Dirac Operators on Noncommutative Principal Circle Bundles

Abstract

We study spectral triples over noncommutative principal U (1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low dimensional noncommutative tori are analysed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U (1)-bundle S 3 θ → S 2 .

Gauge theory for spectral triples and the unbounded Kasparov product

“…Upon making explicit choices of representatives for the gamma matrices, the product operator has the form [26]). 20…”

“…In this way we obtain an explicit description of the noncommutative Hopf fibration in terms of an unbounded KK-product, thus going beyond the projectivity studied in [19,20]. A similar construction, in the context of modular spectral triples, appeared in [27] to construct Dirac operators on a total space carrying a circle action (namely, on SU q (2)).…”

Gluing Non-commutative Twistor Spaces

“…The invariant subalgebra (S 3 θ ) U (1) , which corresponds to the base of the fibration, is generated by the elements X = βα, X * = α * β * and Y = αα * − 1 2 with the relations XY = Y X, Y X * = X * Y and Y 2 +XX * = 1 4 , hence it is the algebra of functions of a commutative 2-dimensional sphere. This Hopf fibration S 1 ֒→ S 3 θ → S 2 is considered from the point of view of spectral triples and Dirac operators in [20].…”

Gluing Noncommutative Twistor Spaces

“…2-dimensional sphere. This Hopf fibration S 1 ֒→ S 3 θ → S 2 is considered from the point of view of spectral triples and Dirac operators in [20].…”