Homotopy moment maps
Homotopy moment maps
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DOI：
10.1016/j.aim.2016.08.012
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发表时间：
2013-04
期刊：
影响因子：
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通讯作者：
M. Callies;Yael Fregier;Christopher L. Rogers;M. Zambon
中科院分区：
文献类型：
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作者：
M. Callies;Yael Fregier;Christopher L. Rogers;M. Zambon
与任何具有 > 1 次闭合形式的流形相关联的是“可观测量的 L∞ 代数”，它充当辛流形上函数泊松代数的高/同伦模拟。为了研究这些流形上的李群作用，我们引入了同伦矩图理论。这样的映射是从群的李代数到可观测量的 L∞-态射，它提升了无穷小作用。我们建立了同伦矩图和等变德拉姆上同调之间的关系，并分析了此类图存在性的阻碍理论。这使我们能够轻松、明确地构造大量示例。这些包括有关环空间和平面连接模空间上的群作用的结果。还与其他人之前在经典场论、代数体理论和 dg 几何方面的工作建立了关系。此外，我们使用我们的理论在几何上构造各种 L∞ 代数作为李代数的更高中心扩展，类似于 Kostant 的量子化理论。特别是，所谓的“弦李2代数”就是这样产生的。
Associated to any manifold equipped with a closed form of degree> 1 is an ‘L∞-algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L∞-morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L∞-algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’arises this way.