Monopoles and Solitons in Fuzzy Physics

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

Current Algebra and Conformal Field Theory on a Figure Eight

We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also found to satisfy Kirchhoff's law at the junction. We obtain the associated conserved charges, and show that they generate a Virasoro algebra. Its conformal anomaly (central charge) is computed for special values of the parameters characterizing the network.Comment: 39 pages; Latex with 1 figure included in encapsulated postscript format. psbox.tex require

Current Oscillations, Interacting Hall Discs and Boundary CFTs

In this paper, we discuss the behavior of conformal field theories interacting at a single point. The edge states of the quantum Hall effect (QHE) system give rise to a particular representation of a chiral Kac-Moody current algebra. We show that in the case of QHE systems interacting at one point we obtain a ``twisted'' representation of the current algebra. The condition for stationarity of currents is the same as the classical Kirchoff's law applied to the currents at the interaction point. We find that in the case of two discs touching at one point, since the currents are chiral, they are not stationary and one obtains current oscillations between the two discs. We determine the frequency of these oscillations in terms of an effective parameter characterizing the interaction. The chiral conformal field theories can be represented in terms of bosonic Lagrangians with a boundary interaction. We discuss how these one point interactions can be represented as boundary conditions on fields, and how the requirement of chirality leads to restrictions on the interactions described by these Lagrangians. By gauging these models we find that the theory is naturally coupled to a Chern-Simons gauge theory in 2+1 dimensions, and this coupling is completely determined by the requirement of anomaly cancellation.Comment: 32 pages, LateX. Uses amstex, amssymb. Typos corrected. To appear in Int. J. Mod. Phys.