论文标题
从Hurwitz数字到Feynman图：在日志重力中计数根树
From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
论文作者
论文摘要
我们表明，临界拓扑上的重力的对数扇区的分区函数代表功能组成的一系列扩展，可以表示为在根树上的总和。我们的工作通过解释（潜在的）汉堡的$τ$ functions of（潜在的）汉堡的$τ$ functions of Logfratity soptition of Function of fa aim a di brun di di di di di di di di di di di di di di di，我们的工作带来了数学物理学，组合hopf代数和植根的树木之间的联系。植根树和Feynman图的Connes-Kreimer Hopf代数。特别是，出现在分区函数中的赫维兹数是植根树的同构类别的系数。在我们的发现和确定的结果之间，在统计物理学文献中绘制了一个相似之处，该文献涉及某些在树木上具有淬火障碍的系统，与非线性偏微分方程相关，该方程接纳了行驶波解决方案。鉴于对原木重力观察到的疾病的进一步描述，这应该特别令人感兴趣。
We show that the partition function of the logarithmic sector of critical topologically massive gravity which represents a series expansion of composition of functions, can be expressed as a sum over rooted trees. Our work brings a connection between integrable hierarchies of mathematical physics, combinatorial Hopf algebras and rooted trees, by explaining how the $τ$-functions of the (potential) Burgers and KP integrable hierarchies appearing in the partition function of log gravity conceal the Hopf algebra of composition of functions, known as the Faà di Bruno algebra, of the same type as the celebrated Connes-Kreimer Hopf algebra of rooted trees and Feynman diagrams. In particular, the Hurwitz numbers appearing in the partition function arise as coefficients of isomorphism classes of rooted trees. A parallel is drawn between our findings and established results in the statistical physics literature concerning certain systems with quenched disorder on trees, associated to nonlinear partial differential equations admitting traveling wave solutions. This should be of particular interest in view of a further description of the disorder observed in log gravity.
