从 Hurwitz 数到 Feynman 图：对数引力中有根树的计数
From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
AI总结：
本文证明临界拓扑大质量引力对数扇区的配分函数可表示为有根树求和，揭示其中 Burgers 和 KP 可积层级的 τ 函数隐藏着 Faà di Bruno 代数，并将 Hurwitz 数解释为有根树同构类的系数，从而连接可积层级、组合 Hopf 代数与有根树。
摘要：
我们证明，临界拓扑大质量引力的对数扇区的配分函数——它表示函数复合的级数展开——可以表示为对有根树的求和。我们的工作在数学物理中的可积层级、组合 Hopf 代数与有根树之间建立了联系，通过解释对数引力配分函数中出现的（势）Burgers 和 KP 可积层级的 τ 函数如何隐藏着函数复合的 Hopf 代数，即 Faà di Bruno 代数，它与著名的有根树和 Feynman 图的 Connes-Kreimer Hopf 代数属于同一类型。特别地，配分函数中出现的 Hurwitz 数作为有根树同构类的系数出现。我们将这些发现与统计物理文献中关于树上具有淬火无序的某些系统的既有结果进行了类比，这些系统与允许行波解的非线性偏微分方程相关联。鉴于对对数引力中所观察到的无序的进一步描述，这一点应当具有特别的兴趣。
英文摘要：
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.
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