QG from SymQRG: AdS$_3$/CFT$_2$ Correspondence as Topological Symmetry-Preserving Quantum RG Flow

Abstract

By analyzing the non-perturbative RG flows that explicitly preserve given symmetries, we demonstrate that they can be expressed as quantum path integrals of the $\textit{SymTFT}$ in one higher dimension. When the symmetries involved include Virasoro defect lines, such as in the case of $T\bar{T}$ deformations, the RG flow corresponds to the 3D quantum gravitational path integral. For each 2D CFT, we identify a corresponding ground state of the SymTFT, from which the Wheeler-DeWitt equation naturally emerges as a non-perturbative constraint. These observations are summarized in the slogan: $\textbf{SymQRG = QG}$. The recently proposed exact discrete formulation of Liouville theory in [1] allows us to identify a universal SymQRG kernel, constructed from quantum $6j$ symbols associated with $U_q(SL(2,\mathbb{R}))$. This kernel is directly related to the quantum path integral of the Virasoro TQFT, and manifests itself as an exact and analytical 3D background-independent MERA-type holographic tensor network. Many aspects of the AdS/CFT correspondence, including the factorization puzzle, admit a natural interpretation within this framework. This provides the first evidence suggesting that there is a universal holographic principle encompassing AdS/CFT and topological holography. We propose that the non-perturbative AdS/CFT correspondence is a $\textit{maximal}$ form of topological holography.