[Paper Review] Approximate CFTs and Random Tensor Models

Alexandre Belin, Jan de Boer+3 more|arXiv (Cornell University)|Aug 7, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper introduces approximate conformal field theories (CFTs) as ensembles of CFT data satisfying conformal symmetry and locality constraints approximately, enabling a statistical description of quantum chaotic systems. It proposes a random tensor model in AdS₃ governed by the Virasoro 6j-symbol that realizes a maximum ignorance ensemble compatible with crossing symmetry and a primary gap, offering a holographic realization of pure gravity as an ensemble of approximate CFTs.

ABSTRACT

A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of `CFT data'. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of `CFT data' satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS$_3$ as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

Motivation & Objective

• To develop a framework for ensembles of CFT data that model quantum chaotic systems, analogous to random matrix ensembles in many-body quantum systems.
• To define approximate CFTs as CFT data satisfying conformal symmetry and locality constraints up to small deviations, enabling statistical description of chaotic CFTs.
• To construct a random tensor model in AdS₃ that realizes a maximum ignorance ensemble respecting conformal symmetry, crossing invariance, and a primary gap to the black hole threshold.
• To provide a holographic realization of pure gravity in AdS₃ as a random matrix/tensor model governed by the Virasoro 6j-symbol.
• To address the absence of natural ensembles in higher-dimensional CFTs by proposing a statistical framework compatible with quantum chaos and holography.

Proposed method

• Introduce the concept of approximate CFTs as CFT data satisfying conformal symmetry and locality constraints up to small, controlled deviations.
• Construct ensembles of approximate CFTs by imposing successively higher moments of the CFT constraints, ensuring consistency with semi-classical gravity.
• Propose a random tensor model in AdS₃ whose dynamics are governed by the Virasoro 6j-symbol, encoding the fusion rules of the CFT in a random, ensemble-averaged way.
• Use the Virasoro 6j-symbol to implement crossing symmetry and conformal block decomposition in a statistical ensemble, ensuring consistency with the analytic bootstrap.
• Define the ensemble as the maximum ignorance model compatible with conformal symmetry, crossing invariance, and a primary gap to the black hole threshold.
• Demonstrate that individual members of the ensemble are indistinguishable from true CFTs for low-energy probes, consistent with semi-classical gravity.

Experimental results

Research questions

• RQ1 How can one define a statistical ensemble of CFTs that captures the ergodic behavior of quantum chaotic systems in higher dimensions, analogous to random matrix ensembles in quantum chaos?
• RQ2 What constraints must be imposed on CFT data to define a consistent ensemble of approximate CFTs that are indistinguishable from true CFTs in semi-classical gravity?
• RQ3 Can a random tensor model in AdS₃ realize a holographic dual of pure gravity by encoding the Virasoro 6j-symbol and satisfying crossing symmetry and a primary gap?
• RQ4 How does the inclusion of wormholes and ensemble averaging in higher-dimensional gravity relate to the statistical distribution of black hole microstates and unitarity?
• RQ5 What is the role of the Virasoro 6j-symbol in constructing a maximum ignorance ensemble for AdS₃ gravity that respects conformal symmetry and crossing invariance?

Key findings

• Approximate CFTs exist generically as CFT data satisfying conformal symmetry and locality constraints up to small deviations, providing a statistical framework for quantum chaotic CFTs.
• Ensembles of approximate CFTs are naturally realized in holography, as individual members are indistinguishable from true CFTs under low-energy probes governed by semi-classical gravity.
• The proposed random tensor model in AdS₃ is governed by the Virasoro 6j-symbol, encoding the fusion rules and ensuring consistency with crossing symmetry and conformal block decomposition.
• The model realizes a maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black hole threshold, making it a candidate for pure gravity in AdS₃.
• The framework provides a holographic realization of ensemble averages in higher-dimensional quantum gravity, extending the duality between JT gravity and matrix models to higher dimensions.
• The construction suggests that semi-classical gravity may describe only the mean and higher moments of a statistical ensemble of CFTs, consistent with quantum chaos and the factorization puzzle.