带普通角流形上的AKSZ下降
AKSZ Descent on Manifolds with Ordinary Corners
AI总结：
本文在显式形式映射空间假设下，发展带普通角流形的AKSZ构造的逐面公式，建立全复形定理，验证四维BF理论的余二维抵消，给出奇异角数据约化准则，为AKSZ下降扩展到Joyce广义角提供基准。
AI中文摘要：
在显式形式映射空间假设下，我们针对带普通角的紧致定向流形，发展了经典AKSZ构造的逐面公式。余维r数据——承载r-1次闭2-形式、r次作用量及上同调向量场的映射空间，以及关联相邻层的修正巴塔林-维尔可维斯基（BV）/巴塔林-弗拉德金-维尔可维斯基（BFV）哈密顿恒等式——均属于Cattaneo-Mnev-Reshetikhin的极大扩展BV-BFV理论。本文新增的核心内容是基于整个面偏序集的组织方式：面上的哈密顿缺陷是其所有余一维面的原像的拉回之和，权重为定向关联数，由此单层恒等式的边界项可分解为带符号的连通片段。通过面关联复形组织这些缺陷可得到一个全复形定理：经阶乘归一化的逐面迁移是上链映射，因此目标闭形式可迁移至上闭链，且因带符号的面微平方为零，两次迭代的缺陷消失。我们针对四维BF理论在M=Γ×[0,1]²上，显式验证了所有四个余二维抵消情形。我们还建立了奇异角数据的约化准则：若原始余二维后代是预辛的，且其约化狄拉克结构是泊松双矢量的图，则移位余切构造可给出典范的严格二次角理论。从约化泊松双矢量到严格角理论的过渡已在文献[CFT2026]中记载，本文明确了其适用假设及与面关联结构的关系。该构造为AKSZ下降扩展到Joyce广义角提供了严谨的普通角基准。
英文摘要：
Under an explicit formal mapping-space hypothesis, we develop a facewise formulation of the classical AKSZ construction on compact oriented manifolds with ordinary corners. The codimension-$r$ data---a mapping space carrying a closed two-form of degree $r-1$, an action of degree $r$, and a cohomological vector field---and the modified Batalin--Vilkovisky/Batalin--Fradkin--Vilkovisky Hamiltonian identity relating consecutive strata are those of the maximally extended BV--BFV theory of Cattaneo--Mnev--Reshetikhin. What is added here is the organization over the entire face poset: the Hamiltonian defect on a face is the sum of the pullbacks of the primitives on its codimension-one faces, weighted by the orientation incidence numbers, so that the boundary term of the single-stratum identity is resolved into its connected pieces with signs. Organizing these defects by the face incidence complex yields a total-complex theorem: factorially normalized facewise transgression is a cochain map, so closed target forms transgress to cocycles, and the twice-iterated defect vanishes because the signed face differential squares to zero. We verify all four codimension-two cancellations explicitly for four-dimensional BF theory on $M=Γ\times[0,1]^2$. We also establish a reduction criterion for singular corner data. If a raw codimension-two descendant is presymplectic and its reduced Dirac structure is the graph of a Poisson bivector, the shifted cotangent construction gives a canonical strict degree-two corner theory. The passage from a reduced Poisson bivector to a strict corner theory is already recorded in \cite{CFT2026}; what is isolated here is the hypothesis under which it applies, and its relation to the face-incidence structure. The construction provides a rigorous ordinary-corner benchmark for extensions of AKSZ descent to Joyce generalized corners.
