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

Abstract

For the noncommutative torus $\mathcal{T}$, in case of the N.C. parameter $\theta =\frac{Z}{n}$, we construct the basis of Hilbert space ${\ca$H}_n$intermsof$θ$functionsofthepositions$z_i$of$n$solitons.Thewrappingaroundthetorusgeneratesthealgebra${\cal A}_n$,whichisthe$Z_n \times Z_n$Heisenberggroupon$θ$functions.Wefindthegenerators$g$ofanlocalelliptic$su(n)$,w$transform covariantly by the global gauge transformation of $\mathcal{A}$By acting on ${\mathcal{H}}_n$ we establish the isomorphism of ${\mathcal{A}}_n$$g$. We embed this $g$ into the $L$-matrix of the elliptic Gaudin and$modelstogivethedynamics.Themomentmapofthistwistedcotangent$su_n({\cal T})$bundleismatchedtothe$D$-equationwithFayet-Illiopoulossourceterm,sothedynamicsoftheN.C.solitonsbecomesthatofthebrane.Thegeometricconfiguration$(k, u)$ofth$spectral curve $\det |L(u)-k|=0$ describes the brane configuration, with the dynamical variables $z_i$ of N.C. solitons as$moduli$T^{\otimes n} / S_n$.Furthermore,intheN.C.Chern-SimonstheoryforthequantumHalleffect,theconstrainequationwithquasiparticlesourceisidentifiedalsowiththemomentmapeqaution$the N.C. $s{u}_n(\mathcal{T})$ cotangent bundle with marked points. The eigenfunction of the Gaudin differential $L$-operators as the Laughli$wavefunction is solved by Bethe ansatz.

25 pages, plain latex, no figures

References in corpus (1)

• Elliptic Ruijsenaars-Schneider and Calogero-Moser Models Represented by Sklyanin Algebra and sl(n) Gaudin Algebra