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

In this letter, we introduce Fractional Quantum Hall Effect (FQHE) Skyrmions in the Chern-Simons effective field theory description, and we present a new derivation of the FQHE Skyrmions properties, namely charge and spin, which results from considerations at the edge of the Hall sample. At the boundary, we demand anomaly cancellation for the chiral edge currents, as well as, allow for the possibility of Skyrmion creation and annihilation. For the Skyrmion charge and spin, we get the values eνN Sky and νN Sky /2, respectively, where e is electron charge, ν is the filling fraction and N Sky is the Skyrmion winding number. We also add terms to the action so that the classical spin fluctuations in the bulk satisfy the standard equations of a ferromagnet and find that spin waves propagate with the classical drift velocity of the electron.The FQHE admits a Landau-Ginzburg description in terms of a complex doublet of bosonic fields Ψ = ψ 1 ψ 2 and a statistical Chern-Simons gauge field. 1,2 The Chern-Simons coupling is chosen such that each "bosonized" electron carries an odd number of elementary flux units, yielding fermionic statistics. The Landau-Ginzburg ground state is given by ψ 1 = √ ρ 0 , ψ 2 = 0, where the ground state density of electrons ρ 0 is equal to the filling fraction ν times the external magnetic field divided by the elementary flux unit 2π/e. The corresponding wave function, when expressed in position space, is nothing but the Laughlin wave function (see Ref.2 for a nice and detailed derivation of this connection). The lowest lying excitations around the ground state are described by the quasiparticle and quasihole Laughlin wave functions. The presence of a Zeeman interaction naively precludes any dynamics for the spin degrees of freedom, and consequently the quasiparticles and quasiholes are fully polarized. In the Landau-Ginzburg description, those excitations are associated with vortices. 2 More specifically, they are field configurations where the phase of ψ 1 has a nonzero winding, and ψ 2 = 0 everywhere. The magnitude of ψ 1 , which is the square root of the density of electrons, vanishes at points associated with the origin of the vortices. 2627 Mod. Phys. Lett. A 1998.13:2627-2635. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 02/03/15. For personal use only.