交换图与两个Symanzik多项式之比的变化
The exchange graph and variations of the ratio of the two Symanzik polynomials
摘要：
量子场论中的相关函数是用费曼振幅来计算的，费曼振幅是与图形相关联的有限维积分。被积函数是与费曼图相关联的第一和第二Symanzik多项式之比的指数，它们分别用图的生成树和生成2-森林来描述。在以前的文件与布洛赫，布尔戈斯和Fresán，我们有关这一比例的渐近阿基米德高度配对度零因子退化家庭的黎曼曲面。受此启发，在本文中，我们认为在有界扰动下的图形的几何形状的两个Symanzik多项式的比率的变化。这是一个自然的问题与理论的幂零和SL 2轨道在霍奇理论。我们的主要结果是比的变化的有界性。为此，我们定义了交换图的一个给定的图，它编码的交换属性之间的生成树和生成2-森林的图。我们提供了这个图的连通分量的描述，并使用它来证明我们的结果的变化的有界性。
英文摘要：
Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fresán, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory. Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a description of the connected components of this graph, and use this to prove our result on boundedness of the variations.
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