Heun Connection Formulae from Liouville Correlators

We perform a detailed study of a class of irregular correlators in Liouville Conformal Field Theory, of the related Virasoro conformal blocks with irregular singularities and of their connection formulae. Upon considering their semi-classical limit, we provide explicit expressions of the connection matrices for the Heun function and a class of its confluences. Their calculation is reduced to concrete combinatorial formulae from conformal block expansions. Since Heun functions solve wave equations on various black holes backgrounds, we exploit our result to give exact expressions for different observables in black hole physics such as greybody factors, quasinormal modes and Love numbers. In the context of anti de Sitter black holes, we use our connection formulas in order to give novel exact expressions for thermal correlators of the boundary theory.