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 .

The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing on brownian motion over a d-dimensional curved manifold. We use the local frame defined by the so called Riemann normal coordinates to derive a general formula for the meansquare geodesic distance (MSD) at the short-time regime. This formula is written in terms of O(d) invariants that depend on the Riemann curvature tensor. We study the n-dimensional sphere case to validate these results. We also show that the diffusion for positive constant curvature is slower than the diffusion in a plane space, while the diffusion for negative constant curvature turns out to be faster. Finally the two-dimensional case is emphasized, as it is relevant for the single particle diffusion on biomembranes.

The many-particle Langevin equation, written in local coordinates, is used to derive a Brownian dynamics simulation algorithm to study the dynamics of colloids moving on curved manifolds. The predictions of the resulting algorithm for the particular case of free particles diffusing along a circle and on a sphere are tested against analytical results, as well as with simulation data obtained by means of the standard Brownian dynamics algorithm developed by Ermak and McCammon [J. Chem. Phys. 69, 1352 (1978)] using explicitly a confining external field. The latter method allows constraining the particles to move in regions very tightly, emulating the diffusion on the manifold. Additionally, the proposed algorithm is applied to strong correlated systems, namely, paramagnetic colloids along a circle and soft colloids on a sphere, to illustrate its applicability to systems made up of interacting particles.

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A theoretical analysis of active motion on curved surfaces is presented in terms of a generalization of the telegrapher equation. Such a generalized equation is explicitly derived as the polar approximation of the hierarchy of equations obtained from the corresponding Fokker-Planck equation of active particles diffusing on curved surfaces. The general solution to the generalized telegrapher equation is given for a pulse with vanishing current as initial data. Expressions for the probability density and the mean squared geodesic displacement are given in the limit of weak curvature. As an explicit example of the formulated theory, the case of active motion on the sphere is presented, where oscillations observed in the mean squared geodesic displacement are explained.