Solitons on Noncommutative Torus as Elliptic Calogero Gaudin Models, Branes and Laughlin Wave Functions

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$, we construct the basis of Hilbert space {\caH}_n$in terms of$\theta$functions of the positions$z_i$of$n$solitons. The wrapping around the torus generates the algebra${\cal A}_n$, which is the$Z_n \times Z_n$Heisenberg group on$\theta$functions. We find the generators$g$of an local elliptic$su(n)$, w$transform covariantly by the global gauge transformation of ${\cal A}$By acting on ${\cal H}_n$ we establish the isomorphism of ${\cal A}_n$$g$. We embed this $g$ into the $L$-matrix of the elliptic Gaudin and$models to give the dynamics. The moment map of this twisted cotangent$su_n({\cal T})$bundle is matched to the$D$-equation with Fayet-Illiopoulos source term, so the dynamics of the N.C. solitons becomes that of the brane. The geometric configuration$(k, u)$of th$spectral curve ${\rm det}|L(u) - k| = 0$ describes the brane configuration, with the dynamical variables $z_i$ of N.C. solitons as$moduli$T^{\otimes n} / S_n$. Furthermore, in the N.C. Chern-Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map eqaution$the N.C. $su_n({\cal T})$ cotangent bundle with marked points. The eigenfunction of the Gaudin differential $L$-operators as the Laughli$wavefunction is solved by Bethe ansatz.