Null infinity as $SU(2)$ Chern-Simons theories and its quantization

TL;DR

This paper develops a novel approach to quantize future null infinity in asymptotically flat spacetimes using $SU(2)$ Chern-Simons theories, linking it to horizon entropy and microstate counting.

Contribution

It introduces a new quantization framework for null infinity based on Chern-Simons theories, connecting symplectic structures to horizon entropy calculations.

Findings

Null infinity's symplectic structure equals two $SU(2)$ Chern-Simons theories with opposite levels.

Quantization yields an entropy proportional to the cross-section area.

Results align with the universal entropy formula for isolated horizons.

Abstract

This paper studies the quantization of the future null infinity ($\mathscr{I}^+$) of an asymptotically flat spacetime. Based on the observation by Ashtekar and Speziale that $\mathscr{I}^+$ can be regarded as a weakly isolated horizon, we adopt the quantization framework developed for weakly horizon to quantize $\mathscr{I}^+$. We first show that the symplectic structure of $\mathscr{I}^+$ is equivalent to the sum of the symplectic structures of two $SU(2)$ Chern-Simons theories with opposite levels. Based on this observation, we apply Chern-Simons quantization approach to quantize $\mathscr{I}^+$. Finally, we compute the entropy of $\mathscr{I}^+$ by counting the microstates, showing that it is proportional to the area of $\tilde{\Delta}$, a spacelike cross-section of $\mathscr{I}^+$. Our result is consistent with the universal entropy formula in the framework of (weakly) isolated horizon.