Solitons on Noncommutative Torus as Elliptic Algebras and Elliptic Models

Abstract

For the noncommutative torus $\mathcal{T}$, in case of the N.C. parameter $\theta =\frac{Z}{n}$ and the area of $\mathcal{T}$ is an integer, we construct the basis of Hilbert space ${\mathcal{H}}_n$ in terms of $\theta$ functions of the positions $z_i$ of $n$ solitons. The loop wrapping around the torus generates the algebra ${\mathcal{A}}_n$. We show that ${\mathcal{A}}_n$ is isomorphic to the $Z_n\times Z_n$ Heisenberg group on $\theta$ 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 ${\mathcal{A}}_n$. By acting on ${\mathcal{H}}_n$ we establish the isomorphism of ${\mathcal{A}}_n$ 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 $\theta$ generic case, we introduce the crossing parameter $\eta$ related with $\theta$ and the modulus of $\mathcal{T}$. The dynamics of solitons is determined by the transfer matrix $T$ of the elliptic quantum group ${\mathcal{A}}_{\tau ,\eta }$, equivalently by the elliptic Ruijsenaars operators $M$. The eigenfunctions of $T$ found by Bethe ansatz appears to be twisted by $\eta$.

26 pages, plain latex, no figure. Rewritten version. Some references added

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