Equivalence of two approaches to Yang–Mills on noncommutative torus

Abstract

There are two notions of Yang-Mills action functional in noncommutative geometry. We show that for noncommutative n-torus both these notions agree. We also prove a structure theorem on the Hermitian structure of a finitely generated projective modules over spectrally invariant subalgebras of C -algebras.

Connes’ calculus for the quantum double suspension

“…module E over A can be written as pA n where p ∈ M n (A) is a self-adjoint idempotent i.e. a projection, and hence has a Hermitian structure on it induced from the canonical structure on A n (Lemma (2.2) in [13]). …”

“…However, note that  E = (p ⊗ u)(Σ 2 A) n is same as (p ⊗ u)(A ⊗ S) n because u is a rank one projection operator. We recall Theorem (3.3) from [13]. Goal of this section is to prove the following theorem.…”

Subadditivity and additivity of the noncommutative Yang-Mills action functional

“…Latter Connes formulated this notion more formally in the language of K-cycles or spectral triples in ( [9]), and investigated the case of noncommutative two-torus in great detail which suggests extensions of Yang-Mills theoretic techniques in the study of noncommutative differential (and possibly holomorphic) geometry of 'vector bundles' on C * -algebras. It turns out that these two notions of Yang-Mills in noncommutative geometry, the older one for the C * -dynamical systems (due to Connes-Rieffel in [13]) and the more formal one in the context of spectral triples (due to Connes in [9]), are equivalent for the case of noncommutative n-tori ( [3]) and the quantum Heisenberg manifolds ( [4]). However, the general case remains unanswered.…”

“…It would not be an exaggeration to say that Yang-Mills is an important and active area of research in noncommutative geometry, and over the years it has been studied by various authors (e.g. [13], [24], [25], [18], [21], [3], [4]).…”

“…we always consider even spectral triples. This is explained at the beginning of Section [3]. Now, if E 1 and E 2 are two Hermitian f.g.p modules over A 1 and A 2 respectively, and ∇ j ∈ C(E j ) for j = 1, 2, then ∇ := ∇ 1 ⊗ 1 + 1 ⊗ ∇ 2 is a connection on E = E 1 ⊗ E 2 .…”