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What Liouville, Painlevé e Virasoro have to say about black holes?

Bruno Carneiro da Cunha
Federal University of Pernambuco
Tuesday, July 9, 2019 - 14:00

A number of interesting problems in physics can be phrased in terms of connection problems of ordinary differential equations.  These are deeply connected to a class of
Riemann-Hilbert problems, of isomonodromy type,
which was recently solved using techniques from conformal field theory.
In this talk, I’ll summarize my efforts to understand the simplest non-trivial case, which comprises the connection problem for the Heun equation. These arise in a number of problems in physics, such as black hole scattering and the construction of
uniformizing maps in complex analysis.
The relevant tau function in this case is the famous
Painlevé transcendents of types V and VI,
which can be expanded in terms of
Nekrasov functions at c=1
or Fredholm determinants.
I will show that the accessory parameters of the Heun equation can be obtained from
zeros of the tau function,
and discuss analytical and numerical implementations of the method in order to calculate
quasi-normal modes of black holes.

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