Graded Geometry, Q‐Manifolds, and Microformal Geometry
Graded Geometry, Q‐Manifolds, and Microformal Geometry
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分级几何、Q-流形和微形式几何
DOI：
10.1002/prop.201910023
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
T. Voronov
中科院分区：
文献类型：
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作者：
T. Voronov
我们给出了梯度几何和微形式几何的阐述，以及Q流形的语言。Q‐流形是具有奇数零平方向量场的超流形。它们可以看作是李代数的非线性模拟（与偶、奇泊松流形并行），是“非线性同调代数”的基础，是描述代数和几何结构的有力工具。这种语言与分级流形的语言相结合，分级流形是在结构轴中具有额外Z级的超流形。“微形式几何”是一个涉及“厚”或“微形式”态射的新概念，它推广了普通光滑映射，但其关键特征是相应的函数回调是非线性的。特别地，同伦泊松超流形的“泊松厚态射”推导出同伦泊松括号的L∞‐态射。有一个基于特殊类型傅里叶积分算子的量子版本，适用于Batalin-Vilkovisky几何。虽然文本主要是说明性的，但有些结果是新的或以前没有发表过的。
We give an exposition of graded and microformal geometry, and the language of Q‐manifolds. Q‐manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non‐linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of “non‐linear homological algebra”, and a powerful tool for describing algebraic and geometric structures. This language goes together with that of graded manifolds, which are supermanifolds with an extra Z ‐grading in the structure sheaf. “Microformal geometry” is a new notion referring to “thick” or “microformal” morphisms, which generalize ordinary smooth maps, but whose crucial feature is that the corresponding pullbacks of functions are nonlinear. In particular, “Poisson thick morphisms” of homotopy Poisson supermanifolds induce L∞ ‐morphisms of homotopy Poisson brackets. There is a quantum version based on special type Fourier integral operators and applicable to Batalin–Vilkovisky geometry. Though the text is mainly expository, some results are new or not published previously.
L-无穷大映射和扭曲
DOI：
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发表时间：
2011
期刊：
Homology, Homotopy and Applications
影响因子：
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作者：
Chuang J, Lazarev A
通讯作者：
Chuang J, Lazarev A