Introduction to Quantum Principal Bundles
by Micho Durdevich
Table of Contents
IntroductionIn 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
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
is commutative. Here
is the coproduct map
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,
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