Introduction to Noncommutative (Riemannian Spin) Geometry

Prof. Ludwik Dąbrowski

SISSA, 2012

• Description:
  
  These lectures concentrate on the latest layer of Noncommutative Geometry: Riemannian and Spin. It is encoded in terms of a spectral triple and its main ingredient, the Dirac operator. The canonical spectral triple on a Riemannian and spin manifold will be described starting with basic notions of multilinear algebra and differential geometry. Its basic properties, and then additional requirements that completely characterize this operator will be presented. They are essential for a further fascinating generalization to noncommutative spaces by A. Connes. In the course of the account some previous levels of NCG will be mentioned regarding the (differential) topology and calculus (like the equivalence between (locally compact) topological spaces and C*-algebras, and between vector bundles and finite projective modules, projectors and K theory, the Hochschild and cyclic cohomology, noncommutative integral, pseudodifferential calculus and local index formula). In the last part the concept of symmetries will be described in terms of Hopf algebras and quantum groups and applied to equivariant spectral triples. By concentrating the material in relatively few lectures some well established topics (e.g. the index theory) will not be discussed. Just an indispensable minimum from the well known theory of the (elliptic) Laplace operator will be used. Such a selection among the wealth of available material hopefully will lead us fast to some of the active and interesting fields of current research.

• Plan:
  1. Introduction.
  2. Exterior and Clifford algebras. Spin groups. Spinors.
  3. Spin structures.
  4. Dirac operator.
  5. Some analytic properties. Spectral triple.
  6. Other (seven) characteristic features:
  - dimension (finite summability)
  - regularity (smoothness)
  - finiteness & projectivity
  - reality
  - first order
  - orientation
  - Poincare duality
  7. Statement of the 'reconstruction theorem' of A. Connes.
  8. Other notions in N.C. vein: bundles, connections, pseudodifferential calculus, geodesic flows...
  
  If time permits:
  9. Examples: noncommutative tori, spheres, finite dimensional spectral triple
  10. Symmetries
  - Group actions (isometries, diffeomorphisms)
  - Hopf algebras and equivariant spectral triples
  

• References
  1. A. Connes, Noncommutative Geometry, Academic Press, 1994.
  2. A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys., 182 (1996) 155-176.
  3. A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives http://www.alainconnes.org/en/downloads.php
  4. L. Dabrowski, Group Actions on Spinors, Lecture Notes Bibliopolis, Napoli, 1988
  5. T. Friedrich, Dirac operators in Riemannian Geometry, Graduate Studies in Mathematics, vol 25. AMS, Providence, Rhode Island, 2000
  6. J. M. Gracia-Bondia, J. C. Varilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhauser Advanced Texts, Birkhauser, Boston, MA, 2001.
  7. P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, CRC Press, Boca Raton, USA, 1995.
  8. N. Higson, The residue index theorem (Lecture notes for the 2000 Clay Institute symposium on NCG); The local index formula in noncommutative geometry (Trieste lecture notes); http://www.math.psu.edu/higson/ResearchPapers.html.
  9. G. Landi, An Introduction to Noncommutative Spaces and their Geometries, Springer, Lecture Notes in Physics, 1997.
  10. H. Lawson, M. Michelsohn, Spin geometry, Princeton University Press 1989.
  11. R.J. Plymen, Strong Morita equivalence, spinors and symplectic spinors, J. Oper. Theory 16