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

By analyzing non-perturbative RG flows that explicitly preserve given topological symmetries, we show that each of their coarse-graining steps can be expressed as a quantum path integral of the SymTFT in one higher dimension.

When the symmetries involved include the Virasoro defect lines, such as in the case of $T\bar{T}$ deformations, this RG kernel is the 3D quantum gravitational path integral.

For 2D CFTs whose structure constants are under control, we identify the corresponding ground state of the SymTFT, from which the Wheeler-DeWitt equation emerges as the non-perturbative no-flux constraint: the gravitational path integral acts on this state as the projector that renders symmetry-preserving coarse-graining exact.

These observations are summarized in the slogan: $\textbf{SymQRG = QG}$.

The exact discrete formulation of Liouville theory in \cite{Chen:2024unp} allows us to identify a universal SymQRG kernel, constructed from quantum $6j$ symbols of $U_q(SL(2,\mathbb{R}))$: it provides a discrete realization of two copies of the Virasoro TQFT on $\Sigma\times I$, and manifests as an exact and analytic 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.

We propose that the non-perturbative AdS$_3$/CFT$_2$ correspondence is a \textit{maximal} form of topological holography, in which the physical boundary is fixed by kinematics alone.