I review results from recent investigations of anomalies in fermion--Yang Mills systems in which basic notions from noncommutative geometry (NCG) where found to naturally appear. The general theme is that derivations of anomalies from quantum field t
heory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficient. This paper is intended as selfcontained introduction to this line of ideas, including a review of some basic facts about anomalies. I first explain the notions from NCG needed and then discuss several different anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current algebras, Chern--Simons terms from effective fermion actions in arbitrary odd dimensions. I also discuss the descent equations which summarize much of the geometric structure of anomalies, and I describe that these have a natural generalization to NCG which summarize the corresponding structures on the level of quantum field theory. Contribution to Proceedings of workshop `New Ideas in the Theory of Fundamental Interactions, Szczyrk, Poland 1995; to appear in Acta Physica Polonica B.
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 pri
ncipal $SU(N)$ 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.
I discuss examples where basic structures from Connes noncommutative geometry naturally arise in quantum field theory. The discussion is based on recent work, partly collaboration with J. Mickelsson.
