Résumé: A random multitrace matrix model can provide us with a new vision to geometry emergence. Furthermore, it yields a new model of quantum gravity in two dimensions. On this basis, the present research seeks to report on Monte Carlo (MC) simulations of the quartic phi-four matrix model with cubic term. We construct the phase diagram of this model. Following that, topology change is observed during the transition between the real quartic matrix model and noncommutative scalar phi-four field theory. The dimension of the geometry is calculated from the critical exponents of the Ising transition. The free propagator of the theory is determined from Wigner semicircle law and found to be in agreement with the collected Monte Carlo data.

Résumé: In this paper, the geometry of quantum gravity is quantized in the sense of being noncommutative (first quantization) but it is also quantized in the sense of being emergent (second quantization). A new mechanism for quantum geometry is proposed in which noncommutative geometry can emerge from “one-matrix multitrace scalar matrix models” by probing the statistical physics of commutative phases of matter. This is in contrast to the usual mechanism in which noncommutative geometry emerges from “many-matrix singletrace Yang–Mills matrix models” by probing the statistical physics of noncommutative phases of gauge theory. In this novel scenario, quantized geometry emerges in the form of a transition between the two phase diagrams of the real quartic matrix model and the noncommutative scalar phi-four field theory. More precisely, emergence of the geometry is identified here with the emergence of the uniform-ordered phase and the corresponding commutative (Ising) and noncommutative (stripe) coexistence lines. The critical exponents and Wigner’s semicircle law are used to determine the dimension and the metric, respectively. Arguments from the saddle point equation, from Monte Carlo simulation and from the matrix renormalization group equation are provided in support of this scenario.

Résumé: In this paper, the geometry of quantum gravity is quantized in the sense of being noncommutative (first quantization) but it is also quantized in the sense of being emergent (second quantization). A new mechanism for quantum geometry is proposed in which noncommutative geometry can emerge from “one-matrix multitrace scalar matrix models” by probing the statistical physics of commutative phases of matter. This is in contrast to the usual mechanism in which noncommutative geometry emerges from “many-matrix singletrace Yang–Mills matrix models” by probing the statistical physics of noncommutative phases of gauge theory. In this novel scenario, quantized geometry emerges in the form of a transition between the two phase diagrams of the real quartic matrix model and the noncommutative scalar phi-four field theory. More precisely, emergence of the geometry is identified here with the emergence of the uniform-ordered phase and the corresponding commutative (Ising) and noncommutative (stripe) coexistence lines. The critical exponents and Wigner’s semicircle law are used to determine the dimension and the metric, respectively. Arguments from the saddle point equation, from Monte Carlo simulation and from the matrix renormalization group equation are provided in support of this scenario.

Résumé: A detailed Monte Carlo calculation of the phase diagram of bosonic mass-deformed IKKT Yang–Mills matrix models in three and six dimensions with quartic mass deformations is given. Background emergent fuzzy geometries in two and four dimensions are observed with a fluctuation given by a noncommutative gauge theory very weakly coupled to normal scalar fields. The geometry, which is determined dynamically, is given by the fuzzy spheres and respectively. The three and six matrix models are effectively in the same universality class. For example, in two dimensions the geometry is completely stable, whereas in four dimensions the geometry is stable only in the limit , where M is the mass of the normal fluctuations. The behaviors of the eigenvalue distribution in the two theories are also different. We also sketch how we can obtain a stable fuzzy four-sphere in the large N limit for all values of M as well as models of topology change in which the transition between spheres of different dimensions is observed. The stable fuzzy spheres in two and four dimensions act precisely as regulators which is the original goal of fuzzy geometry and fuzzy physics. Fuzzy physics and fuzzy field theory on these spaces are briefly discussed.

Résumé: We study a six matrix model with global S O ( 3 ) × S O ( 3 ) symmetry containing at most quartic powers of the matrices. This theory exhibits a phase transition from a geometrical phase at low temperature to a Yang-Mills matrix phase with no background geometrical structure at high temperature. This is an exotic phase transition in the same universality class as the three matrix model but with important differences. The geometrical phase is determined dynamically, as the system cools, and is given by a fuzzy sphere background S 2 N × S 2 N , with an Abelian gauge field which is very weakly coupled to two normal scalar fields.