Quantum Principal Bundles

Table of Contents

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

The formalism of Connections

The fundamental concept of connections on a quantum principal bundle is defined as follows. Let left-invariants be the space of left-invariant elements [W3] of the first-order calculus Gamma. This space is the analog of the dual space of the Lie algebra of the structure group G. A connection on P is every first-order hermitian linear map connection-map satisfying

connection-definition

where adjoint-action-sum and adjoint-action is the corresponding quantum adjoint action of G.

The first summand in the above formula corresponds to the pseudotensoriality property of connections. The second summand plays the role of the classical requirement that connections map fundamental vector fields into their generators [KN]. It can be shown [D3] that every quantum principal bundle admits a connection.

In classical geometry, fixing a connection means that we can speak about vertical differential forms (relative to a given connection). Horizontality is an intrinsic property but verticality is connection-dependent. Furthermore, the whole algebra of differential forms splits into the tensor product of horizontal and vertical forms.

A similar situation holds in quantum theory. As explained in [D3], every connection omega on a quantum principal bundle P naturally induces a horizontal-vertical splitting of the form

hor-ver-decomposition

where left-invariant-forms is the algebra of all left-invariant forms on the group (the left-invariant part of gamma-wedge, equivalently the subalgebra of gamma-wedge generated by gamma-inv). In contrast to the classical case however, the map mu-omega is not an algebra isomorphism: it is generally only left-linear over quantum horizontal forms.

With the help of the decomposition mu-omega we can now define the horizontal projection operator horizontal-projection simply by annihilating the vertical components of differential forms. And having the operator of horizontal projection, we can define the covariant derivative

covariant-derivative-map
and the curvature tensor
curvature-tensor
the same way as in the classical geometry.

It can be shown [D3] that all basic classical identities with the curvature and covariant derivative have quantum counterparts. In particular, we have a quantum Bianchi identity

q-bianchi-identity

and a generalized Leibniz rule for the covariant derivative

generalized-leibniz-rule

where phi-psi-in-qhors. The above purely quantum terms vanish, if the connection omega is sufficiently compatible with the geometrical structure of the bundle, in the appropriate sense. Such connections are called regular connections. By definition, in classical geometry all connections are regular.

It is also possible to prove the first structure equation

first-structure-eq
where the brackets lie-brackets are induced from the appropriate [W3][D3] analog of the Lie commutator for G.
[Previous Segment]: Differential Calculus
[Next Segment]: Quantum Characteristic Classes