从操作数和PROPs到费曼过程

The equivalence between Feynman transform and Verdier duality

The equivalence between dg duality and Verdier duality has been established for cyclic operads earlier. We propose a generalization of this correspondence from cyclic operads and dg duality to twisted modular operads and Feynman transform. Specifically, for each twisted modular operad $\mathcal{P}$ (taking values in dg-vector spaces over a field $k$ of characteristic 0), there is a certain sheaf $\mathcal{F}$ associated with it on the moduli space of stable metric graphs such that the Verdier dual sheaf $D\mathcal{F}$ is associated with the Feynman transform $F\mathcal{P}$ of $\mathcal{P}$. In…

x物理学的操作

Operads for x-physics

对运算代数的基本内容作了简明扼要的介绍，这对运算物理学的研究有一定的参考价值。也阐明了Gerstenhaber代数如何与线性预运算（comp代数）相关联。简要地讨论了它们与力学的关系。提出了费曼图是可观测量的假设。

Feynman图与运算代数的极小模型

Feynman diagrams and minimal models for operadic algebras

本文构造了微分分次算子的cobar-结构上代数的显式极小模型。这个极小模型的结构映射用装饰树上的和表示。我们引入了适当的概念，一个同伦等价的运算代数，并表明，我们的极小模型同伦等价于原来的代数。这些结果推广了A-无穷代数的一些著名结果，并给出了概念上的解释。此外，我们表明，这些结果结转到代数的情况下，模块操作，在树的总和得到取代一般费曼图的总和。作为我们的工作的副产品，我们证明规范独立的Kontsevich的“双建设”生产图上同调类从可收缩微分分次Frobenius代数。

模运算的对偶Feynman变换

Dual Feynman transform for modular operads

本文引入并研究了模运算的对偶费曼变换的概念。这推广并给出了一个概念性的解释Kontsevich的对偶结构产生图上同调类从一个可收缩微分分次Frobenius代数。一个模运算的对偶费曼变换在真空图上计算时，确实是Getzler和Kapranov引入的费曼变换的线性对偶。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简单的描述。我们讨论了对偶Feynman变换的进一步推广，其代数不一定是可收缩的。这自然会产生类似于Boardman-Vogt拓扑树复形的双色有色图复形。

Regular patterns, substitudes, Feynman categories and operads

We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem (exploiting Power's General Coherence Result) which allows to reduce to the case where the structure …

Decorated Feynman Categories

In [KW14], the new concept of Feynman categories was introduced to simplify the discussion of operad--like objects. In this present paper, we demonstrate the usefulness of this approach, by introducing the concept of decorated Feynman categories. The procedure takes a Feynman category $\mathfrak F$ and a functor $\mathcal O$ to a monoidal category to produce a new Feynman category ${\mathfrak F}_{dec {\mathcal O}}$. This in one swat explains the existence of non--sigma operads, non--sigma cyclic operads, and the non--sigma--modular operads of Markl as well as all the usual candidates simply fr…