SOLITONS ON NONCOMMUTATIVE TORUS AS ELLIPTIC CALOGERO–GAUDIN MODELS, BRANES AND LAUGHLIN WAVE FUNCTIONS

BO-YU HOU, DAN-TAO PENG, KANG-JIE SHI, RUI-HONG YUE
2003 International Journal of Modern Physics A  
For the noncommutative torus T, in case of the N.C. parameter θ = Z/n, we construct the basis of Hilbert space H_n in terms of θ functions of the positions z_i of n solitons. The wrapping around the torus generates the algebra A_n, which is the Z_n × Z_n Heisenberg group on θ functions. We find the generators g of an local elliptic su(n), wtransform covariantly by the global gauge transformation of ABy acting on H_n we establish the isomorphism of A_ng. We embed this g into the L-matrix of the
more » ... lliptic Gaudin andmodels to give the dynamics. The moment map of this twisted cotangent su_n( 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 thspectral curve det|L(u) - k| = 0 describes the brane configuration, with the dynamical variables z_i of N.C. solitons asmoduli T^⊗ 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( T) cotangent bundle with marked points. The eigenfunction of the Gaudin differential L-operators as the Laughliwavefunction is solved by Bethe ansatz.
doi:10.1142/s0217751x03014228 fatcat:gknlgusxofgabhnsd564q4c7fq