  We derive a family of weighted scalar curvature monotonicity formulas for
generalized Ricci flow, involving an auxiliary dilaton field evolving by a
certain reaction-diffusion equation motivated by renormalization group flow.
These scalar curvature monotonicities are dual to a new family of Perelman-type
energy and entropy monotonicity formulas by coupling to a solution of the
associated weighted conjugate heat equation. In the setting of Ricci flow, we
further obtain a new family of convex Nash entropies and pseudolocality
principles.
