The (Secret?) homological algebra of the Batalin-Vilkovisky approach
The (Secret?) homological algebra of the Batalin-Vilkovisky approach
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Batalin-Vilkovisky 方法的（秘密？）同调代数
DOI：
10.1090/conm/219/03076
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
J. Stasheff
中科院分区：
文献类型：
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作者：
J. Stasheff
这是一个“上同调物理”的综述，这个短语最初出现在规范理论中的反常现象中。微分形式至少早在高斯（Gauss，1833年）就已经隐含在物理学中了。他的电磁定义的链接数），并更明显地在狄拉克的磁粒子（1931年）。磁荷由第一陈数给出。这样，特征类（以及由此产生的李代数和李群的上同调）就被引入物理学。由Fade'ev和Popov引入的“幽灵”被纳入了后来被称为BRST上同调。后来鬼被重新解释为李代数上同调的Chevalley-Eilenberg复形的生成元。上同调物理学还利用群论上同调，代数变形理论，特别是同调代数的一个新的扩展，将李代数上同调与Koszul-Tate决议相结合，这是谈话的重点。这两种上同调的协同组合出现在约束泊松代数的上同调约化的Batalin-Fradkin-Vilkovisky方法中。在Batalin-Vilkovisky方法中，量子化粒子拉格朗日量和弦场论的拉格朗日量，也发展出类似的“奇”版本。一个修正主义者认为巴塔林-维尔科维奇机械承认它的一部分，作为重建同调代数与一些强大的新思想做梦也想不到的纪律。
This is a survey of `Cohomological Physics', a phrase that first appeared in the context of anomalies in gauge theory. Differential forms were implicit in physics at least as far back as Gauss (1833) (cf. his electro-magnetic definition of the linking number), and more visibly in Dirac's magnetic monopole (1931). The magnetic charge was given by the first Chern number. Thus were characteristic classes (and by implication the cohomology of Lie algebras and of Lie groups) introduced into physics. The `ghosts' introduced by Fade'ev and Popov were incorporated into what came to be known as BRST cohomology. Later the ghosts were reinterpreted as generators of the Chevalley-Eilenberg complex for Lie algebra cohomology. Cohomological physics also makes use of group theoretic cohomology, algebraic deformation theory and especially a novel extension of homological algebra, combining Lie algebra cohomology with the Koszul-Tate resolution, the major emphasis of the talk. This synergistic combination of both kinds of cohomology appeared in the Batalin-Fradkin-Vilkovisky approach to the cohomological reduction of constrained Poisson algebras. An analogous `odd' version was developed in the Batalin-Vilkovisky approach to quantizing particle Lagrangians and Lagrangians of string field theory. A revisionist view of the Batalin- Vilkovisky machinery recognizes parts of it as a reconstruction of homological algebra with some powerful new ideas undreamt of in that discipline.