Geometry in quantum theory III: Differential geometry of modules and rings

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

Any smooth real manifold X is homeomorphic to the real spectrum of the real ring C(X) of smooth real functions on X provided with the Gelfand topology. Furthermore, the sheaf CX of germs of functions from C(X) on this topological space fixes a unique smooth manifold structure on X such that it is the sheaf of smooth functions on X. The pair (X,CX) exemplifies a local-ringed space. One can associate to any commutative ring A the particular local-ringed space, called an affine scheme, on the spectrum Spec(A) of A endowed with the Zariski topology.

Given a connected smooth manifold X and the ring C(X) of smooth real functions on X, the well-known Serre -- Swan theorem states  that a C(X)-module is finitely generated projective iff it is isomorphic to the module of sections of some vector bundle over X. Moreover, this  isomorphism is a categorical. A variant of the Serre -- Swan theorem for Hilbert modules over non-commutative C*-algebras holds.

Let K be a commutative ring, A a 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 the functor Q->diff(P,Q) are the jet modules JP of P. Using the first order jet module, one also restarts the notion of a connection on an A-module P. For instance, 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 supergeometry, connections on graded modules  over a graded commutative ring and graded local-ringed spaces are defined.

In non-commutative geometry, different definitions of a differential operator on modules over a non-commutative ring have been suggested. Roughly speaking, the difficulty lies in the fact that, if d is a derivation of a non-commutative ring A, the product ad, where a is from A, need not be so. There are also different definitions of a connection on modules over a non-commutative ring.

Let K be a commutative ring, A a (commutative or non-commutative) K-ring, and Z(A) the center of A. Derivations of A make up a Lie K-algebra d(A). Let us consider the Chevalley -- Eilenberg complex of K-multilinear morphisms of d(A) to A, seen as a d(A)-module. Its subcomplex of Z(A)-multilinear morphisms is a differential graded algebra, called the Chevalley -- Eilenberg differential calculus over A.

If A is the real ring C(X) of smooth real functions on a smooth manifold X, the module d C(X) of its derivations is the Lie algebra of vector fields on X and the Chevalley -- Eilenberg differential calculus over C(X) is exactly the algebra of exterior forms on a manifold X where the Chevalley -- Eilenberg coboundary operator d coincides with the familiar exterior differential. 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.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric and Algebaic Topological Methods in Quantum Mechanics (WS, 2005)
G.Sardanashvily, G.Giachetta, What is a geometry in quantum theory, arXiv: hep-th/0401080