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Employing the conformal welding technique, we find an exact expression forthe Full Counting Statistics of energy transfers in a class of inhomogeneousnonequilibrium states of a (1+1)-dimensional unitary Conformal Field Theory.The expression involves the Schwarzian action of a complex field obtained bysolving a Riemann-Hilbert type problem related to conformal welding of infinitecylinders. On the way, we obtain a formula for the extension of characters ofunitary positive-energy representations of the Virasoro algebra to 1-parametergroups of circle diffeomorphisms and we develop techniques, based on theanalysis of certain classes of Fredholm operators, that allow to control theleading asymptotics of such extensions for small real part of the modularparameter \tau

Full counting statistics of energy transfers in inhomogeneous nonequilibrium states of (1+1)D CFT