Introduction to Quantum Principal Bundles

by Micho Durdevich

Table of Contents

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


Introduction

In diversity of mathematical concepts and theories a fundamental role is played by those giving a unified treatment of different and at a first sight mutually independent circles of problems.

As far as classical differential geometry is concerned, such a fundamental role is given to the theory of principal bundles. Various basic concepts of theoretical physics are also naturally expressible in the language of principal bundles. Classical gauge theory and general relativity theory are paradigmic examples.

But classical geometry is just a very special case of a much deeper quantum geometry.

So it is natural to ask what would be the analogs of principal bundles in quantum geometry. And it is reasonable to expect that such quantum principal bundles would play a similar fundamental role in quantum geometry, as is the role of classical principal bundles in classical geometry.

During last years, I have been developing a general theory of quantum principal bundles, where quantum groups play the role of structure groups and general quantum spaces play the role of the base manifolds. The main formalism is presented in papers [D2][D3], and a very brief exposition (without proofs) can be found in [D1].

All my scientific papers mentioned here are available for a direct download (in Tex, PostScript and PDF) from the main download page. Here we shall discuss basic ideas of the theory, trying to speak informally and paying a special attention to interesting purely quantum phenomenas appearing in the game.

It is not difficult to incorporate the basic geometrical idea of a principal bundle, into the noncommutative context. Let G be a compact matrix quantum group [W2], represented by a Hopf *-algebra smooth-A (playing the role of polynomial functions over G). Let us consider a quantum space P represented by a *-algebra smooth-B. Let us assume that G acts on P on the right. This means that the appropriate *-homomorphism coaction-map is given (giving a coaction of the Hopf-algebra smooth-A on smooth-B). The main requirement is that the diagram

main-diagram

is commutative. Here coproduct is the coproduct map in smooth-A. Following classical theory, we would like to express the idea that the action of G is 'free'. It turns out that the freeness condition is expressible as the property that for every element-a-in-A there exist elements elements-bq such that

sum-freeness

Now having the action of G on P we can define the base manifold M as the corresponding 'orbit space'. At the dual level, this means that the *-algebra V representing M is just the fixed point algebra for the action F. More precisely,

base-manifold-def

Geometrically, the idea is that smooth functions on M are just smooth functions on P constant along the action orbits.

In such a way, we arrive to quantum principal bundles.


[Next Segment]: Differential Caluclus on Quantum Principal Bundles