My new book on differential geometry of modules and rings

G. Sardanashvily, “Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory" (Lambert Academic Publishing, Saarbrucken, 2012) #

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). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians.

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 connection.

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.