Communications in Mathematical Physics

, Volume 220, Issue 2, pp 301-331

First online:

Strong Connections and Chern–Connes Pairing¶in the Hopf–Galois Theory

  • Ludwik DąbrowskiAffiliated withScuola Internazionale Superiore di Studi Avanzati, Via Beirut 2–4, 34014 Trieste, Italy.¶E-mail:
  • , Harald GrosseAffiliated withInstitute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria.¶E-mail:
  • , M. HajacAffiliated withMathematical Institute, Polish Academy of Sciences, ul. Śniadeckich 8, Warsaw, 00-950, Poland

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We reformulate the concept of connection on a Hopf–Galois extension BP in order to apply it in computing the Chern–Connes pairing between the cyclic cohomology HC 2 n (B) and K 0 (B). This reformulation allows us to show that a Hopf–Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz–Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern–Connes pairing for the super line bundles associated to a super Hopf fibration.