Higher Dimenssional Quantum Hall Effect, Solitons On Noncommutative Torus And Integrable Models

After shortly reviewed the Haldane's description of the quantum Hall effect on 2-sphere S2 and Zhang and Hu's generalization to 4-sphere 54. we obtained the corresponding noncommutative algebra An and the Moyal structure of the Hilbert space. Then by taking use of the method proposed by Susskind and Polychronakos, we gave out the non-commutative Chern-Simons theory to describe the incompressible quantum Hall fluid on 52 and its algebraic sturcture. We also discussed the generalization of the description of the incompressile quantum Hall fluid to S4.On the noncommutative torus T, restricting in the case that the noncommutative parameter and the area A of T being an integer, we obtained n solitons on T by orbifolding the total torus T n times into Tn. Then we constructed the basis of Hilbert space Hn in terms of 8 functions of the positions Zi of n solitons. The Wi wrapping around the torus generates the algebra An, which is the Zn x Zn Heisenberg group on 0 functions. We found the generators g of an local elliptic su(ri) G(su(n)), which transform covariantly by the global gauge transformation of An- By acting on Hn we establish the isomorphism of An and ?, we embeded this g into the L-matrix of the elliptic Gaudin and Calogero-Moser models to give the dynamics. The moment map of this twisted cotangent sun(T) bundle is matched to the D-equation with Fayet-Illiopoulos source term, so the dynamics of the noncommutative solitons becomes that of the brane. The geometric configuration (k, u) of the spectral curve det|L(u) ?k\ = 0 describes the brane configuration, with the dynamical variables Zj of the noncommutative solitons as the moduli T畁/Sn. Furthermore, in the noncommutative Chern-Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is also identified with the moment map eqaution of the noncommutative sun(T) cotangent bundle with marked points. The eigenfunctions of the Gaudin differential //-operators as the Laughlin wavefunction are solved by Bethe ansatz.For the integrable model, after reviewed the construction of one important integrable model, the 1-D SU(n) Hubbard model, we constructed the tetrahedral Zamolodchikovalgebra, then proved the R-matrix of the model satifies the Yang-Baxter equation. In consequence, we also present a generalizations of the 1-D SU(n) Hubbard model.