Dual Feynman transform for modular operads
Dual Feynman transform for modular operads
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模运算的对偶费曼变换
DOI：
10.4310/cntp.2007.v1.n4.a1
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
A. Lazarev
中科院分区：
文献类型：
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作者：
J. Chuang;A. Lazarev
引入并研究了模歌剧的对偶Feynman变换的概念。这推广和解释了Kontsevich的对偶结构，从可收缩的可微分次Frobenius代数产生图上同调类。当在真空图上求值时，模算子的对偶Feynman变换实际上是Getzler和Kapranov引入的Feynman变换的线性对偶。与费曼变换形成鲜明对比的是，对偶概念允许通过生成元和关系进行极其简单的表示；这导致了对其代数的明确和容易的描述。我们讨论了对偶Feynman变换的进一步推广，其代数不一定是可压缩的。这自然产生了一个类似于Boardman-Vogt拓扑树复合体的双色图复合体。
We introduce and study the notion of a dual Feynman transform of a modular operad. This generalizes and gives a conceptual explanation of Kontsevich's dual construction producing graph cohomology classes from a contractible differential graded Frobenius algebra. The dual Feynman transform of a modular operad is indeed linear dual to the Feynman transform introduced by Getzler and Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman transform, the dual notion admits an extremely simple presentation via generators and relations; this leads to an explicit and easy description of its algebras. We discuss a further generalization of the dual Feynman transform whose algebras are not necessarily contractible. This naturally gives rise to a two-colored graph complex analogous to the Boardman-Vogt topological tree complex.