标题： 分数自旋，展开和全息论：I. 对偶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： 在本工作中，以及在配套论文arXiv:2403.02301中，我们基于AKSZ形式主义开启了一种全息对偶的方法。 作为第一步，我们通过获得4D高自旋引力（HSG）和3D颜色共形高自旋引力（CCHSG）——即与共形高自旋规范场和颜色规范场耦合的颜色共形物质场——作为单一父理论的两种不同且经典一致的约化，从而改进了Vasiliev在arXiv:1203.5554中的全息对偶提案。 后者在物理上由一个取值于Vasiliev高自旋代数的分数自旋扩展的平坦超连接组成。 4D HSG和3D CCHSG约化由对偶结构群和二形式上同调元素所表征，它们在共同父模型中的嵌入为从圆柱体上的多维AKSZ路径积分在对偶边界条件下推导全息关系提供了依据，相关内容将分别发表。 在本工作中，我们i) 构造了底层的非交换几何，作为一个在一对共形粒子的厄米模块中表示的元对称算子代数；ii) 识别了一个离散模群，该群来源于一阶量子化系统的扭曲边界条件，并连接了二阶量子化系统的不同边界条件；iii) 识别了表征HSG和CCHSG约化的环路、结构群和二形式上同调元素，并等价了对偶的第二陈类。 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 a 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 4D HSG and 3D 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.