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.

Figures (73)

• Figure 1: An illustration of the SymQRG coarse-graining procedure for the tensor network state sum representation of field theories in Sec. \ref{['tensor network']}. The orange circle in the middle of the bulk indicates the topological boundary. The second step is the crucial step where we use the topological invariance of the bulk path integral (corresponding to 2D CFT crossing) to perform re-triangulation on $\ket{\Psi}_{\Lambda}$. In the third step, we contract the state $_{\Lambda}\bra{\Omega}$ defined on the red edges with the triangle to turn it into the state $_{\Lambda'}\bra{\Omega'}$ defined on the longer green edge, realizing a symmetry-preserving block-spin transformation. Before the rescaling procedure, we land on the bottom right diagram where the field theory lattice gets coarse-grained and we probes deeper into the UV direction of the bulk. After the rescaling procedure, the overlap corresponds to a deformed path integral, and is equivalent to imposing another finite-cutoff boundary condition further towards the IR direction in the bulk. In the CFT, the fact that we move towards the IR is represented by the smaller holes after rescaling.
• Figure 2: A logical map of the tensor network SymQRG. The sandwich construction in the study of generalized symmetries allow us to express the CFT path integral as an overlap of 3D states, where the bra state corresponds to a "physical boundary condition" that controls the kinematics (conformal blocks) in the 2D CFT; the ket state is a ground state of the SymTFT, which corresponds to a "topological boundary condition" attached to the 3D bulk, and governs the dynamics (OPE coefficients) of the 2D CFT. Both states admit tensor network representations due to the locality inherent in field theories. The underlying Hilbert space is provided by the Levin-Wen string net model based on the Moore-Seiberg data of the 2D CFT. Using the crossing symmetry or re-triangulation invariance of the $\ket{\Psi}_{\Lambda}$ state, we can naturally obtain a topological symmetry-preserving RG procedure that progresses into the bulk of the SymTFT. The crossing symmetry ensures that the "Wheeler-DeWitt" equation, or the no-flux constraint, is satisfied.
• Figure 3: Comparison between three different RG procedures. Wilson-Kadanoff RG keeps all the couplings during each RG step, and the couplings are classical and non-dynamical. Quantum RG projects onto a subset of simple operators during each RG step, which makes the corresponding couplings quantum and dynamical in the emergent bulk. SymQRG projects onto simple operators classified based on symmetry charges, and the bulk generated is the symmetry-perserving SymTFT.
• Figure 4: An illustration of the sandwich construction, where any $D$-dimensional field theory is equivalent to collapsing the two boundaries of a $D+1$-dimensional SymTFT, which governs the topological symmetries of the original theory. A topological boundary condition and a physical boundary condition are specified on the two boundaries. The precise meaning of $\mathcal{C}$ and $\mathcal{M_C}$ in the context of 2D CFTs will be explained in \ref{['roleofpsi module category']}.
• Figure 5: Comparison between Feynman's path integral \ref{['Feynman']} in quantum mechanics on the left and the topological symmetry-preserving quantum RG \ref{['kernelequation1']} for quantum field theories. They both take a form of path integral in an extra dimension that sums over all possible paths connecting $x$ at different time and source $J$ at different RG time. The "bulk" in \ref{['kernelequation1']} is a space of 2D field theories with symmetries.