A Homotopy Theoretic Proof of the BV Identity in Loop Homology
A Homotopy Theoretic Proof of the BV Identity in Loop Homology
复制标题
循环同调中BV恒等式的同伦理论证明
DOI：
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发表时间：
2007
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通讯作者：
Hirotaka Tamanoi
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文献类型：
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作者：
Hirotaka Tamanoi
Chas和Sullivan利用链和链同伦证明了闭的有限维光滑流形上的自由环空间的同调中存在一个Batalin-Vilkovisky代数结构。这种代数结构包括一个称为循环积的结合积、一个称为循环括号的李括号和一个称为BV运算符的平方0运算符。Cohen和Jones从谱的角度给出了环积的同伦理论描述。本文给出了循环括号的显式同伦理论描述，并利用这种刻画给出了连接循环积、循环括号和BV算子的BV恒等式的同调证明。证明是基于这样一个观察，即在自由循环空间中，循环括号和BV导数是由相同的循环给出的，只是它们因循环的参数化而不同。
Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological proof of the BV identity connecting the loop product, the loop bracket, and the BV operator. The proof is based on an observation that the loop bracket and the BV derivation are given by the same cycle in the free loop space, except that they differ by parametrization of loops.