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

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

2002-04-19

arxiv.org/abs/hep-th/0204163v1

Int.J.Mod.Phys. A18 (2003) 2477-2500

Physics

High Energy Physics

High Energy Physics - Theory

25 pages, plain latex, no figures

Scientific paper

10.1142/S0217751X03014228

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$, we construct the basis of Hilbert space ${\ca$H}_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.

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