Q流形的导出结合代数与循环同调
Derived Associative Algebras and Cyclic Cohomology of Q-manifolds
AI总结：
本文在Q流形框架下重新研究Loday的导出积，将格拉斯曼奇积解释为平移分次后的标准积，得到Q流形对应的导出结合代数，并将Connes循环同调应用于该代数，提出Q流形的导出循环同调，还明确了其在相关理论中的意义。
AI中文摘要：
本文在Q流形（即配备了“平方为零”的奇向量场的超流形）的框架下重新研究了Loday的导出积。我们将格拉斯曼奇积解释为平移分次后的标准积，这意味着任意Q流形都对应一个导出（非交换）结合的ℤ₂分次代数。我们将Connes的循环同调应用于该导出结合代数，得到了一种不同于标准同调的Q流形上新同调，称之为Q流形的导出循环同调。导出上循环可被解释为BV–BFV–BRST形式论中的经典BRST不变泛函，或在正则叶状结构中通过其叶状李代数胚得到的广义Ruelle–Sullivan流。
英文摘要：
Loday's derived product is revisited in the setting of Q-manifolds, i.e., supermanifolds equipped with an odd vector field that `squares to zero'. We interpret the Grassmann odd product as a standard product upon shifting the grading, which implies the existence of a derived (noncommutative) associative $\mathbb{Z}_2$-graded algebra associated with any Q-manifold. We apply Connes' cyclic cohomology to the derived associative algebra, giving a new cohomology on Q-manifolds distinct from the standard cohomology: we refer to this as the derived cyclic cohomology of a Q-manifold. The derived cyclic cocycles are interpreted as classically BRST-invariant functionals within the BV--BFV--BRST formalism or generalised Ruelle--Sullivan currents when applied to regular foliations via their foliation Lie algebroids.
