5,818 research outputs found

    Non-commutative geometry of 4-dimensional quantum Hall droplet

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    We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid's theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy S4S^{4} appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in this non-commutative geometry, are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of SO(5) angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of SO(5) angular momentum, are given. It is discussed that some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory.Comment: latex 22 pages, no figures. some references added. some results are clarifie

    Higher Dimensional Geometries from Matrix Brane constructions

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    Matrix descriptions of even dimensional fuzzy spherical branes S2kS^{2k} in Matrix Theory and other contexts in Type II superstring theory reveal, in the large NN limit, higher dimensional geometries SO(2k+1)/U(k)SO(2k+1)/U(k), which have an interesting spectrum of SO(2k+1)SO(2k+1) harmonics and can be up to 20 dimensional, while the spheres are restricted to be of dimension less than 10. In the case k=2k=2, the matrix description has two dual field theory formulations. One involves a field theory living on the non-commutative coset SO(5)/U(2)SO(5)/U(2) which is a fuzzy S2S^2 fibre bundle over a fuzzy S4S^4. In the other, there is a U(n) gauge theory on a fuzzy S4S^4 with O(n3) {\cal O}(n^3) instantons. The two descriptions can be related by exploiting the usual relation between the fuzzy two-sphere and U(n) Lie algebra. We discuss the analogous phenomena in the higher dimensional cases, developing a relation between fuzzy SO(2k)/U(k)SO(2k)/U(k) cosets and unitary Lie algebras.Comment: 28 pages (Harvmac big) ; version 2 : minor typos fixed and ref. adde

    Interval linear systems as a necessary step in fuzzy linear systems

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    International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

    Noncommutative Vortices and Flux-Tubes from Yang-Mills Theories with Spontaneously Generated Fuzzy Extra Dimensions

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    We consider a U(2) Yang-Mills theory on M x S_F^2 where M is an arbitrary noncommutative manifold and S_F^2 is a fuzzy sphere spontaneously generated from a noncommutative U(N) Yang-Mills theory on M, coupled to a triplet of scalars in the adjoint of U(N). Employing the SU(2)-equivariant gauge field constructed in arXiv:0905.2338, we perform the dimensional reduction of the theory over the fuzzy sphere. The emergent model is a noncommutative U(1) gauge theory coupled adjointly to a set of scalar fields. We study this model on the Groenewald-Moyal plane and find that, in certain limits, it admits noncommutative, non-BPS vortex as well as flux-tube (fluxon) solutions and discuss some of their properties.Comment: 18+1 pages, typos corrected, published versio
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