[Paper Review] Quantum Gravity and Random Tensors

Razvan Gurău, Vincent Rivasseau|arXiv (Cornell University)|Jan 24, 2024

Black Holes and Theoretical Physics4 citations

TL;DR

This paper proposes random tensor models as a framework for quantum gravity in higher dimensions, generalizing random matrix theory to tensor fields. It establishes a large N $1/N$ expansion governed by melonic graphs, derives conformal field theories with melonic fixed points, and identifies asymptotically free bosonic theories in 4D via imaginary couplings, offering new tools for AdS/CFT and quantum gravity.

ABSTRACT

Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss in the first part of this note the applications of such models to quantum gravity. In a second part we review tensor field theories, that is standard field theories in $\mathbb{R}^d$ but with tensor fields, which lead to a new family of large $N$ conformal field theories relevant for the study of the $AdS/CFT$ correspondence.

Motivation & Objective

• To extend random matrix theory to higher-order tensors for modeling random higher-dimensional geometries in quantum gravity.
• To develop tensor field theories that yield large N conformal field theories relevant for the AdS/CFT correspondence.
• To identify universal features in random tensor models, particularly the melonic large N limit, as a starting point for non-perturbative quantum gravity.
• To explore the renormalization group flow in tensor field theories and identify fixed points with conformal invariance.
• To investigate the possibility of asymptotically free bosonic field theories in 4D using imaginary couplings in the tensor model action.

Proposed method

• Utilizes a $1/N$ expansion of random tensor models, where $N$ is the Hilbert space dimension, indexed by a non-topological degree invariant.
• Applies the melonic graph family as the leading-order contribution in the $1/N$ expansion, providing a universal and analytically tractable sector.
• Constructs tensor field theories in $\mathbb{R}^d$ with quartic interactions, generalizing the SYK model to higher-order tensor fields.
• Derives beta functions for couplings using renormalization group techniques, including the cubic beta function for the tetrahedral coupling.
• Applies conformal partial wave techniques to compute OPE coefficients and verify conformal invariance at fixed points.
• Uses non-perturbative resummation of diagrams to confirm the $F$-theorem and verify monotonicity of sphere free energy.

Experimental results

Research questions

• RQ1 Can random tensor models provide a consistent framework for quantum gravity in dimensions greater than two?
• RQ2 What is the role of melonic graphs in the large $N$ limit of tensor models, and how do they govern the $1/N$ expansion?
• RQ3 Do tensor field theories admit conformal fixed points that can be used to model CFTs relevant for AdS/CFT?
• RQ4 Can asymptotically free bosonic field theories be realized in 4D via imaginary couplings in tensor models?
• RQ5 What are the invariant quantities—such as spectra or eigenvalues—characterizing random tensors, and how do they relate to physical observables?

Key findings

• The $1/N$ expansion of random tensors is indexed by a non-topological degree invariant, generalizing the genus expansion of random matrices.
• The melonic graphs dominate the large $N$ limit and give rise to a universal, analytically tractable sector of random tensor models.
• Tensor field theories in $\mathbb{R}^d$ exhibit melonic fixed points and conformal invariance, with OPEs dominated by bilinear primaries.
• The beta function for the tetrahedral coupling in $d=4$ is cubic, and flipping its sign via imaginary coupling yields an asymptotically free bosonic theory.
• The model respects the $F$-theorem: the sphere free energy decreases monotonically from UV to IR, confirmed via conformal partial wave resummation.
• The spectrum of the tensor resolvent provides a viable alternative to tensor eigenvalues, though its relation to physical eigenvalues remains non-trivial.