[Nakahara GTP] 13.4 The Wess-Zumino consistency condition

 This article is one of Manifold, Differential Geometry, Fibre Bundle.

13.4.1 The Becchi–Rouet–Stora operator and the Faddeev–Popov ghost 

Let $W[\mathcal{A}]$ be the effective action of the Weyl fermion in the complex representation $\mathbf{r}$ of the gauge group $G$. In the previous section, we observed that the change of $W[\mathcal{A}]$ under an infinitesiomal gauge transformation $\delta_v\mathcal{A}=-\mathcal{D}v$ is given by $$\begin{align}\delta_vW[\mathcal{A}]=-\int (\mathcal{D}_\mu v)^\alpha \frac{\delta}{\delta \mathcal{A}_\mu^\alpha}W[\mathcal{A}]=\int v^\alpha\mathcal{D}_\mu \langle j^\mu\rangle_\alpha.\end{align}$$

We introduce the BRS operator $\mathcal{S}$ and the Faddeev-Popov ghost $\omega$. Let $\Omega^m(G)$ be the set of mapes from $S^m$ to $G$. In addition to the ordinary exterior derivative $d$, we introduce another exterior derivative $\mathcal{S}$ on $\Omega^m(G)$ which we call the Becchi-Roues-Stora (BRS) opeartor. In general, $\mathcal{S}$ is defined on an infinite-dimensional space but we may also consider the restriction of $\mathcal{S}$ to a finite-dimensional compect subspace of $\Omega^m(G)$, such as $S^n$, parametrized by $\lambda^\alpha$. Then $\mathcal{S}$ may be written as $\mathcal{S}\equiv d\lambda^\alpha\, \partial/\partial\lambda^\alpha$. We require that $d$ and $\mathcal{S}$ be anti-derivatives, $$\begin{align}d^2=\mathcal{S}^2=d\mathcal{S}^2+\mathcal{S}d=0.\end{align}$$ If we defined $\Delta\equiv d+\mathcal{S}$, $\Delta$ is clearly nilpotent, $$\begin{align}\Delta^2=d^2+d\mathcal{S}+\mathcal{S}d+\mathcal{S}^2=0.\end{align}$$ Under the action of $g=g(x,\lambda^\alpha)$, $\mathcal{A}$ transforms as $$\begin{align}\mathcal{A}\rightarrow \mathbf{A}\equiv g^{-1}(\mathcal{A}+d)g.\end{align}$$ Note that $\mathcal{A}$ is independent of $\lambda$ while $\mathbf{A}$ depends on $\lambda$ through $g$. Define the Faddeev-Popov (FP) ghost by $$\begin{align}\omega\equiv g^{-1}\mathcal{S}g.\end{align}$$ The actions of $\mathcal{S}$ on $\mathbf{A}$ and $\omega$ are found to be $$\begin{align}\mathcal{S}\mathbf{A}=\mathcal{S}[g^{-1}(\mathcal{A}+d)g]=-g^{-1}\mathcal{S}g\mathbf{A}-g^{-1}\mathcal{A}\mathcal{S}g+g^{-1}\mathcal{S}(dg)\nonumber\\ =-\omega\mathbf{A}-(\mathbf{A}-g^{-1}dg)\omega-g^{-1}d(\mathcal{S}g)\nonumber\\ =-\omega\mathbf{A}-\mathbf{A}\omega-d\omega\equiv -\mathcal{D}_\mathbf{A}\omega \\ \mathcal{S}\omega= -g^{-1}\mathcal{S}gg^{-1}\mathcal{S}g=-\omega^2.\end{align}$$ It is easy to verify that $\mathcal{S}$ is nilpotent on $\mathbf{A}$ and $\omega$ and, hence, on any polynomial of $\mathbf{A}$ and $\omega$ as it should be; see exercise 13.1. Define the field strength of $\mathbf{A}$ by $$\begin{align}\mathbf{F}\equiv d\mathbf{A}+\mathbf{A}^2=g^{-1}\mathcal{F}g.\end{align}$$ We also define $$\begin{align}\mathbb{A}\equiv g^{-1}(\mathcal{A}+\Delta)g=\mathbf{A}+g^{-1}\mathcal{S}g=\mathbf{A}+\omega\\ \mathbb{F}\equiv \Delta \mathbb{A}+\mathbb{A}^2=g^{-1}\mathcal{F}g=\mathbf{F}\end{align}$$ where (10) follows since $\mathcal{F}=d\mathcal{A}+\mathcal{A}^2=\Delta\mathcal{A}+\mathcal{A}^2$ (note that $\mathcal{S}\mathcal{A}=0$). It is found from theorem 10.1 that $\mathbb{A}$ is an Ehresmann connection on the principal bundle and $\mathbb{F}$ its associated curvature two-form.

The existence of a non-Abelian anomaly implies that $W[\mathbf{A}]$ does not vanish under the action of the BRS operator $\mathcal{S}$ ($\omega$ roughly corresponds to $v$; see (6)), $$\begin{align}\mathcal{S}W[\mathbf{A}]=G[\omega,\mathbf{A}].\end{align}$$ Since $W[\mathbf{A}]$ is independent of $\omega$, $\mathcal{S}$ acts through $\mathbf{A}$ only. Before we write down the Wess-Zumino consistency condition for the non-Abelian anomaly, we stop here and consider the physical meaning of the BRS opeartor and FP ghost.

Exercise 13.1. Verify form (6,7) that the actions of $\mathcal{S}$ on $\mathbf{A}$ and $\omega$ are nilpotent, $$\begin{align}\mathcal{S}^2\mathbf{A}=0\quad \mathcal{S}^2\omega=0.\end{align}$$

13.4.2 The BRS operator, FP ghost and moduli space 

To find the physical meaning of $\mathcal{S}$ and $\omega$, we need to examine the topology of the gauge fields. Let $\mathfrak{A}$ be the space of all gauge potential configurations on $S^m$. For definiteness, we take $m=4$ but the generalization to arbitrary $m$ is obvious. The topology of $\mathfrak{A}$ is trivial since, for any gauge potential configurations $\mathcal{A}_1$ and $\mathcal{A}_2$, the combination $t\mathcal{A}_1+(1-t)\mathcal{A}_2$ ($0\le t\le 1$) is again a gauge potential on $S^4$. Note, however, that $\mathfrak{U}$ does not descrivle the physical configuration space of the gauge thoery. We have to identify thoes field configurations which are connected by $G$-gauge tranformations. Let $\mathfrak{G}$ be the space of all gauge tranformations on $S^4$ ($\mathfrak{G}=\Omega^4(G)$ in our previous notation). Then the physical configuration space must be identified with $\mathfrak{A}/\mathfrak{G}$, called the moduli space of the gauge theory. We have seen in section 10.5 that the gauge field configuration on $S^4$ is classified by the transition function $g:S^3\rightarrow G$, $S^3$ being the equator of $S^4$. In the present case, $\mathfrak{A}/\mathfrak{G}$ is classified by the transition function on the equator $S^3\rightarrow G$ and, hence, $$\begin{align}\mathfrak{A}/\mathfrak{G}\simeq \Omega^3(G).\end{align}$$ Thus, each connected component of $\mathfrak{A}/\mathfrak{G}$ is labelled by the instanton number $k$. This component is denoted by $\Omega^4_k(G)$. 

We note that the space $\mathfrak{A}$ has a natural projection $\pi:\mathfrak{A}\rightarrow \mathfrak{A}/\mathfrak{G}$ and can be made into a fibre bundle whose fibre is $\mathfrak{G}$, see figure 13.1. Let $a\in\mathfrak{A}$ be a representative of the class $[a]\in\mathfrak{A}/\mathfrak{G}$ and let $$\begin{align}\mathcal{A}(x)=g^{-1}(x)(a(x)+d)g(x)\end{align}$$ be an element of $\mathfrak{A}$ in $[a]$. We denote the exterior derivative operator in $\mathfrak{A}$ by $\delta$, which is a functional variation and should not be confused with the usual derivative $d$. If $\delta$ is applied on (14), we find that $$\begin{align}\delta\mathcal{A}=-g^{-1}\delta g\mathcal{A}+g^{-1}\delta ag-g^{-1}a\delta g-g^{-1}d(\delta g)\nonumber\\ g^{-1}\delta a g-d(g^{-1}\delta g)-g^{-1}\delta g\mathcal{A}-\mathcal{A}g^{-1}\delta g\nonumber\\ =g^{-1}\delta a g-\mathcal{D}_\mathcal{A}(g^{-1}\delta g)\end{align}$$ where $\mathcal{D}_\mathcal{A}=d+[\mathcal{A},\ ]$. The first term of (15) represents the derivative of $\mathcal{A}$ along $\mathfrak{A}/\mathfrak{G}$ while the second represents that along the fivre; see figure 13.1. The BRS transformation $\mathcal{S}$ is obtained by restricting the variation $\delta$ along the fibre, $$\begin{align}\mathcal{S}\mathcal{A}\equiv \delta \mathcal{A}|_{fibre}=-\mathcal{D}_\mathcal{A}\omega\end{align}$$ where the FP ghost $\omega$ is $g^{-1}\mathcal{S}g\equiv g^{-1}\delta g|_{fibre}$. We also find that $$\begin{align}\mathcal{S}\omega =\delta\omega|_{fibre}=-g^{-1}\mathcal{S}gg^{-1}\mathcal{S}g=-\omega^2\end{align}$$ which repreduces (6). 

13.4.3 The Wess–Zumino conditions 

Exercise 13.1 shows that $\mathcal{S}$ is nilpotent on any polynomial $f$ on $\mathcal{A}$ and $\omega$, $$\begin{align}\mathcal{S}^2f(\omega,\mathbf{A})=0.\end{align}$$ The nilpotency is required by the interpretation of $\mathcal{S}$ as an exterior derivative operator. In particular, we should have $$\begin{align}\mathcal{S}G[\omega,\mathbf{A}]=\mathcal{S}^2W[\mathbf{A}]=0.\end{align}$$ This condition is called the Wess-Zumino consistency condition (WZ condition) and can be used to determine the non-Abelian anomaly. If the anomaly $G$ is mathematically well defined, $G$ should satisfy the WZ condition. This condition is so strong that once the first term of $G[\omega,\mathbf{A}]$ is given, the anomaly is completely pinned down.

13.4.4 Descent equations and solutions of WZ conditions

Stora and Zumino constructed the solution of WZ conditions as follows. The Abelian anomaly in $(2l+2)$-dimensional space is given by $$\begin{align}\mbox{ch}_{l+1}(\mathbf{F})=\frac{1}{(l+1)!}\mbox{tr}\left( \frac{i\mathbf{F}}{2\pi}\right)^{l+1}\end{align}$$ where $\mathbf{F}=d\mathbf{A}+\mathbf{A}^2$, $\mathbf{A}=g^{-1}(\mathcal{A}+d)g$ as before. Let $Q_{2l+1}(\mathbf{A},\mathbf{F})$ be the Chern-Simons form of $\mbox{ch}_{l+1}(\mathbf{F})$, $$\begin{align}\mbox{ch}_{l+1}(\mathbf{F})=dQ_{2l+1}(\mathbf{A},\mathbf{F}).\end{align}$$ Since the algebraic structure of the triplet $(\Delta,\mathbb{A},\mathbb{F})$ is exactly the same as that of $(d,\mathbf{A},\mathbf{F})$, we also have $$\begin{align}\mbox{ch}_{l+1}(\mathbb{F})=dQ_{2l+1}(\mathbb{A},\mathbb{F})=\Delta Q_{2l+1}(\mathbb{A}+\omega,\mathbb{F}).\end{align}$$ where we have noted that $\mathbb{A}=\mathbf{A}+\omega$ and $\mathbb{F}=\mathbf{F}$. If we expand $Q_{2l+1}(\mathbb{A},\mathbb{F})=Q_{2l+1}(\mathbb{A}+\omega,\mathbb{F})$ in powers of $\omega$, we have $$\begin{align}Q_{2l+1}(\mathbb{A},\mathbb{F})=Q^0_{2l+1}(\mathbf{A},\mathbf{F})+Q^1_{2l}(\omega,\mathbf{A},\mathbf{F})+Q^2_{2l-1}(\omega,\mathbf{A},\mathbf{F})+\cdots+Q^{2l+1}_0(\omega,\mathbf{A},\mathbf{F})\end{align}$$ where $Q^s_r$ is $s$th order in $\omega$ and $r+s=2l+1$.

We now note that $\mbox{ch}_{l+1}(\mathbb{F})=\mbox{ch}_{l+1}(\mathbf{F})$ since $\mathbb{F}=\mathbf{F}=g^{-1}\mathcal{F}g$. In terms of the Chern-Simons forms, this can be expressed as $$\begin{align}\Delta Q_{2l+1}(\mathbb{A},\mathbb{F})=dQ_{2l+1}(\mathbf{A},\mathbf{F}).\end{align}$$ Substituting (23) into (24), we have $$\begin{align}(d+\mathcal{S})[Q^0_{2l+1}(\mathbf{A},\mathbf{F})+Q^1_{2l}(\omega,\mathbf{A},\mathbf{F})+\cdots+Q_0^{2l+1}(\omega,\mathbf{A},\mathbf{F})]=dQ^0_{2l+1}(\mathbf{A},\mathbf{F}).\end{align}$$ If we collect terms of the same order in $\omega$, we have the 'descent equations' $$\begin{align} \mathcal{S}Q^0_{2l+1}(\mathbf{A},\mathbf{F})+dQ^1_{2l}(\omega,\mathbf{A},\mathbf{F})=0 \\ \mathcal{S}Q^1_{2l}(\omega,\mathbf{A},\mathbf{F})+dQ^2_{2l-1}(\omega,\mathbf{A},\mathbf{F})=0 \\ \vdots\nonumber\\ \mathcal{S}Q_1^{2l}(\omega,\mathbf{A},\mathbf{F})+dQ_0^{2l+1}(\omega,\mathbf{A},\mathbf{F})=0 \\ \mathcal{S}Q_0^{2l+1}(\omega,\mathbf{A},\mathbf{F})=0.\end{align}$$ Note here that $\mathcal{S}$ increases the degree of $\omega$ by one, see (6,7). Let us look at the $2l$-form $Q^1_{2l}(\omega,\mathbf{A},\mathbf{F})$. If we put $$\begin{align}G[\omega,\mathbf{A},\mathbf{F}]\equiv \int_MQ^1_{2l}(\omega,\mathbf{A},\mathbf{F})\end{align}$$ $G[\omega,\mathbf{A},\mathbf{F}]$ satisfies the WZ condition, $$\mathcal{S}G[\omega,\mathbf{A},\mathbf{F}]=\int_M\mathcal{S}Q^1_{2l}(\omega,\mathbf{A},\mathbf{F})=-\int_MdQ^2_{2l-1}(\omega,\mathbf{A},\mathbf{F})=-\int_{\partial M}Q^2_{2l-1}(\omega,\mathbf{A},\mathbf{F})=0$$ where we have assumed that $M$ has no boundary and use has been made of (27). This shows taht once $Q^1_{2l}(\omega,\mathbf{A},\mathbf{F})$ is obtained, the anomaly $G[\omega,\mathbf{A},\mathbf{F}]$ is easily found.

Proposition 13.1. $Q^1_{2l}$ defined here is given by $$\begin{align}Q^1_{2l}(\omega,\mathcal{A},\mathcal{F})=\left( \frac{i}{2\pi}\right)^{l+1}\frac{1}{(l-1)!}\int_0^1\delta t(1-t)\, \mbox{str}[\omega d(\mathcal{A}\mathcal{F}^{l-1}_t)].\end{align}$$ (Note: In this proof, we tentatively drop the noramlization factor $(i/2\pi)^{l+1}$ to simplify the expressions, This factor will be recovered at the very end.)

Proof. We start with (11.5.6), $$Q_{2l}(\mathcal{A}+\omega,\mathcal{F})=\frac{1}{l!}\int_0^1\delta t\, \mbox{str}[(\mathcal{A}+\omega)\hat{\mathcal{F}}^l_t)]$$ where $$\hat{\mathcal{F}}^l_t\equiv t\mathcal{F}+(t^2-t)(\mathcal{A}+\omega)^2\\ =\mathcal{F}_t+(t^2-t)\{\mathcal{A},\omega\}+(t^2-t)\omega^2\\ \mathcal{F}_t\equiv d(t\mathcal{A})+(t\mathcal{A})^2.$$ If we substitute $\hat{\mathcal{F}}^l_t$ into $Q_{2l+1}$ and collect terms of first order in $\omega$, we have: $$\frac{1}{l!}\int^1_0\delta t\, \mbox{tr}[\omega \mathcal{F}^l_t+(t^2-t)(\mathcal{A}[\mathcal{A},\omega]\mathcal{F}^{l-1}_t+\mathcal{A}\mathcal{F}_t[\mathcal{A},\omega]\mathcal{F}^{l-2}_t\\ +\cdots+\mathcal{A}\mathcal{F}^{l-1}_t[\mathcal{A},\omega])]\\ =\frac{1}{l!}\int \delta t\, \mbox{str}[\omega\mathcal{F}^l_t+(t^2-t)\mathcal{A}(\mathcal{F}^{l-1}_t[\mathcal{A},\omega]]+\mathcal{F}^{l-2}_t[\mathcal{A},\omega]\mathcal{F}_t+\cdots)]\\ =\frac{1}{l!}\int \delta t\, \mbox{str}[\omega\mathcal{F}^l_t+(t^2-t)l\mathcal{A}[\mathcal{A},v]\mathcal{F}^{l-1}_t]\\ =\frac{1}{l!}\int \delta t\, \mbox{str}[\omega\mathcal{F}^l_t+l(t^2-t)([\mathcal{A},\mathcal{A}]\omega\mathcal{F}^{l-1}_t+\mathcal{A}\omega[\mathcal{A},\mathcal{F}^{l-1}_t])]\\ =\frac{1}{l!}\int \delta t\, \mbox{str}[\omega\{ \mathcal{F}^l_t+l(t^2-t)(t[\mathcal{A},\mathcal{A}]\mathcal{F}^{l-1}_t-\mathcal{A}[\mathcal{A}_t,\mathcal{F}^{l-1}_t])\} ]$$ where $\mbox{str}$ is the symmetrized trace defined by (11.1.8). Now we use $$\mathcal{D}_t\mathcal{F}_t^{l-1}\equiv d\mathcal{F}^{l-1}_t+[\mathcal{A}_t,\mathcal{F}^{l-1}_t]=0\\ \frac{\partial \mathcal{F}_t}{\partial t}=d\mathcal{A}+t[\mathcal{A},\mathcal{A}]$$ to change the final line of the previous equation to $$\frac{1}{l!}\int \delta t\, \mbox{str}\left[\omega\left\{ \mathcal{F}^l_t+l(t^2-t)\left[ \left( \frac{\partial \mathcal{F}_t}{\partial t}-d\mathcal{A}\right) \mathcal{F}^{l-1}_t+\mathcal{A}d\mathcal{F}^{l-1}_t\right]\right\} \right] \\ =\frac{1}{l!}\int \delta t\, \mbox{str}\left[ \omega \left\{ \mathcal{F}^l_t+l(1-t)d(\mathcal{A}\mathcal{F}^{l-1}_t)+(t-1)\frac{\partial \mathcal{F}^l_t}{\partial t}\right\} \right].$$ Integrating by parts, we find that $$Q^1_{2l}(\omega,\mathcal{A},\mathcal{F})=\frac{1}{(l-1)!}\int \delta t(1-t)\, \mbox{str}[\omega d (\mathcal{A}\mathcal{F}^{l-1}_t)].$$ If we recover tha normalization, we finally have $$Q^1_{2l}(\omega,\mathcal{A},\mathcal{F})=\left( \frac{i}{2\pi}\right)^{l+1}\frac{1}{(l-1)!}\int_0^1\delta t(1-t)\, \mbox{str}[\omega d(\mathcal{A}\mathcal{F}^{l-1}_t)].$$

For $m=2l=2$ and $m=4$, we have $$\begin{align}Q^1_2(\omega,\mathbf{A},\mathbf{F})=\left(\frac{i}{2\pi}\right)^2\mbox{tr}(\omega d\mathbf{A})\\ Q^1_4(\omega,\mathbf{A},\mathbf{F})=\frac{1}{6} \left(\frac{i}{2\pi}\right)^3\mbox{str}(\omega d(\mathbf{A}d\mathbf{A}+\frac{1}{2}\mathbf{A}^3)).\end{align}$$ These results are also verified by direct computations. Up to the normalization factor, (33) yields the non-Abelian anomaly in four-dimensional space; see (13.3.?).