Connes’ calculus for the quantum double suspension

Abstract

a b s t r a c tGiven a spectral triple (A, H, D) Connes associated a canonical differential graded algebra Ω • D (A). However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes' calculus for its quantum double suspension. This allows one to compute Ω • D for spectral triples obtained by iterated quantum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes' calculus for a large family of spectral triples.

Subadditivity and additivity of the noncommutative Yang-Mills action functional

“…So, Dirac dga can be thought of as noncommutative space of forms. However, this dga is very hard to compute and not much of computation is known in the literature except ( [6], [3], [4], [5]). Using this Dirac dga, Connes extended the classical notion of Yang-Mills action functional to the noncommutative geometry framework in ( [9]).…”

“…(2) If one starts with an odd spectral triple (A, H, D), i,e. without the grading operator, then one can construct an even spectral triple (A, H = H ⊗C 2 , D, γ) using any two 2×2 Pauli spin matrices such that Ω • D (A) ∼ = Ω • D (A) as dgas (Lemma 2.7 in [5]). Therefore, when working with Dirac dga, one can w.l.o.g.…”

“…Section [4] contains an instance of additivity of Y-M. Section [5], [6], [7] discuss examples where our hypothesis is validated. These include the case of compact spin manifolds, matrix algebras, noncommutative n-torus and the quantum Heisenberg manifolds.…”