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 $\mu l=1$.

21 pages, extended version, 2 figures added, references added

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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