Holograph in noncommutative geometry: Part 1

In this paper, we consider the holograph principle emergent from noncommutative geometry, based on the spectral action principle. We show that under some appropriate conditions, the gravity theory on a manifold with boundary could be equivalent to a gauge theory $SU(N)$ on the boundary. Then an expression for $N$ with the geometrical quantities of the manifold is given. Based on this result, we find that the volume of the manifold and the boundary have some discrete structure. Applying the result to the black hole, we get that the radium of the Schwarzschild black hole is quantized. We also find an explanation why the extremal RN-black hole has zero temperature but with finite entropy.