Centro Brasileiro de Pesquisas Físicas - CBPF/MCT
III Escola de Verão do CBPF - Fev 2001

Curso: Geometria Não-Comutativa / Noncommutative geometry

- Luis Alvarez-Gaumé (CERN)
- Ion Vasile Vancea (IFT-SP)

We will introduce several basic and advanced aspects of noncommutative geometry and noncommutative field theories its connections with strong interactions and string theories.

Part I: Mathematical Introduction
I. BASIC ALGEBRAIC OBJECTS: Modules, Algebras, Ideals, *-algebras, C*-algebras, Banach-algebras. Definitions and essential properties.
II. ALGEBRAIC STRUCTURE OF COMMUTATIVE (DIFFERENTIAL) GEOMETRY: Algebras of functions on manifolds, Multilinear applications, Forms, Bundles vs. modules, maybe de Rham cohomology.
III. NONCOMMUTATIVE GEOMETRY: Dirac operator on Euclidean manifold, Spectral Triple, Noncommulative differential calculus, Forms, maybe Fredholm cohomology.
     a) A NC Quantum Mechanical Systems

     b) BRST calculus of NC Chern-Simons anomaly
     c) NC Supergeometry and Q-Algebra

Part II: Topics in Noncommulative Field Theories
I. INTRODUCTORY MATERIAL: A summary of Noncommutative Geometry. *-algebras, *-products an examples out of physics applications. Origin of noncommutative field theories from String Theory. (As basics references for this lecture one can use G. Landi´s book, A. Connes book and J. Madore´s book all three called "Non-commutative Geometry". I will also use in part the paper by N. Seiberg and E. Witten hep-th/9908142)
II. TOPICS IN NCQFT: Here I would to look at properties of perturbative field theory in this context, explore the UV/IR mixing. Problems with causality and unitarity. Some simple exact results where many of the physical properties of these theories can be exhibited (there are many references on this subject, Iwill use among otheres Minwalla et al 9912072 and 0006219, 0006209 and references therein)
III. NONCOMMUTATIVE GAUGE THEORIES: Formulation. Observables. Wilson and Polyakov loops. A quick review of ´tHooft´s theory of confinement in terms of twisted tori. Morita equivalence. Physical interpration. Problems of the future (Here there are many papers and one possible paper to consider is 0004147. But many other papers will be reviewed specially in the subject of Morita equivalence)
IV. SOLITONS AND INSTANTOS IN NONCOMMUTATIVE FIELD THEORIES: Contributions to open string tachyon potential (the basic reference are tehe papers by Strominger et al. J. Harvey et al, and Kutasov, Marino and Moore, apart from the seminal work by Sen)

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