Homological reduction of constrained Poisson algebras
Homological reduction of constrained Poisson algebras
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约束泊松代数的同调约简
DOI：
10.4310/jdg/1214459757
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发表时间：
1996
影响因子：
2.5
通讯作者：
J. Stasheff
中科院分区：
文献类型：
--
作者：
J. Stasheff
由Batalin-Fradkin-Vilkovisky发展的"经典BRST构造“是一种同调构造，用于通过在约束轨迹上消失的函数的理想$I$来约化泊松流形$W$上的光滑函数的泊松代数$P = C^\infty（W）$。如果$I$在Poisson括号下闭合，则该理想称为一级理想;几何学家将约束轨迹称为各向同性。物理学家的模型是泊松代数P$的微分泊松代数扩展;它的微分包含一个重新发明了理想I$的Koszul复形的部分和一个看起来像Cartan-Chevalley-Eilenberg微分的部分。
本论文关注纯粹的同调（泊松）代数结构，使用“模型”的概念，从理性同伦理论和同调扰动理论的技术，建立一些基本的结果，解释经典的BRST-BFV结构的数学存在。虽然通常的治疗BFV的基础依赖（个人的约束）和名义上有限维，我注意避免假设有限维和工作更恒定的理想。特别是，技术适用于“不规则”的情况下（理想的是不是由一个规则的约束序列），虽然几何解释是不太完整。
The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$ is closed under the Poisson bracket; geometers refer to the constraint locus as coisotropic. The physicists' model is crucially a differential Poisson algebra extension of a Poisson algebra $P$; its differential contains a piece which reinvented the Koszul complex for the ideal $I$ and a piece which looks like the Cartan-Chevalley-Eilenberg differential.
The present paper is concerned purely with the homological (Poisson) algebraic structures, using the notion of ``model'' from rational homotopy theory and the techniques of homological perturbation theory to establish some of the basic results explaining the mathematical existence of the classical BRST-BFV construction. Although the usual treatment of BFV is basis dependent (individual constraints) and nominally finite dimensional, I take care to avoid assumptions of finite dimensionality and work more invariantly in terms of the ideal. In particular, the techniques are applied to the `irregular' case (the ideal is not generated by a regular sequence of constraints), although the geometric interpretation is less complete.