Geometria Não-Comutativa /
- 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
IV. ANOTHER NONCOMMUTATIVE GEOMETRY *-DEFORMATION OF ALGEBRA OF
V. MORE ADVANCED TOPICS AND EXEMPLES:
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
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.
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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