Bootstrapping Noncommutative Geometry with Dirac Ensembles

TL;DR

The work advances a nonperturbative program for quantum geometry by treating finite Dirac operators from real spectral triples as fluctuating random variables in Dirac ensembles. It develops a bootstrap framework that combines Schwinger–Dyson equations, large-$N$ factorization, and Hankel/positivity constraints to bound moments and couplings without solving models explicitly, linking to spectral geometry via Laplace eigenvalues. Concrete models (cubic and quartic, single- and multi-trace, and fermionic variants) illustrate how positivity- and loop-equation constraints carve out allowed regions in moment/coupling space and reveal rich phase structure. Thematically, the paper unifies noncommutative-geometry-inspired quantum gravity with random-matrix/bootstrap techniques, and it sketches bridges to the Standard Model through almost-commutative geometries and Yang–Mills–Higgs Dirac ensembles, highlighting both conceptual insights and directions for rigorous analysis and numerical exploration.

Abstract

This paper surveys a bootstrap framework for random Dirac operators arising from finite spectral triples in noncommutative geometry. Motivated by a toy model for quantum gravity to replace integration over metrics by integration over Dirac operators, we give an overview of multitrace and multimatrix random matrix models built from spectral triples and analyze them in the large $N$ limit using positivity constraints on Hankel moment matrices. In this setting, the bootstrap philosophy, originating in the S-matrix program and revived in modern conformal bootstrap theory, reappears as a rigorous analytic tool for extracting spectral data from consistency alone, without solving the model explicitly. We explain how Schwinger-Dyson equations, factorization at large $N$, and the noncommutative moment problem lead to finite-dimensional semidefinite programs whose feasible regions encode the allowed pairs of coupling constants and moments. Connections with spectral geometry, in particular the study of Laplace eigenvalues, are also discussed, illustrating how bootstrapping provides a unified mechanism for deriving bounds in both commutative and noncommutative settings.

Figures (7)

• Figure 1: Bootstrapped regions of possible solutions to the loop equations for the cubic type $(1,0)$ Dirac ensemblehessam2022fromnoncom. Each colour corresponds to a different number of constraints derived from positivity of some principal minors of the specified size. The analytic solution found in hessam2023double is plotted in red for comparison.
• Figure 2: Bootstrapped region of solution space for the type $(1,0)$ quartic Dirac ensemble hessam2022bootstrapping. The different coloured regions denote different combinations of constraints applied.
• Figure 3: Bootstrapped region of solution space for the type $(1,0)$ quartic Dirac ensemble, allowing for asymmetric solutions. This plot was generated using ten SDE's and a submatrix of the Hankel matrix of size ten. There is a clear fork in the solutions space, where each outer prong corresponds to a subspace of possible asymmetric solutions.
• Figure 4: Bootstrapped region of solution space for the type $(0,1)$ quartic Dirac ensemble, allowing for asymmetric solutions. These were found using eight SDE's and a submatrix of the Hankel matrix of size 8.
• Figure 5: Bootstrapped region of solution space for the type $(2,0)$ quartic Dirac ensemble hessam2022bootstrapping. The different coloured regions denote different combinations of constraints applied. The red curve is the conjectured analytic solution.