Solitons on Noncommutative Torus as Elliptic Algebras and Elliptic Models

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$ and the area of ${\cal T}$ is an integer, we construct the basis of Hilbert space ${\cal H}_n$ in terms of $\theta$ functions of the positions of $n$ solitons. The Wilson loop wrapping the solitons around the torus generates the algebra ${\cal A}_n$. We find that ${\cal A}_n$ is isomorphic to the $Z_n \times Z_n$ Heisenberg group on $\theta$ functions. We find the explicit form for the solitons local translation operators, show that it is the generators $g$ of an elliptic $su(n)$, which transform covariantly by the global gauge transformation of the Wilson loop in ${\cal A}_n$. Then by acting on ${\cal H}_n$ we establish the isomorphism of ${\cal A}_n$ and $g$. Then it is easy to give the projection operators corresponding to the solitons and the ABS construction for generating solitons. We embed this $g$ into elliptic Gaudin and C.M. models to give the dynamics. For $\theta$ generic case, we introduce the crossing parameter $\eta \sim \theta \tau$, here $\tau$ is the modulus of ${\cal T}$. We suggest that ${\cal A}_n$ turns to be the elliptic quantum group ${\cal A}_{\tau, \eta}$. The dynamics of solitons is determined by the transfer matrix $T$, equivalently by the elliptic Ruijsenaars operators $M$. The eigenfunctions of $T$ found by Bethe ansatz appears to be twisted by $\eta$ under actions of Wilson loop. The affine Weyl reflection of solitons and permutation $S_n$ of the 2-branes are described also