Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of discrete physics and matrix models and discuss mathematically coherent discretizations of monopoles and solitons using fuzzy physics and noncommutative geometry. A fuzzy sigma-model action for the two-sphere fulfilling a fuzzy Belavin-Polyakov bound is also put forth.Comment: 17 pages, Latex. Uses amstex, amssymb.Spelling of the name of one Author corrected. To appear in Commun.Math.Phy

We propose a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its Cartesian products. Its relation to the Ginsparg-Wilson approach is also clarified.The nonperturbative formulation of chiral gauge theories is a long standing program in particle physics. It seems clear that one should regularize these theories with all symmetries intact. There is a major problem associated with conventional lattice approaches to this program, with roots in topological features: The Nielsen-Ninomiya theorem 1 states that if we want to maintain chiral symmetry, then under plausible assumptions, one cannot avoid the doubling of fermions in the usual lattice formulations.Recently a novel approach to discrete physics has been developed. It works with quantum fields on a "fuzzy space" M F obtained by treating the underlying manifold M as a phase space and quantizing it. 2-9 Topological features, chiral anomalies and σ-models have been successfully developed in this framework, 4,10,11 using the cyclic cohomology of Connes. 12,13 In this letter, we propose a solution of the fermion doubling problem for M = S 2 using fuzzy physics. An alternative approach can be found in Ref. 4. There have also been important developments 14,15 in the theory of chiral fermions and anomalies in the usual lattice formulations. We will show that there are striking relationships between our approach and these developments. *

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 .

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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...