G. Sardanashvily, “Lectures on Differential Geometry of Modules
and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken
Differential geometry of
smooth vector bundles can be formulated in algebraic terms of modules over
rings of smooth function. Generalizing this construction, one can define the
differential calculus, differential operators and connections on modules over
arbitrary commutative, graded commutative and non-commutative rings. For
instance, this is the case of quantum theory, supergeometry and non-commutative
geometry, respectively. The book aims to summarize the relevant material on
this subject. Some basic applications to quantum theory are considered.
The book is based on the
graduate and post graduate courses of lectures given at the Department of
Theoretical Physics of Moscow State University (Russia )
and the Department of Mathematics and Physics of University of Camerino (Italy
Contents
1
Commutative geometry: 1.1 Commutative algebra, 1.2 Differential operators
on modules and rings, 1.3 Connections on modules and rings, 1.4 Differential calculus
over a commutative ring, 1.5 Local-ringed spaces, 1.6 Differential geometry
of C(X)-modules, 1.7 Connections on local-ringed spaces.
2
Geometry of quantum systems: 2.1 Geometry of Banach manifolds, 2.2 Geometry of
Hilbert manifolds, 2.3 Hilbert and C*-algebra bundles, 2.4 Connections on
Hilbert and C*--algebra bundles, 2.5 Instantwise quantization, 2.6 Berry
3
Supergeometry: 3.1 Graded tensor calculus, 3.2 Graded differential calculus
and connections, 3.3 Geometry of graded manifolds, 3.4 Supermanifolds, 3.5
Supervector bundles, 3.6 Superconnections.
4
Non-commutative geometry: 4.1 Modules over C*-algebras, 4.2 Non-commutative differential calculus, 4.3 Differential operators in non-commutative geometry, 4.4 Connections in
non-commutative geometry, 4.5 Matrix geometry, 4.6 Connes’ non-commutative
geometry, 4.7 Differential calculus over Hopf
algebras.
5
Appendix. Cohomology: 5.1 Cohomology of complexes, 5.2 Cohomology of Lie
algebras, 5.3 Sheaf cohomology.
