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Solitons on Noncommutative Torus as Elliptic Algebras and Elliptic Models

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $θ= \frac{Z}{n}$ and the area of ${\cal T}$ is an integer, we construct the basis of Hilbert space ${\cal H}_n$ in terms of $θ$ functions of the positions $z_i$ of $n$ solitons. The loop wrapping around the torus generates the algebra ${\cal A}_n$. We show that ${\cal A}_n$ is isomorphic to the $Z_n \times Z_n$ Heisenberg group on $θ$ functions. We find the explicit form for the local operators, which is the generators $g$ of an elliptic $su(n)$, and transforms covariantly by the global gauge transformation of the Wilson loop in ${\cal A}_n$. 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 the $L$-matrix of the elliptic Gaudin and C.M. models to give the dynamics. For $θ$ generic case, we introduce the crossing parameter $η$ related with $θ$ and the modulus of ${\cal T}$. The dynamics of solitons is determined by the transfer matrix $T$ of the elliptic quantum group ${\cal A}_{τ, η}$, equivalently by the elliptic Ruijsenaars operators $M$. The eigenfunctions of $T$ found by Bethe ansatz appears to be twisted by $η$.