2002/04/19 by Bo-Yu Hou, BO-YU HOU, Dan-Tao Peng +5 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #Cotangent bundle #Eigenfunction #Elliptic function #Gauge theory #Noncommutative algebraic geometry #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Torus #Wave function #hep-th
paper · pdf · doi:10.1142/s0217751x03014228
published in International Journal of Modern Physics A 18(14), 2477-2500 (World Scientific) · 25 pages, plain latex, no figures
arxiv created 2002/04/19 · openalex publication_date 2003/06/10 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
For the noncommutative torus [Formula: see text], in the case of the noncommutative parameter [Formula: see text], 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 [Formula: see text], which is the Z n × Z n Heisenberg group on θ functions. We find the generators g of a local elliptic su (n), which transform covariantly by the global gauge transformation of [Formula: see text]. By acting on ℋ n we establish the isomorphism of [Formula: see text] and g. We embed this g into the L-matrix of the elliptic Gaudin and Calogero–Moser models to give the dynamics. The moment map of this twisted cotangent [Formula: see text] bundle is matched to the D-equation with the Fayet–Illiopoulos source term, so the dynamics of the noncommutative solitons become that of the brane. The geometric configuration (k, u) of the spectral curve det |L(u) - k| = 0 describes the brane configuration, with the dynamical variables z i of the noncommutative solitons as the moduli T ⊗ n /S n . Furthermore, in the noncommutative Chern–Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map equation of the noncommutative [Formula: see text] cotangent bundle with marked points. The eigenfunction of the Gaudin differential L-operators as the Laughlin wave function is solved by Bethe ansatz.