Quantum Principal Bundles

Table of Contents

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

Quantum Characteristic Classes

We shall now outline the construction of characteristic classes and the Weil homomorphism for quantum principal bundles. A detailed theory is presented in [D8][D9]. It turns out that there exist two very different natural ways of incorporating classical Weil theory into the quantum context.

Regular Connections

The first methods works for quantum bundles admitting regular connections. As we mentioned in the previous section, for such connections the covariant derivative

satisfies the graded Leibniz rule, and the standard form of the Bianchi identity holds.

As explained in [W3], there exists a natural braid operator

This map is a proper replacement for the classical transposition. It intertwines the adjoint action of the structure quantum group on

Let be the braided-symmetric algebra built over . In other words, it is a *-algebra generated by the vector space and the braided-symmetricity quadratic relations

. It turns out [D3] that the curvature map

admits a unique extension to a *-homomorphism

. This map intertwines the natural adjoint action of G on

and the right action

. In particular, this means that

where

is the subalgebra of adjointly-invariant elements.

It can be shown that the image of is contained in closed elements of . Moreover, if we pass to cohomology classes of . it turns out that such a factorized map does not depend of the choice of a regular connection . In such a way we obtain the quantum Weil homomorphism as an intrinsic map

. As we already mentioned, the space

plays the role of the dual space of the Lie algebra of G. Accordingly,

plays the role of the polynomial functions over the Lie algebra of G and

plays the role of the invariant polynomials for G. This is the quantum analog of universal chracteristic classes.

General Bundles

In the case of general quantum principal bundles (where we are not interested whether or not the bundle admits regular connections) the previous construction does not work, and another approach is necessary. The main idea is to define quantum characteristic classes as generated by those generic algebraic expressions, built from a given connection

and its differential

, that are closed elements of

More preciesly, we start from the free differential algebra built over . It is possible to introduce a differential version of the adjoint action, as a graded-differential *-homomorphism Here is the graded tensor product of graded-differential algebras. By construction every connection extends uniquely to a graded-differential *-homomorphism This homomorphism intertwines the maps ad and F, and hence it maps the ad-invariants into . To simplify the notation we have denoted various extended maps by the same symbols as the original versions.

The algebra is a graded-differential *-subalgebra of and the restricted map is a graded-differential *-homomorphim--so we can pass to cohomology classes. It turns out that the cohomology map does not depend of the choice of a connection and we arrive to an intrinsic *-homomorphism

In this context, the algebra plays the role of universal characteristic classes for G. Various very interesting purely quantum phenomenas appear. At first we have a variety of quantum principal bundles with a very non-trivial topological structure and in many cases without any classical limit. In particular, there exist examples of quantum principal bundles with non-trivial odd-dimensional characteristic classes. This is impossible in classical geometry, where all characteristic classes are expressed via the curvature tensor (this is also impossible in the quantum-regular case). Secondly, it is very interesting to consider quantum bundles with classical structure groups over classical manifolds, and to analyze their characteristic classes. Since we have an additional freedom in constructing a differential calculus over the bundle and the group, new cohomology classes (that are not interpretable as characteristic classes in the classical sense) are now included in the framework of quantum characteristic classes. It is also very interesting to analyze relations between regular and general universal characteristic classes. In general, two algebras of universal characteristic classes will be different. The difference between them is encoded in the algebraic properties of the braid operator

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