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Kerr Scattering Coefficients via Isomonodromy

We study the scattering of a massless scalar field in a generic Kerr background. Using a particular gauge choice based on the current conservation of the radial equation, we give a generic formula for the scattering coefficient in terms of the composite monodromy parameter $σ$ between the inner and the outer horizons. Using the isomonodromy flow, we calculate $σ$ exactly in terms of the Painlevé V $τ$-function. We also show that the eigenvalue problem for the angular equation (spheroidal harmonics) can be calculated using the same techniques. We use recent developments relating the Painlevé V $τ$-function to Liouville irregular conformal blocks to claim that this scattering problem is solved in the combinatorial sense, with known expressions for the $τ$-function near the critical points.