Abstract

The theme of this talk is to see what AdS-CFT brings to the table in terms of understanding curved spacetime QFT . The idea is to explore how general properties of holographic CFTs with a gravity dual can be understood in terms of a purely geometric problem, and studied using geometric tools. We consider holographic CFTs on static spacetimes, where the spatial part is compact but otherwise general. We assume the dual gravitational description is given by pure gravity with a negative cosmological constant (ie. the universal sector). Firstly I will discuss how one can show for a 2+1 dimensional CFT that the vacuum (or Casimir) energy is always negative, unless the space the CFT lives on has constant curvature, when the energy may also vanish. This argument does not assume a smooth bulk geometry (although any singularities have to be ‘good’). Secondly I will show that in any dimension the spectrum of CFT fluctuations associated to scalar operators is bounded from below by the minimum scalar curvature of the space the CFT lives on. Normalised by this quantity the `gap’ in the spectrum of a scalar operator is minimized when the CFT is living on a sphere. In fact this also holds for a conformally coupled free scalar. Optimistically one might even conjecture such a bound holds more generally for CFTs.