Descent equations of Yang--Mills anomalies in noncommutative geometry

Consistent Yang--Mills anomalies $intom_{2n-k}^{k-1}$ ($ninN$, $ k=1,2, ldots ,2n$) as described collectively by Zuminos descent equations $deltaom_{2n-k}^{k-1}+ddom_{2n-k-1}^{k}=0$ starting with the Chern character $Ch_{2n}=ddom_{2n-1}^{0}$ of a principal $SU(N)$ bundle over a $2n$ dimensional manifold are considered (i.e. $intom_{2n-k}^{k-1}$ are the Chern--Simons terms ($k=1$), axial anomalies ($k=2$), Schwinger terms ($k=3$) etc. in $(2n-k)$ dimensions). A generalization in the spirit of Connes noncommutative geometry using a minimum of data is found. For an arbitrary graded differential algebra $CC=bigoplus_{k=0}^infty CC^{(k)}$ with exterior differentiation $dd$, form valued functions $Ch_{2n}: CC^{(1)}to CC^{(2n)}$ and $om_{2n-k}^{k-1}: underbrace{CC^{(0)}timescdots times CC^{(0)}}_{mbox{{small $(k-1)$ times}}} times CC^{(1)}to CC^{(2n-k)}$ are constructed which are connected by generalized descent equations $deltaom_{2n-k}^{k-1}+ddom_{2n-k-1}^{k}=(cdots)$. Here $Ch_{2n}= (F_A)^n$ where $F_A=dd(A)+A^2$ for $AinCC^{(1)}$, and $(cdots)$ 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 $int$ on it. Examples for graded differential algebras with such integrations are given and thereby noncommutative generalizations of Yang--Mills anomalies are found.