Descent equations of Yang--Mills anomalies in noncommutative geometry

Abstract

Consistent Yang--Mills anomalies $\int\om_{2n-k}^{k-1}$ ($n\in \mathbb{N}$, $k=1,2,\dots ,2n$) as described collectively by Zumino's descent equations $δ\om_{2n-k}^{k-1}+\dd\om_{2n-k-1}^{k}=0$ starting with the Chern character $Ch_{2n}=\dd\om_{2n-1}^{0}$ of a principal $\SU(N)$ bundle over a $2n$ dimensional manifold are considered (i.e.\ $\int\om_{2n-k}^{k-1}$ 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)}\times\cdots \times \CC^{(0)}}_{\mbox{\small $(k-1)$ times}} \times \CC^{(1)}\to \CC^{(2n-k)}$ are constructed which are connected by generalized descent equations $δ\om_{2n-k}^{k-1}+\dd\om_{2n-k-1}^{k}=(\cdots)$. Here $C{h}_{2n}=(F_A{)}^n$ where $F_A=\dd(A)+A^2$ for $A\in\CC^{(1)}$, 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.

20 pages, latex, no figures (A few minor errors corrected)

References in corpus (5)

• Fermion Current Algebras and Schwinger Terms in 3+1 Dimensions
• (3+1)-Dimensional Schwinger Terms and Non-commutative Geometry
• Non--commutative Integration Calculus
• Elementary Derivation of the Chiral Anomaly
• On anomalies and noncommutative geometry

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• Scattering matrix in external field problems
• Chiral anomalies in noncommutative gauge theories
• Anomalies and Schwinger terms in NCG field theory models
• Consistent Anomalies in Translation-Invariant Noncommutative Gauge Theories
• Quantum Gauge Theories and Noncommutative Geometry
• star-Cohomology, Third Type Chern Character and Anomalies in General Translation-Invariant Noncommutative Yang-Mills