Descent equations of Yang-Mills anomalies in noncommutative geometry

Abstract

Consistent Yang-Mills anomalies ∫ ω 2 n− k k−1 ( n ϵ N, k = 1,2,…, 2 n) as described collectively by Zumino's descent equations δω 2 n− k k−1 + d ω 2 n− k−1 k = 0 starting with the Chern character Ch 2 n = d ω 2 n−1 0 of a principal SU ( N) bundle over a 2 n-dimensional manifold are considered (i.e. ∫ ω 2 n− k k−1 are the Chern-Simons terms ( k = 1), axial anomalies ( k = 2), Schwinger terms ( k = 3), etc. in (2 n − k) dimensions). A generalization in the spirit of Conne's non-commutative geometry using a minimum of data is found. For an arbitrary graded differential algebraa Ω = ⊗ k = 0 ∞Ω (k) with exterior differentiation d, form-valued functions ▪ are constructed which are connected by generalized descent equations δω 2 n− k k−1 + d ω 2 n− k−1 k = (…). Here Ch 2 n = ( F A ) n , where F A = d( A) + A 2 for A ϵ Ω (1) , and (…) is not zero but a sum of graded commutators which vanish under integrations (traces). The problem of constructing Yang-Mills anomalies on a given graded differential algebra is thereby reduced to finding an interesting integration ∫ on it. Examples for graded differential algebras with such integrations are given, and thereby noncommutative generalizations of Yang-Mills anomalies are found.