Current-algebra cocycles and conditional chiral anomaly cancellation — Examples

Let $\mathfrak{g}=\mathbb{R}t$ be the one-dimensional Abelian real Lie algebra and take $V_{+}=V_{-}=\mathbb{C}$. For $q\in \mathbb{R}$, define complex-linear representations by ${\rho }_{\pm }(zt)=\pm qz{\mathrm{id}}_{\mathbb{C}}$ for $z\in \mathbb{C}$, and attach the left-handed bookkeeping label to both. This opposite-charge pair is called vectorlike at the representation-data level here. It does not construct spinor fields or a charge-conjugation map; an interpretation as a field pair requires a separately specified field model and convention. The coefficient sums satisfy $q^3+(-q{)}^3=0$ and $q+(-q)=0$. With the no-half anticommutator convention $\{A,B\}=AB+BA$, the cubic coefficient is $d=2(q^3+(-q{)}^3)=0$, and the separate linear representation coefficient is $\mathrm{\ell }=q+(-q)=0$. This verifies cancellation of these two algebraic coefficients in that interface only; it does not identify them with a regulated anomaly, prove an all-orders theorem, or establish a nonperturbative gauge theory or cancellation of global anomalies.

Four-dimensional chiral anomaly coefficient interface