From a string theory point of view the most natural gauge action on the fuzzy sphere S 2 L is the Alekseev-Recknagel-Schomerus action which is a particular combination of the YangMills action and the Chern-Simons term . The differential calculus on the fuzzy sphere is 3−dimensional and thus the field content of this model consists of a 2-dimensional gauge field together with a scalar fluctuation normal to the sphere . For U (1) gauge theory we compute the quadratic effective action and shows explicitly that the tadpole diagrams and the vacuum polarization tensor contain a gauge-invariant UV-IR mixing in the continuum limit L−→∞ where L is the matrix size of the fuzzy sphere. In other words the quantum U (1) effective action does not vanish in the commutative limit and a noncommutative anomaly survives . We compute the scalar effective potential and prove the gauge-fixingindependence of the limiting model L = ∞ and then show explicitly that the one-loop result predicts a first order phase transition which was observed recently in simulation . The one-loop result for the U (1) theory is exact in this limit . It is also argued that if we add a large mass term for the scalar mode the UV-IR mixing will be completely removed from the gauge sector . It is found in this case to be confined to the scalar sector only. This is in accordance with the large L analysis of the model . Finally we show that the phase transition becomes harder to reach starting from small couplings when we increase M .

We study a three matrix model with global SO(3) symmetry containing at most quartic powers of the matrices. We find an exotic line of discontinuous transitions with a jump in the entropy, characteristic of a 1st order transition, yet with divergent critical fluctuations and a divergent specific heat with critical exponent α = 1/2. The low temperature phase is a geometrical one with gauge fields fluctuating on a round sphere. As the temperature increased the sphere evaporates in a transition to a pure matrix phase with no background geometrical structure. Both the geometry and gauge fields are determined dynamically. It is not difficult to invent higher dimensional models with essentially similar phenomenology. The model presents an appealing picture of a geometrical phase emerging as the system cools and suggests a scenario for the emergence of geometry in the early universe.Our understanding of the fundamental laws of physics has evolved to a very geometrical one. However, we still have very little insight into the origins of geometry itself. This situation has been undergoing a significant evolution in recent years and it now seems possible to understand classical geometry as an emergent concept. The notion of geometry as an emergent concept is not new, see for example [1] for an inspiring discussion and [2, 3] for some recent ideas. We examine such a phenomenon in the context of noncommutative geometry [4] emerging from matrix models, by studying a surprisingly rich three matrix model [5,6,7]. The matrix geometry that emerges here has received attention as an alternative setting for the regularization of field theories [8,9,10,11] and as the configurations of D0 branes in string theory [12,13]. Here, however, the geometry emerges as the system cools, much as a Bose condensate or superfluid emerges as a collective phenomenon at low temperatures. There is no background geometry in the high temperature phase. The simplicity of the model, in this study, allows for a detailed examination of such an exotic transition. We suspect the asymmetrical nature of the transition may be generic to this phenomenon.We consider the most general single trace Euclidean action (or energy) functional for a three matrix model invariant under global SO(3) transformations containing no higher than the fourth power of the matrices. This model is surprisingly rich and in the infinite matrix limit can exhibit many phases as the parameters are tuned. We find that generically the model has two clearly distinct phases, one geometrical the other a matrix phase. Small fluctuations in the geometrical phase are those of a Yang-Mills theory and a scalar field around a ground state corresponding to a round two-sphere. In the matrix phase there is no background spacetime geometry and the fluctuations are those of the matrix entries around zero. In this note we focus on the subset of parameter space where, in the large matrix limit, the gauge group is Abelian.For finite but large N , at low temperature, the model exhibits fluctuations around a fuzz...

We present, theoretical predictions and Monte Carlo simulations, for a simple three matrix model that exhibits an exotic phase transition. The nature of the transition is very different if approached from the high or low temperature side. The high temperature phase is described by three self interacting random matrices with no background spacetime geometry. As the system cools there is a phase transition in which a classical two-sphere condenses to form the background geometry. The transition has an entropy jump or latent heat, yet the specific heat diverges as the transition is approached from low temperatures. We find no divergence or evidence of critical fluctuations when the transition is approached from the high temperature phase. At sufficiently low temperatures the system is described by small fluctuations, on a background classical two-sphere, of a U (1) gauge field coupled to a massive scalar field. The critical temperature is pushed upwards as the scalar field mass is increased. Once the geometrical phase is well established the specific heat takes the value 1 with the gauge and scalar fields each contributing 1/2.