Quantum Principal Bundles

Table of Contents

Introduction
Differential Calculus
Connections Formalism
Characteristic Classes
Quantum Classifying Spaces
Quantum Frame Bundles
Associated Bundles
Gauge Transformations
Literature

Differential Calculus

A careful analysis shows that the most appropriate approach to the foundations of differential calculus on quantum principal bundles is the axiomatic one. Actually, this becomes transparent already in the foundations of differential calculus on quantum groups [W3]. What we have to do is to introduce axiomatically a class of graded-differential *-algebras, representing differential forms over a given quantum principal bundle P.

But before introducing differential calculus on P, it is necessary to introduce differential calculus on the structure group G. Following [W3] let us assume that Gamma is a bicovariant *-calculus over G and let gamma-wedge be an appropriate graded-differential *-algebra built over Gamma (for example we can assume that gamma-wedge is the braided-exterior algebra of [W3] or the universal envelope described in [D1]-Appendix B).

The elements of Gamma play the role of first-order differential forms over G and the elements of gamma-wedge are interpreted as all possible differential forms.

We shall assume that the calculus on P is based on an arbitrary graded-differential *-algebra q-diff-forms satisfying the following three conditions:

  1. We have zero-order-forms. This means that 0th-order forms are just functions on the bundle;
  2. The differential algebra q-diff-forms is generated by smooth-B. In other words, the spaces homogeneous-qforms are linearly spanned by the elements of the form
    homogeneous-qforms
    where bi-in-B and q-differential is the differential. This is a kind of a minimality condition for q-diff-forms and it ensures uniqueness of many entities associated to the calculus;
  3. The action map F admits (necessarily unique, grade-preserving and hermitian) extension
    f-extended
    which is a differential algebra homomorphism. In the above expression we used the graded tensor product of graded-differential algebras.
All the above conditions are satisfied in the classical theory, where all the algebras are commutative and we play with the standard differential forms. In classical geometry differential calculus is considered as something intrinsically associated to the space--and consequently only the classical calculus is considered. The situation is very different in non-commutative geometry. For a given quantum principal bundle P we will have a variety of different calculi q-diff-forms. In particular it is possible to construct a non-classical differential calculus over a classical principal bundle P. Such quantum-type differential structures play a very interesting role in the study of characteristic classes and topological properties of classical spaces (see below). A general phenomena is that in quantum geometry there exist no a unique and distinguished way of constructing differential calculus on quantum spaces. Differential calculus is context-dependent.

It is always possible to construct a differential algebra satisfying the above listed conditions--for example as a trivial solution we can define q-diff-forms as the universal differential envelope [C] of B. Let us assume that a differential calculus q-diff-forms is fixed. Then we can naturally define two important algebras: The first one is the graded *-algebra of quantum horizontal forms, defined by

q-diff-horforms-def
The second one is a graded-differential *-algebra q-diff-forms-base representing the calculus on the base M. It is defined as the F-fixed point subalgebra of q-diff-forms. It is a subalgebra of hor(P). It is easy to see [D3] that the map F restricted to horizontal forms gives a map acting-on-qhors which is the right action of the structure group G on horizontal forms.
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