Introduction to my book “Lectures 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) #

Geometry of classical mechanics and field theory mainly is differential geometry of finite-dimensional smooth manifolds and fibre bundles.

At the same time, the standard mathematical language of quantum mechanics and field theory has been long far from geometry. In the last twenty years, the incremental development of new physical ideas in quantum theory has called into play advanced geometric techniques, based on the deep interplay between algebra, geometry and topology.

Geometry in quantum systems speaks mainly the algebraic language of rings, modules and sheaves due to the fact that basic ingredients in the differential calculus and differential geometry of smooth manifolds (except non-linear differential operators) can be restarted in a pure algebraic way.

Let X be a smooth manifold and C(X) a ring of smooth real functions on X. A key point is that, by virtue of the well-known Serre--Swan theorem, a C(X)-module is finitely generated and projective if and only if it is isomorphic to a module of sections of some smooth vector bundle over X. Moreover, this isomorphism is a categorial equivalence. Therefore, differential geometry of smooth vector bundles can be adequately formulated in algebraic terms of a ring C(X), its derivations and the Koszul connections.

In a general setting, let K be a commutative ring, A an arbitrary commutative K-ring, and P, Q some A-modules. The K-linear Q-valued differential operators on P can be defined. The representative objects of functors Q-> Dif(P,Q) are jet modules JP of P. Using the first order jet module J^1P, one also restarts the notion of a connection on an A-module P.

As was mentioned above, if P is a C(X)-module of sections of a smooth vector bundle Y-> X, we come to the familiar notions of a linear differential operator on Y, the jets of sections of Y-> X and a linear connection on Y-> X.

In quantum theory, Banach and Hilbert manifolds, Hilbert bundles and bundles of C*-algebras over smooth manifolds are considered. Their differential geometry also is formulated as geometry of modules, in particular, C(X)-modules.

Let K be a commutative ring, A a (commutative or non-commutative) K-ring, and Z(A) the center of A. Derivations of A constitute a Lie K-algebra DA. Let us consider the Chevalley-Eilenberg complex of K-multilinear morphisms of DA to A, seen as a DA-module. Its subcomplex O*(A) of Z(A)-multilinear morphisms is a differential graded algebra, called the Chevalley-Eilenberg differential calculus over A. If A is an R-ring C(X) of smooth real functions on a smooth manifold X, the module DC(X) of its derivations is a Lie algebra of vector fields on X, and the Chevalley-Eilenberg differential calculus over C(X) is exactly an algebra of exterior forms on a manifold $X$ so that the Chevalley-Eilenberg coboundary operator d coincides with an exterior differential, i.e., O*(A) is the familiar de Rham complex. In a general setting, one therefore can think of elements of the Chevalley-Eilenberg differential calculus over an algebra A as being differential forms over A.

Similarly, the differential calculus over a Grassmann-graded commutative ring is constructed. This is the case of supergeometry. In supergeometry, connections on graded manifolds and supervector bundles are defined as those on graded modules over a graded commutative ring and graded local-ringed spaces.

Note that a Grassmann-graded commutative ring is a particular non-commutative ring. However, the definition of its derivations differs from the non-commutative Leibniz rule. Therefore, supergeometry is not particular non-commutative geometry.

Non-commutative geometry also is developed as a generalization of the calculus in commutative rings of smooth functions. In a general setting, any non-nommutative K-ring A over a commutative ring K can be called into play. One can consider the above mentioned Chevalley-Eilenberg differential calculus over A. However, the definition of differential operators on modules over commutative rings fails to be straightforwardly extended to the non-commutative ones. A key point is that A-module morphisms fail to be zero order differential operators if A is non-commutative. In this case, several nonequivalent definitions of differential operators have been suggested. Accordingly, there are also different definitions of a connection on modules over a non-commutative ring.

The present book aims to summarize the relevant material on the differential calculus and differential geometry on modules and rings. Some basic applications to quantum models are considered.

For the sake of convenience of the reader, several topics on cohomology are compiled in Appendix.

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.