Introduction to Quantum Principal Bundles

Introduction

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 (playing the role of polynomial functions over G). Let us consider a quantum space P represented by a *-algebra . Let us assume that G acts on P on the right. This means that the appropriate *-homomorphism is given (giving a coaction of the Hopf-algebra on ). The main requirement is that the diagram

is commutative. Here is the coproduct map in . 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 there exist elements such that

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 representing M is just the fixed point algebra for the action F. More precisely,