Energy flow and fluctuations in non-equilibrium conformal field theory on star graphs

We consider non-equilibrimn quantum steady states in conformal field theory (CFT) on star-graph configurations. with a particular, simple connection condition at the vertex of the graph. These steady states occur after a large time as a result, of initially thermalizing the legs of the graph at different temperatures. and carry energy flows. Using purely Virasoro algebraic calculations we evaluate the exact scaled cumulant, generating function for these flows. We show that this function satisfies a generalization of the usual non-equilibrium fluctuation relations. This extends results by two of the authors to the case of more than two legs. It also provides an alternative derivation centered on Virasoro algebra operators rather than local fields. hence an alternative regularization scheme thus confirming the validity and universality of the scaled cumulant generating, function. Our derivation shows how the usual Virasoro algebra leads. in large volumes, to a continuous-index Virasoro algebra for which we develop diagrammatic principles, winch may be of interest in other non-equilibrium contexts as well. Finally, our results shed light on the Poisson-process interpretation of the long-time energy transfer in CFT.