DSI-QG: Discrete Symmetry Quantum Gravity

• DSI-QG is a discrete symmetry-inherited quantum renormalization group framework that links 2D CFT topological defects with nonperturbative 3D AdS gravity.
• It unifies continuous T0 deformation and discrete tensor network state-sum formulations to encode Virasoro and modular symmetry data in the RG flow.
• The approach builds a holographic dictionary by leveraging quantum 6j symbols and Frobenius algebra structures to connect boundary CFT properties with bulk gravitational dynamics.

DSI-QG refers to the discrete symmetry-inherited quantum renormalization group (RG) approach to quantum gravity, as developed in "QG from SymQRG: AdS$_3$/CFT$_2$ Correspondence as Topological Symmetry-Preserving Quantum RG Flow" (Bao et al., 16 Dec 2024). The paradigm asserts that constructing a discrete, symmetry-preserving quantum RG kernel for a two-dimensional conformal field theory (CFT)—incorporating all topological defect lines (notably Virasoro)—is mathematically equivalent to defining the nonperturbative bulk path integral of three-dimensional AdS quantum gravity. This approach unifies continuous (T$\bar{T}$ deformation) and discrete (tensor network state-sum) formulations and leads to a background-independent, symmetry-driven holographic correspondence.

1. Topological Symmetry Preservation and RG Kernels

The foundational principle of DSI-QG is that the complete network of topological symmetry defect lines in a CFT—such as those associated with the Virasoro algebra—encodes all the nonlocal fusion and crossing information needed to fully classify local and nonlocal field transformations. The quantum RG procedure (SymQRG) explicitly projects operations onto the symmetry sectors defined by these defects, ensuring that only symmetry-allowed combinations are preserved under coarse-graining.

The RG kernel resulting from this symmetry-preserving flow can be recast as the quantum path integral of a Symmetry Topological Field Theory (SymTFT) in one higher dimension. When the set of symmetries includes the full Virasoro network, the corresponding SymTFT is identified as the three-dimensional AdS quantum gravity path integral. The equivalence is captured in the concise slogan:

$\text{SymQRG (Virasoro lines)} = \text{QG}$

This identification holds continuously—for RG flows realized via T$\bar{T}$ deformation—and discretely, via tensor network state sums assembled from modular and crossing data.

2. T$\bar{T}$ Deformation and Wheeler–DeWitt Constraints

The continuous realization employs T$\bar{T}$ deformation of a 2D CFT as the RG flow generator. T$\bar{T}$, an irrelevant double-trace operator, drives the theory toward the UV, but when interpreted as RG evolution, it reorganizes the operator content in a way that reflects quantum geometric averaging.

Iterative applications of T$\bar{T}$ deformation correspond to a family of integral transforms. The kernel of these transforms is precisely the 3D quantum gravitational path integral. This construction automatically promotes the background metric of the CFT to a dynamical quantum variable. The ground state of the corresponding SymTFT satisfies a nonperturbative constraint that is identified as the Wheeler–DeWitt equation, encoding the diffeomorphism invariance and Hamiltonian constraint typically imposed in canonical quantum gravity.

3. Discrete Tensor Network Construction and Quantum 6j Symbols

In the discrete formulation, the RG transformation is implemented as a tensor network constructed from state sums over topological and physical boundary conditions in three dimensions. The partition function of the 2D CFT is expressed as an overlap between two ground states: one, $|\Psi\rangle$, built from boundary operators and dynamical OPE coefficients (Levin–Wen or Turaev–Viro model), and one, $\langle\Omega|$, assembled from Virasoro conformal blocks.

Coarse-graining consists of inserting tensor layers weighted by quantum $6j$ symbols of $U_q(SL(2,\mathbb{R}))$, reflecting the quantum group extension of the fusion algebra. In the semiclassical regime, these $6j$ symbols reproduce the geodesic lengths and hyperbolic volumes characteristic of AdS$_3$ geometry (and BTZ black holes). The modularity and crossing symmetry of the CFT become topological constraints on the 3D bulk—specifically, the Wheeler–DeWitt constraint arises as the consistency condition for triangulation invariance.

4. Algebraic Structure and Topological Boundary Conditions

The sandwich construction—where the CFT is realized as an overlap between topological and physical boundaries of SymTFT—relies on the full algebraic specification of boundary conditions. This is parameterized by symmetric special Frobenius algebras (in the Moore–Seiberg category framework), which dictate the structure of topological lines, boundary condition changes, and operator content.

Different choices of boundary algebra correspond to different bulk duals, determining which AdS$_3$ geometries are summed over. Algebras controlling condensation (anyon or higher-form gauging) provide a natural resolution to the factorization puzzle in AdS/CFT: the topological boundary uniquely fixes duality relations and enforces the required constraints for wormhole factorization.

5. Holography and Universality

DSI-QG supplies the first evidence for a universal holographic principle in which topological symmetry preservation in RG flows implies maximal topological holography. The discrete RG kernel, constructed to inherit all topological and conformal symmetries of the boundary CFT, yields a complete, nonperturbative path integral for quantum gravity in the bulk. In the limit where the number of primary fields becomes large (as in Liouville theory), the bulk sum recovers Einstein classical gravity.

The approach unifies disparate aspects of AdS/CFT and topological holography, suggesting that the "maximal" holographic correspondence is governed by the preservation of all topological defects and symmetry structures in RG flows. This framework encodes the full algebraic (Moore–Seiberg and Frobenius algebra) data needed for a precise holographic dictionary and offers a nonperturbative, background-independent construction of gravity duals for 2D CFTs.

6. Mathematical Representations

Key mathematical objects and relations include:

• Quantum $6j$ symbols associated with $U_q(SL(2,\mathbb{R}))$
• The state sum representation for the partition function via tensor networks
• Frobenius algebraic data and module category structure governing boundary conditions
• Wheeler–DeWitt constraint equations as consistency conditions on SymTFT ground states

The integrability and modular invariance inherent to the DSI-QG framework guarantee topological invariance of the bulk path integral across triangulations.

7. Implications and Open Directions

This approach bridges continuous and discrete perspectives on quantum gravity, providing an explicit, finite-$N$ nonperturbative construction of holographic duals. It suggests new pathways for the paper of factorization and wormhole puzzles, quantum error correction in tensor networks, and the precise role of higher categorical symmetries in the emergence of bulk geometry.

A plausible implication is that future work on symmetry-preserving quantum RG flows in more general dimensions and with different symmetry categories may yield further universal holographic correspondences, extending "DSI-QG" principles beyond AdS$_3$/CFT$_2$.

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The DSI-QG framework thus synthesizes algebraic, tensor network, and RG principles to provide a symmetry-maximal, nonperturbative, and topologically controlled realization of the AdS/CFT correspondence.