Universal fine grained asymptotics of weakly coupled Quantum Field Theory.

We give a proof that in any weakly coupled quantum field theory with a finite group global symmetry $\mathrm{G}$, on a spatial manifold $S^{d-1}$, at sufficiently high energy, the density of states $\rho_\alpha(E)$ for each irreducible representation $\alpha$ of $\mathrm{G}$ obeys a universal formula as conjectured by Harlow and Ooguri. This generalizes similar results that were previously obtained in $(1+1)$-D to higher spacetime dimension. The basic idea of the proof relies on the approximate existence of a Hilbert series in the weakly coupled regime, and is also applicable to the calculation of twisted supersymmetric indices. We further compare and contrast with the scenario of continuous $\mathrm{G}$, where we prove a universal, albeit different, behavior. We discuss the role of averaging in the density of states.