From rieffel@Math.Berkeley.EDU Thu Nov 11 10:53:01 2004 Date: Wed, 10 Nov 2004 09:03:21 -0800 (PST) From: Marc A. Rieffel To: bzumino@lbl.gov, origa@socrates.Berkeley.EDU Subject: monopoles Dear Bruno and Ori, During my talk last week I was asked for names of physicists who had written about monopoles over matrix algebras, and I could not remember good references since it was some months since I had looked at that, being in the meantime buried in trying to do the related mathematics. I append below the abstracts of some representative papers. Please pass these on to anyone else attending the seminar that you may know would be interested. It was a pleasure to talk in the seminar, and I enjoyed the various questions asked during my talk. Marc ------------------- hep-th/0407089 Finite Gauge Theory on Fuzzy CP^2 Authors: Harald Grosse, Harold Steinacker Comments: 42 pages + 17 pages appendix, 1 figure Report-no: LMU-TPW 04-05, UWTHPh-2004-16 We give a non-perturbative definition of U(n) gauge theory on fuzzy CP^2 as a multi-matrix model. The degrees of freedom are 8 hermitian matrices of finite size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The model depends on a noncommutativity parameter 1/N, and reduces to the usual U(n) Yang-Mills action on the 4-dimensional classical CP^2 in the limit N -> \infty. We explicitly find the monopole solutions, and also certain U(2) instanton solutions for finite N. The quantization of the model is defined in terms of a path integral, which is manifestly finite. An alternative formulation with constraints is also given, and a scaling limit as R^4_\theta is discussed. ------------------------- math-ph/0105033 Fuzzy Line Bundles, the Chern Character and Topological Charges over the Fuzzy Sphere Authors: Harald Grosse, Christian W. Rupp, Alexander Strohmaier Comments: Latex2e, 13 pages, 1 figure. This paper continues and supersedes math-ph/0103003. v2: Typos corrected Subj-class: Mathematical Physics; Quantum Algebra MSC-class: 81R60; 55R10 Journal-ref: J.Geom.Phys. 42 (2002) 54-63 Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation based differential calculus. The associated Chern numbers (topological charges) arise to be non-integer, in the commutative limit the well known integer Chern numbers of the complex line bundles over the 2-sphere are recovered. ------------------------------ hep-th/0404045 From: Paolo Valtancoli [view email] Date (v1): Tue, 6 Apr 2004 12:56:14 GMT (14kb) Date (revised v2): Thu, 8 Apr 2004 06:15:14 GMT (14kb) Projectors, matrix models and noncommutative monopoles Authors: P. Valtancoli Comments: 22pages, LaTeX, no figures We study the interconnection between the finite projective modules for a fuzzy sphere, determined in a previous paper, and the matrix model approach, making clear the physical meaning of noncommutative topological configurations. ------------------------ hep-th/9811169 Monopoles and Solitons in Fuzzy Physics Authors: S. Baez, A. P. Balachandran, S. Vaidya, B. Ydri Comments: 17 pages, Latex. Uses amstex, amssymb.Spelling of the name of one Author corrected. To appear in Commun.Math.Phys Report-no: SU-4240-688, TIFR/TH/98-44 Subj-class: High Energy Physics - Theory; Mathematical Physics; Quantum Algebra Journal-ref: Commun.Math.Phys. 208 (2000) 787-798 Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of discrete physics and matrix models and discuss mathematically coherent discretizations of monopoles and solitons using fuzzy physics and noncommutative geometry. A fuzzy sigma-model action for the two-sphere fulfilling a fuzzy Belavin-Polyakov bound is also put forth. ---- hep-th/0408111 Duality in Fuzzy Sigma Models Authors: A.P. Balachandran, Giorgio Immirzi Comments: Latex, 9 pages; misprints corrected Report-no: SU-4252 797, DFUP-04-18 Subj-class: High Energy Physics - Theory; Algebraic Geometry Nonlinear `sigma' models in two dimensions have BPS solitons which are solutions of self- and anti-self-duality constraints. In this paper, we find their analogues for fuzzy sigma models on fuzzy spheres which were treated in detail by us in earlier work. We show that fuzzy BPS solitons are quantized versions of `Bott projectors', and construct them explicitly. Their supersymmetric versions follow from the work of S. Kurkcuoglu. --------------------