# Holographic constraints on CFT energies Wiseman found a series of evidence that the free energy decreases when one deforms the spatial sphere or the flat space (periodically) of the QFT. ## Refs - [[2016#Fischetti, Hickling, Wiseman]] - bounds on local energy density - [[2019#Cheamsawat, Gibbons, Wiseman]] - bounds from studying BHs directly ## Non-holographic studies - [[FischettiWiseman2017]] - direct calculations in 2+1 dim CFT on curved backgrounds ## Results - 2+1d QFT (space either $R^2$ or $S^2$) - perturbatively - $\Delta F<0$ for free scalar and Dirac theories - likewise for holographic CFT on flat space - non-perturbatively - also holds at $T=0$ for holographic CFTs on any space - for $T>0$ holds for *energy* for periodic flat space deformations, for free theories it seems to hold (numerically) ## Related topics - [[0231 Bulk solutions for CFTs on non-trivial geometries]]