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We present a study of D = 4 supersymmetric Yang–Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first-/second-order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first-order due to the discontinuity in the action whereas it looks second-order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N. We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a nonpolynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this nonpolynomial law.

The phenomenon of emergent fuzzy geometry and noncommutative gauge theory from Yang–Mills matrix models is briefly reviewed. In particular, the eigenvalue distributions of Yang–Mills matrix models in lower dimensions in the commuting (matrix or Yang–Mills) phase of these models are discussed.

A Gaussian approximation to the bosonic part of M-(atrix) theory with mass deformation is considered at large values of the dimension $d$. From the perspective of the gauge/gravity duality this action reproduces with great accuracy the stringy Hagedorn phase transition from a confinement (black string) phase to a deconfinement (black hole) phase whereas from the perspective of the matrix/geometry approach this action only captures a remnant of the geometric Yang–Mills-to-fuzzy-sphere phase where the fuzzy sphere solution is only manifested as a three-cut configuration termed the “baby fuzzy sphere” configuration. The Yang–Mills phase retains most of its characteristics with two exceptions: (i) the uniform distribution inside a solid ball suffers a crossover at very small values of the gauge coupling constant to a Wigner’s semicircle law, and (ii) the uniform distribution at small values of the temperatures is nonexistent.

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.

A consistent QM/NCG duality is put forward as a model for the ${\mathrm{\text{AdS}}}^2/{\mathrm{\text{CFT}}}_1$ correspondence. This is a duality/correspondence between (1) the dAFF conformal quantum mechanics (QM) on the boundary (which is only “quasi-conformal” in the sense that there is neither an $SO(1,2)$-invariant vacuum state nor there are strictly speaking primary operators), and between (2) the noncommutative geometry of ${\mathrm{\text{AdS}}}_{𝜃}^2$ in the bulk (which is only “quasi-AdS” in the sense of being only asymptotically ${\mathrm{\text{AdS}}}^2$). The Laplacian operators on noncommutative ${\mathrm{\text{AdS}}}_{𝜃}^2$ and commutative ${\mathrm{\text{AdS}}}^2$ have the same spectrum and thus their correlators are conjectured to be identical. These bulk correlation functions are found to be correctly reproduced by appropriately defined boundary quantum observables in the dAFF quantum mechanics. Moreover, these quasi-primary operators on the boundary form a subalgebra of the operator algebra of noncommutative ${\mathrm{\text{AdS}}}_{𝜃}^2$.

In this paper, we present the construction of noncommutative ${\mathrm{\text{AdS}}}_{𝜃}^2$ black hole and its four-dimensional Yang–Mills IKKT-type matrix model which includes two competing Myers term one responsible for the condensation of pure ${\mathrm{\text{AdS}}}_{𝜃}^2$ and the other one responsible for the condensation of the dilaton field. It is argued that the phase diagram of this matrix model features three phases: (1) a gravitational phase (${\mathrm{\text{AdS}}}_{𝜃}^2$ black hole), (2) a geometric phase (${\mathrm{\text{AdS}}}_{𝜃}^2$ background) and (3) a Yang–Mills phase. The Hawking process is therefore seen as an exotic line of discontinuous transitions between the gravitational and geometrical phases. Alternatively, a noncommutative nonlinear sigma model describing the transition of the dilaton field between the gravitational and geometrical phases is also constructed.

The near-horizon noncommutative geometry of black holes, given by ${\mathrm{\text{AdS}}}_{𝜃}^2\times {𝕊}_N^2$, is discussed and the phase structure of the corresponding Yang–Mills matrix models is presented. The dominant phase transition as the system cools down, i.e. as the gauge coupling constant is decreased is an emergent geometry transition between a geometric noncommutative ${\mathrm{\text{AdS}}}_{𝜃}^2\times {𝕊}_N^2$ phase (discrete spectrum) and a Yang–Mills matrix phase (continuous spectrum) with no background geometrical structure. We also find a possibility for topology change transitions in which space or time directions grow or decay as the temperature is varied. Indeed, the noncommutative near-horizon geometry ${\mathrm{\text{AdS}}}_{𝜃}^2\times {𝕊}_N^2$ can evaporate only partially to a fuzzy sphere ${𝕊}_N^2$ (emergence of time) or to a noncommutative anti-de Sitter space–time ${\mathrm{\text{AdS}}}_{𝜃}^2$ (topology change).

The Wilsonian renormalization group approach to matrix models is outlined and applied to multitrace matrix models with an emphasis on the computation of the fixed points which could describe the phase structure of noncommutative scalar phi-four theory.

We show that heavy pure states of gravity can appear to be mixed states to almost all probes. Our arguments are made for AdS₅ Schwarzschild black holes using the field theory dual to string theory in such spacetimes. Our results follow from applying information theoretic notions to field theory operators capable of describing very heavy states in gravity. For certain supersymmetric states of the theory, our account is exact: the microstates are described in gravity by a spacetime "foam", the precise details of which are invisible to almost all probes.

We find a new evidence of a gauge/gravity duality: an explicit correspondence between eigenvalue densities and geometries. We examine the duality between IIA supergravity solutions and SU(2|4) symmetric gauge theories, which are BMN matrix model, N = 8 super Yang-Mills theory on R × S² and N = 4 super Yang-Mills theory on R × S³/Z_k. We show that the corresponding geometries are realized as the solution to the saddle point equations in the gauge theories, and that typical scales of the geometries are characterized by the range of the eigenvalue distribution.