SciPost Phys. 12, 132 (2022) ·
published 19 April 2022
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· pdf
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 $\psi^{new}_{\mu \nu}$ can be regarded as a branch point
field associated to the branch point $\mu l =1$.
