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We study the origin of the recently proposed effective theory of stresstensor exchanges based on reparametrization modes, that has been used toefficiently compute Virasoro identity blocks at large central charge. We firstprovide a derivation of the nonlinear Alekseev--Shatashvili action governingthese reparametrization modes, and argue that it should be interpreted as thegenerating functional of stress tensor correlations on manifolds related to theplane by conformal transformations. In addition, we demonstrate that the rulespreviously prescribed with the reparametrization formalism for computingVirasoro identity blocks naturally emerge when evaluating Feynman diagramsassociated with stress tensor exchanges between pairs of external primaryoperators. We make a few comments on the connection of these results togravitational theories and holography.

Reparametrization modes in 2d CFT and the effective theory of stress tensor exchanges