In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein–Hilbert and of the Palatini–Cartan theories in every space-time dimension greater than two. In the former case one obtains a BV theory for the first-order formulation of Einstein–Hilbert theory, in the latter a BV theory for Palatini–Cartan theory with a partial implementation of the torsion-free condition already on the space of fields. All theories described here are BV versions of the same classical system on cylinders. The AKSZ implementations we present have the advantage of yielding a compatible BV–BFV description, which is the required starting point for a quantization in presence of a boundary.

在本笔记中，AKSZ 构造应用于爱因斯坦-希尔伯特理论和帕拉蒂尼-嘉当理论在每个大于 2 的时空维度上的缩减相空间的 BFV 描述。在前一种情况下，获得爱因斯坦-希尔伯特理论一阶公式的 BV 理论，在后一种情况下，获得帕拉蒂尼-嘉当理论的 BV 理论，其中部分实现了场空间上的无扭转条件。这里描述的所有理论都是圆柱体上相同经典系统的 BV 版本。我们提出的 AKSZ 实现具有产生兼容 BV-BFV 描述的优势，这是在存在边界的情况下进行量化所需的起点。