Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity

Abstract

We discuss integrable aspects of the logarithmic contribution of the partition function of cosmological critical topologically massive gravity. On one hand, written in terms of Bell polynomials which describe the statistics of set partitions, the partition function of the logarithmic fields is a generating function of the potential Burgers hierarchy. On the other hand, the polynomial variables are solutions of the Kadomtsev-Petviashvili equation, and the partition function is a KP $\tau$ function, making more precise the solitonic nature of the logarithmic fields being counted. We show that the partition function is a generating function of Hurwitz numbers, and derive its expression. The fact that the partition function is the generating function of branched coverings gives insight on the orbifold target space. We show that the logarithmic field ${\psi }_{\mu \nu }^{new}$ can be regarded as a branch point field associated to the branch point $\mu l=1$.

arXiv comment21 pages, extended version, 2 figures added, references added

References in corpus (42)

• The Black Hole in Three Dimensional Space Time
• Logarithmic Operators in Conformal Field Theory
• Quantum Gravity Partition Functions in Three Dimensions
• Chiral Gravity in Three Dimensions
• Three-Dimensional Gravity Revisited
• Counting Gauge Invariants: the Plethystic Program
• Bits and Pieces in Logarithmic Conformal Field Theory
• Instability in cosmological topologically massive gravity at the chiral point
• Chiral Gravity, Log Gravity and Extremal CFT
• An algebraic approach to logarithmic conformal field theory
• Topologically Massive Gravity and the AdS/CFT Correspondence
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
• Algebra of differential operators associated with Young diagrams
• Conformal Algebras of 2D Disordered Systems
• Graviton 1-loop partition function for 3-dimensional massive gravity
• Conformal Field Theory at central charge c=0 and Two-Dimensional Critical Systems with Quenched Disorder
• AdS_3/LCFT_2 - Correlators in Cosmological Topologically Massive Gravity
• Holographic applications of logarithmic conformal field theories
• Matrix Models for Random Partitions
• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• 2-Dimensional Turbulence: yet another Conformal Field Theory Solution
• Logarithmic Correlations in Quenched Random Magnets and Polymers
• Constrained KP Hierarchies: Additional Symmetries, Darboux-Bäcklund Solutions and Relations to Multi-Matrix Models
• Virasoro constraints for Kontsevich-Hurwitz partition function
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• A Logarithmic Conformal Field Theory Solution For Two Dimensional Magnetohydrodynamics In Presence of The Alf'ven Effect
• Genus expansion of HOMFLY polynomials
• Quantization of conical spaces in 3D gravity
• Complex Geometry of Matrix Models
• Logarithmic Correlation Functions in Two Dimensional Turbulence
• On the combinatorics of partition functions in AdS3/LCFT2
• The Alf'ven Effect and Conformal Field Theory
• Logarithmic Conformal Field Theory Solutions of Two Dimensional Magnetohydrodynamics
• Moduli space of logarithmic states in critical massive gravities
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Set partitions and integrable hierarchies
• On the combinatorics of several integrable hierarchies
• String Theory and Integrable Systems
• String theory and classical integrable systems
• Formulae for calculating Hurwitz numbers
• Remarks on Combinatorial Aspects of the KP Equation

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• Fragmented perspective of self-organized criticality and disorder in log gravity
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point
• On palindromic numerators of bigraded symmetric orbifold Hilbert series and Kostka-Foulkes polynomials
• Monodromy, Logarithmic Sectors, and Two-Point Functions in Critical Topologically Massive Gravity