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The free spinor field on a fuzzy sphere is described within Watamura approach to Dirac operator. Except of the highest mode, its spectrum is the same but truncated as in the commutative case. We present a simple gauge extension of the model with usual polynomial interaction. The gauge symmetry is exact, and the chiral properties of the field modes are standard except the highest mode.

Noncommutative U(1) gauge theory in four dimensions is shown to be equivalent in some scaling limit to an ordinary nonlinear sigma model in two dimensions. The model in this regime is solvable and the corresponding exact beta function is found. We also show that classical U(n) gauge theory on can be approximated by a sequence of ordinary (d-2)-dimensional Georgi–Glashow models with gauge groups U(n(L+1)), where L+1 is the matrix size of the regularized noncommutative plane .

We present a series of instanton-like solutions to a matrix model which satisfy a self-duality condition and possess an action whose value is, to within a fixed constant factor, an integer l². For small values of the dimension n² of the matrix algebra the integer resembles the result of a quantization condition but as n → ∞ the ratio l/n can tend to an arbitrary real number between zero and one.

We study Myers' world volume effective action of coincident D-branes. We investigate a system of N₀D0-branes in the geometry of Dp-branes with p = 2 or p = 4. The choice of coordinates can make the action simplified and tractable. For p = 4, we show that a certain pointlike D0-brane configuration solving equations of motion of the action can expand to form a fuzzy two-sphere via magnetic moment effect without changing quantum numbers. We compare noncommutative D0-brane configurations with dual spherical D(6 - p)-brane systems. We also discuss the relation between these configurations and giant gravitons in 11 dimensions.

Recently we have introduced a matrix model depending on two coupling constants g² and λ, which contains the fuzzy sphere as a background; to obtain the classical limit g² must depend on N in a precise way. In this paper we show how to obtain the classical solitons of the N → ∞ limit imposing the development ; as a consequence at finite N one obtains a noncommutative version of the solitons for the fuzzy sphere.

We consider a matrix model depending on the parameter λ which permits the fuzzy sphere as a classical background. By expanding the bosonic matrices around this background one recovers a U(1) (U(n)) noncommutative gauge theory on the fuzzy sphere. To check the classical stability of this background, we look for new classical solutions of this model and find them for λ < 1, that make the fuzzy sphere solution unstable for and stable otherwise.

We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two-dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions.

We study the interconnection between the finite projective modules for a fuzzy sphere, determined in a previous paper, and the matrix model approach, making clear the physical meaning of noncommutative topological configurations.

We show how to generalize our method, based on projective modules and matrix models, which enabled us to derive noncommutative monopoles on a fuzzy sphere, to the non-Abelian case, recovering known results in literature. We then discuss a possible candidate for deforming the commutative Chern class to the noncommutative case.

In this paper we review Klimčík's construction of noncommutative gauge theory on the fuzzy supersphere. This theory has an exact SUSY gauge symmetry with a finite number of degrees of freedom and thus in principle it is amenable to the methods of matrix models and Monte Carlo numerical simulations. We also write down in this paper a novel fuzzy supersymmetric scalar action on the fuzzy supersphere.

The fuzzy disk is a discretization of the algebra of functions on the two-dimensional disk using finite matrices which preserves the action of the rotation group. We define a φ⁴ scalar field theory on it and analyze numerically three different limits for the rank of the matrix going to infinity. The numerical simulations reveal three different phases: uniform and disordered phases already present in the commutative scalar field theory and a nonuniform ordered phase as noncommutative effects. We have computed the transition curves between phases and their scaling. This is in agreement with studies on the fuzzy sphere, although the speed of convergence for the disk seems to be better. We have also performed the limits for the theory in the cases of the theory going to the commutative plane or commutative disk. In this case the theory behaves differently, showing the intimate relationship between the nonuniform phase and noncommutative geometry.

We present a new model of quantum gravity as a theory of random geometries given explicitly in terms of a multitrace matrix model. This is a generalization of the usual discretized random surfaces of two-dimensional quantum gravity which works away from two dimensions and captures a large class of spaces admitting a finite spectral triple. These multitrace matrix models sustain emergent geometry as well as growing dimensions and topology change.

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.

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 consider the quantum computational process as viewed by an insider observer: this is equivalent to an isomorphism between the quantum computer and a quantum space, namely the fuzzy sphere. The result is the formulation of a reversible quantum measurement scheme, with no hidden information.

Beginning with a review of the existing literature on the computation of spectral distances on noncommutative spaces like Moyal plane and fuzzy sphere, adaptable to Hilbert–Schmidt operatorial formulation, we carry out a correction, revision and extension of the algorithm provided in [1] i.e. [F. G. Scholtz and B. Chakraborty, J. Phys. A, Math. Theor.46 (2013) 085204] to compute the finite Connes’ distance between normal states. The revised expression, which we provide here, involves the computation of the infimum of an expression which involves the “transverse” $(\mathrm{\Delta }{\rho }_{\perp })$ component of the algebra element in addition to the “longitudinal” component $\mathrm{\Delta }\rho$ of [1], identified with the difference of density matrices representing the states, whereas the expression given in [1] involves only $\mathrm{\Delta }\rho$ and corresponds to the lower bound of the distance. This renders the revised formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a nontrivial task, but under rather generic conditions it turns out that the Connes’ distance is proportional to the Hilbert-Schmidt norm of $\mathrm{\Delta }\rho$, leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [P. Martinetti and L. Tomassini, Commun. Math. Phys.323 (2013) 107–141]. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite-dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere, is never reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of maximal noncommutativity ($n=1/2$), the finite distance is shown to coincide exactly with the above-mentioned lower bound, with the transverse component playing no role. This, however, starts changing from $n=1$ onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our revised algorithm. The contrasting features of these types of noncommutative spaces becomes quite transparent through the analysis, carried out in the eigen-spinor bases of the respective Dirac operators.

We show that the standard Calogero Lax matrix can be interpreted as a function on the fuzzy sphere and the Avan–Talon r-matrix as a function on the direct product of two fuzzy spheres. We calculate the limiting Lax function and r-function when the fuzzy sphere tends to the ordinary sphere and we show that they define an integrable model interpreted as a large N Calogero model by Bordemann, Hoppe and Theisen.