There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.

Fractional quantum Hall (FQH) system at Landau level filling fraction $\nu=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance $\sigma_{xy}=5e^2/2h$. Thermal Hall conductances of the Pf and APf states are quantized at $\kappa_{xy}=7/2$ and $\kappa_{xy}=3/2$ respectively in a proper unit. However, a recent experiment shows the thermal Hall conductance of $\nu=5/2$ FQH state is $\kappa_{xy}=5/2$. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a $\kappa_{xy}=5/2$ phase. In this work, we do perturbative and non-perturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold $\Lambda_1$, which is lowered to an emergent U(1)$\times$U(1) symmetry at energy scales between $\Lambda_1$ and a higher value $\Lambda_2$, and is finally lowered to the composite fermion parity symmetry $\mathbb{Z}_2^F$ above $\Lambda_2$. Based on the emergent symmetries, we propose a specific phase diagram of the disordered $\nu=5/2$ FQH system, and show that a $\kappa_{xy}=5/2$ phase arises at disorder energy scales $\Lambda>\Lambda_1$. Furthermore, we show the gapped double-semion sector of $N_D$ closed domain walls contributes non-local topological degeneracy $2^{N_D-1}$, causing a low temperature peak in the heat capacity. We also implement a non-perturbative method to bootstrap generic topological 1+1D domain walls (2-surface defects) applicable to any 2+1D non-Abelian topological order.

Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and with an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, \emph{always existent} for \emph{any} SPT state in any spacetime dimension $d \geq 2$ of \emph{any} finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological {emergent} $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has \emph{spontaneous symmetry breaking} with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped boundaries/interfaces obtained via \emph{symmetry breaking} (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on \emph{symmetry extension}. More generally, applying our approach to SPT states, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of \emph{symmetry breaking}, \emph{symmetry extension}, and \emph{dynamical gauging}.