统的扭曲边界条件，并连接第二量子化系统不同的边界条件；iii) 识别了表征 HSG 和 CCHSG 约化的平行圈、结构群和二形式上同调元素，并使对偶二阶陈省身类相等。 显示英文摘要

摘要： In this work and in the companion paper arXiv:2403.02301, we initiate an approach to holography based on the AKSZ formalism. As the first step, we refine Vasiliev's holography proposal in arXiv:1203.5554 by obtaining 4D higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG) -- i.e., coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields -- as two distinct and classically consistent reductions of a single parent theory. The latter consists, on-shell, of a flat superconnection valued in a fractional-spin extension of Vasiliev's higher-spin algebra. The HSG and CCHSG reductions are characterized by dual structure groups and two-form cohomology elements, and their embedding in a common parent model provides a rationale for deriving holographic relations from multi-dimensional AKSZ partition functions on cylinders with dual boundary conditions, to appear separately. In this work we i) construct the underlying non-commutative geometry as a metaplectic operator algebra represented in a Hermitian module of a pair of conformal particles; ii) identify a discrete modular group, arising from twisted boundary conditions of the first-quantized system, and connecting different boundary conditions of the second-quantized system; and iii) identify the holonomies, structure groups and two-form cohomology elements that characterize the HSG and CCHSG reductions, and equate the dual second Chern classes.