13.1 Introduction

Before we introduce topological and geometrical methods to anomalies, we give a brief survey of the subject here. Let $\psi$ be a massless Dirac field in four-dimensional space interacting with an external gauge field ${\mathcal{A}}_{\mu }=A_{\mu }^{\alpha }{T}_{\alpha }$ , where $\{T_{\alpha }\}$ is the set of anti-Hermitian generators of the gauge group $G$ which is compact and semisimple (SU(N), for example). The theory is described by the Lagrangian

$$\mathcal{L}=\mathrm{i}\bar{\psi }{\gamma }^{\mu }\left({\partial }_{\mu }-{\mathcal{A}}_{\mu }\right)\psi .\text{\hspace{1em}(13.1)}$$

The Lagrangian is invariant under the usual (local) gauge transformation

$$\psi (x)\to g^{-1}\psi (x){\mathcal{A}}_{\mu }(x)\to g^{-1}\left[{\mathcal{A}}_{\mu }(x)+{\partial }_{\mu }\right]g.\text{\hspace{1em}(13.2)}$$

It also has a global symmetry,

$$\psi (x)\to {\mathrm{e}}^{\mathrm{i}{\gamma }_5\alpha }\psi (x)\bar{\psi }(x)\to \bar{\psi }(x){\mathrm{e}}^{\mathrm{i}{\gamma }_5\alpha }\text{\hspace{1em}(13.3)}$$

called the chiral symmetry. The chiral current $j_5$ derived from this symmetry is

$$j_5^{\mu }\equiv \bar{\psi }{\gamma }^{\mu }{\gamma }_5\psi .\text{\hspace{1em}(13.4)}$$

In general, whether the symmetry of a Lagrangian is retained under quantization is not a trivial question. In fact, it has been shown that the chiral symmetry of $\mathcal{L}$ is destroyed at the quantum level. Adler (1969) and Bell and Jackiw (1969) have shown by computing the triangle diagram with an external axial current and two external vector currents that the naive conservation law ${\partial }_{\mu }{j}_5^{\mu }=0$ is violated,

$$\begin{array}{rl}{\partial }_{\mu }{j}_5^{\mu }& =\frac{1}{16{\pi }^2}{ϵ}^{\kappa \lambda \mu \nu }\mathrm{tr}{\mathcal{F}}_{\kappa \lambda }{\mathcal{F}}_{\mu \nu }\\ & =\frac{1}{4{\pi }^2}\mathrm{tr}\left[{ϵ}^{\kappa \lambda \mu \nu }{\partial }_{\kappa }\left({\mathcal{A}}_{\lambda }{\partial }_{\mu }{\mathcal{A}}_{\nu }+\frac{2}{3}{\mathcal{A}}_{\lambda }{\mathcal{A}}_{\mu }{\mathcal{A}}_{\nu }\right)\right]\end{array}\text{\hspace{1em}(13.5)}$$

where tr is a trace over the group indices. The current $j_5^{\mu }$ which appears in (13.5) has no group index, and, hence, (13.5) is called the Abelian anomaly.

It is interesting to study the behaviour of a current which carries the group index. Consider a Weyl fermion $\psi$ which couples with an external gauge field. The non-Abelian gauge current of the theory also satisfies an anomalous conservation law which defines the non-Abelian anomaly. The action is given by

$$\mathcal{L}\equiv {\psi }^{†}(\mathrm{i}\nabla ){\mathcal{P}}_{+}\psi {\mathcal{P}}_{\pm }=\frac{1}{2}(I\pm {\gamma }^5).\text{\hspace{1em}(13.6)}$$

The Lagrangian has the gauge symmetry

$${\mathcal{A}}_{\mu }\to g^{-1}\left({\mathcal{A}}_{\mu }+{\partial }_{\mu }\right)g\psi \to g^{-1}\psi .\text{\hspace{1em}(13.7)}$$

The corresponding non-Abelian current is

$$j^{\mu \alpha }\equiv {\psi }^{†}{\gamma }^{\mu }{T}^{\alpha }{\mathcal{P}}_{+}\psi .\text{\hspace{1em}(13.8)}$$

It has been shown by Bardeen (1969) and Gross and Jackiw (1972) that, up to the one-loop level, the current is not conserved,

$$({\mathcal{D}}_{\mu }{j}_{\delta }^{\mu }{)}^{\alpha }=\frac{1}{24{\pi }^2}\mathrm{tr}\left[T^{\alpha }{\partial }_{\kappa }{ϵ}^{\kappa \lambda \mu v}\left({\mathcal{A}}_{\lambda }{\partial }_{\mu }{\mathcal{A}}_v+\frac{1}{2}{\mathcal{A}}_{\lambda }{\mathcal{A}}_{\mu }{\mathcal{A}}_v\right)\right].\text{\hspace{1em}(13.9)}$$

At first sight, the RHSs of (13.5) and (13.9) look very similar. However, the difference between the normalization and the numerical factors of $\frac{2}{3}$ and $\frac{1}{2}$ have a deep topological origin. We shall see later that the Abelian anomaly in $(2l+2)$ dimensions and the non-Abelian anomaly in $2l$ dimensions are closely related but in an unexpected manner.

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