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

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