Approximate CFTs and Random Tensor Models

TL;DR

The paper develops a framework to describe chaotic conformal field theories via ensembles of CFT data, introducing approximate CFTs that satisfy bootstrap constraints only to finite tolerance and up to restricted observables. It demonstrates that naive Gaussian ensembles yield large crossing-variance and shows that carefully constructed non-Gaussian moments, guided by crossing kernels and Virasoro data, can suppress these fluctuations, aligning with holographic expectations. A concrete tensor/matrix model for AdS3 gravity is proposed, with a Virasoro 6j-symbol structure that maps to a Virasoro-based simplicial gravity picture and a maximal-ignorance ensemble compatible with conformal symmetry, modular invariance, and a black-hole threshold gap. The work connects bootstrap, quantum chaos, and holography through explicit constructions, offering a route to continuum gravity via a controlled double/ triple-scaling of a tensor model and highlighting the role of wormhole-like correlations in gravitational ensembles.

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.

Figures (7)

• Figure 1: Crossing equation for the 4-point function in a conformal field theory follows from the associativity of the OPE expansion and implies that the conformal block expansions given by expressions \ref{['eq.sblock']} and \ref{['eq.tblock']} should be convergent and equal.
• Figure 2: (a) The connected spectral correlation functions that arise in the field theory are captured by multi-boundary wormholes in the gravitational dual. We show here 'fixed-length' boundary conditions, fixing the inverse temperatures at each disconnected component of the boundary. The energy correlations of \ref{['eq.energyCorrs']} are obtained by inverse Laplace transform. (b) The same geometries also contribute to connected multi-boundary correlation functions, where the boundary conditions include operator insertions.
• Figure 3: A genus-three partition function can be decomposed in several different channels. We show two examples: the skyline channel (left), which is the important non-Gaussianity that we focus on here, and the pillow channel (right). The dominant contribution to the pillow channel comes from a Gaussian contraction which only takes into account the terms with $6=6'$Chandra:2022bqq.
• Figure 4: (a) A bulk geometry that computes the connected contribution of a square of four-point functions. The action of this geometry matches (\ref{['ss']}). (b) Diagrammatic representation of a putative bulk geometry that would contribute to the connected correlations of spectral densities, as estimated in \ref{['eq.ss.den.conn']}. The OPE coefficients on the two sides are not correlated. There is no known saddle that captures this correlation, and this geometry may be off-shell.
• Figure 5: The Feynman rules of the tensor model after resummation. Each quartic vertex is given by a three-simplex, weighted by a Virasoro $6j-$symbol. The propagator glues two faces of these simplices and evaluates to the '$C_0$ formula'. As we show below this re-summed two-point function gives rise to the same quadratic moment as the Gaussian model of Chandra:2022bqq. Interestingly the bare model, before resummation, has the same quadratic and quartic vertices, but with different uphysical $a$-dependent coefficients, and moreover possesses another four-valent vertex with a different index-structure, which we do not show here (see Equation \ref{['eq.V4']}). A similar comment holds for the quadratic terms (see Equation \ref{['eq.TensorPropagator']}).