Holograph in noncommutative geometry: Part 1

TL;DR

This paper explores how holography emerges from noncommutative geometry, linking gravity and gauge theories, and suggests that black hole properties like radius and entropy are quantized due to underlying discrete structures.

Contribution

It demonstrates the emergence of holography from noncommutative geometry and derives quantization conditions for black hole parameters based on geometrical and spectral principles.

Findings

Gravity on manifolds with boundary can be equivalent to boundary gauge theories.

Black hole radius is quantized according to the geometrical structure.

Extremal Reissner-Nordström black holes have zero temperature but finite entropy.

Abstract

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.