From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity

Abstract

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 $\tau$-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.

arXiv commentMajor changes to match published version

References in corpus (33)

• The Black Hole in Three Dimensional Space Time
• Hopf Algebras, Renormalization and Noncommutative Geometry
• Quasiparticle Lifetime in a Finite System: A Non--Perturbative Approach
• Quantum Gravity Partition Functions in Three Dimensions
• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Chiral Gravity in Three Dimensions
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Instability in cosmological topologically massive gravity at the chiral point
• Chiral Gravity, Log Gravity and Extremal CFT
• Topologically Massive Gravity and the AdS/CFT Correspondence
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Combinatorial Hopf algebras in quantum field theory I
• Two interacting Hopf algebras of trees
• From Free Fields to $AdS$ -- III
• Algebra of differential operators associated with Young diagrams
• Conformal Algebras of 2D Disordered Systems
• Graviton 1-loop partition function for 3-dimensional massive gravity
• Ward identities and combinatorics of rainbow tensor models
• Holographic applications of logarithmic conformal field theories
• Matrix Models for Random Partitions
• Rainbow tensor model with enhanced symmetry and extreme melonic dominance
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• Genus expansion of HOMFLY polynomials
• The matrix model for hypergeometric Hurwitz numbers
• Bogolubov's Recursion and Integrability of Effective Actions
• Integrable Renormalization I: the Ladder Case
• Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization
• On Kreimer's Hopf algebra structure of Feynman graphs
• On the combinatorics of partition functions in AdS3/LCFT2
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• On the combinatorics of several integrable hierarchies
• Set partitions and integrable hierarchies

Cited by in corpus (3)

• Fragmented perspective of self-organized criticality and disorder in log gravity
• Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point
• Monodromy, Logarithmic Sectors, and Two-Point Functions in Critical Topologically Massive Gravity