13.4 The Wess-Zumino consistency conditions

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 $r$ of the gauge group $G$ . In the previous section, we observed that the change of $W[\mathcal{A}]$ under an infinitesimal gauge transformation ${\delta }_v\mathcal{A}=-\mathcal{D}v$ is given by

$${\delta }_{v}W[\mathcal{A}]=-\int {\left({\mathcal{D}}_{\mu }v\right)}^{\alpha }\frac{\delta }{\delta {\mathcal{A}}_{\mu }^{\alpha }}W[\mathcal{A}]=\int v^{\alpha }{\mathcal{D}}_{\mu }⟨j^{\mu }{⟩}_{\alpha }.\text{\hspace{1em}(13.58)}$$

Following Stora (1984) and Zumino (1985) we introduce the BRS operator $\mathcal{S}$ and the Faddeev-Popov ghost $\omega$ . Let ${\Omega }^m(G)$ be the set of maps from $S^m$ to $G$ . In addition to the ordinary exterior derivative $\mathrm{d}$ , we introduce another exterior derivative $\mathcal{S}$ on ${\Omega }^m(G)$ which we call the Becchi-Rouet-Stora (BRS) operator. 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 compact subspace of ${\Omega }^m(G)$ , such as $S^n$ , parametrized by ${\lambda }^{\alpha }$ . Then $\mathcal{S}$ may be written as $\mathcal{S}\equiv \mathrm{d}{\lambda }^{\alpha }\partial /\partial {\lambda }^{\alpha }$ . We require that $\mathrm{d}$ and $\mathcal{S}$ be anti-derivatives,

$${\mathrm{d}}^2={\mathbb{S}}^2=\mathrm{d}\mathbb{S}+\mathbb{S}\mathrm{d}=0.\text{\hspace{1em}(13.59)}$$

If we define $\Delta \equiv \mathsf{d}+\mathsf{S}$ , $\Delta$ is clearly nilpotent,

$${\Delta }^2={\mathrm{d}}^2+\mathrm{d}\mathcal{S}+\mathcal{S}\mathrm{d}+{\mathcal{S}}^2=0.\text{\hspace{1em}(13.60)}$$

Under the action of $g=g(x,{\lambda }^{\alpha }),\mathcal{A}$ transforms as

$$\mathcal{A}\to A\equiv g^{-1}(\mathcal{A}+d)g.\text{\hspace{1em}(13.61)}$$

Note that $\mathcal{A}$ is independent of $\lambda$ while $A$ depends on $\lambda$ through $g$ . Define the Faddeev-Popov (FP) ghost by

$$\omega \equiv g^{-1}\mathcal{S}g.\text{\hspace{1em}(13.62)}$$

The actions of $\mathcal{S}$ on $A$ and $\omega$ are found to be

$$\begin{array}{rl}\mathbb{S}\mathbf{A}& =\mathbb{S}[g^{-1}(\mathcal{A}+\mathrm{d})g]=-g^{-1}\mathbb{S}g\mathbf{A}-g^{-1}\mathcal{A}\mathbb{S}g+g^{-1}\mathbb{S}(\mathrm{d}g)\\ & =-\omega A-(A-g^{-1}dg)\omega -g^{-1}d(\mathcal{S}g)\\ & =-\omega A-A\omega -\mathrm{d}\omega \equiv -{\mathcal{D}}_A\omega \end{array}\text{\hspace{1em}(13.63a)}$$

$$\mathbb{S}\omega =-g^{-1}\mathbb{S}g{g}^{-1}\mathbb{S}g=-{\omega }^2.\text{\hspace{1em}(13.63b)}$$

It is easy to verify that $\mathcal{S}$ is nilpotent on $A$ and $\omega$ and, hence, on any polynomial of $A$ and $\omega$ as it should be; see exercise 13.1. Define the field strength of $A$ by

$$\mathbf{F}\equiv \mathrm{d}\mathbf{A}+{\mathbf{A}}^2=g^{-1}\mathcal{T}g.\text{\hspace{1em}(13.64)}$$

We also define

$$\mathbb{A}\equiv g^{-1}(\mathcal{A}+\Delta )g=\mathbf{A}+g^{-1}\mathfrak{S}g=\mathbf{A}+\omega \text{\hspace{1em}(13.65a)}$$

$$\mathbb{F}\equiv \Delta \mathbb{A}+{\mathbb{A}}^2=g^{-1}\mathcal{F}g=F\text{\hspace{1em}(13.65b)}$$

where (13.65b) follows since $\mathcal{F}=\mathrm{d}\mathcal{A}+{\mathcal{A}}^2=\Delta \mathcal{A}+{\mathcal{A}}^2$ (note that $\mathbb{S}\mathcal{A}=0$ ). 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 $W[A]$ does not vanish under the action of the BRS operator $\mathcal{S}$ ( $\omega$ roughly corresponds to $v$ ; see (13.39) and (13.63a)),

$$\mathbb{S}W[\mathbf{A}]=G[\omega ,\mathbf{A}].\text{\hspace{1em}(13.66)}$$

Since $W[A]$ is independent of $\omega$ , $\mathbb{S}$ acts through $A$ 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 operator and the FP ghost.

Exercise 13.1

Verify from (13.63) that the actions of $\mathcal{S}$ on $A$ and $\omega$ are nilpotent,

$${\mathcal{S}}^{2}A=0{\mathcal{S}}^2\omega =0.\text{\hspace{1em}(13.67)}$$

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 (Atiyah and Jones 1978, Singer 1985, Sumitani 1985). 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 $\mathfrak{A}$ does not describe the physical configuration space of the gauge theory. We have to identify those field configurations which are connected by $G$ -gauge transformations. Let $\mathfrak{G}$ be the space of all gauge transformations 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 $g:S^3\to G$ , $S^3$ being the equator of $S^4$ . In the present case, $\mathfrak{A}/\mathfrak{G}$ is classified by the transition function on the equator $S^3\to G$ and, hence,

$$\mathfrak{A}/\mathfrak{G}\simeq {\Omega }^3(G).\text{\hspace{1em}(13.68)}$$

Thus, each connected component of $\mathfrak{A}/\mathfrak{G}$ is labelled by the instanton number $k$ . This component is denoted by ${\Omega }_k^4(G)$ .

We note that the space $\mathfrak{A}$ has a natural projection $\pi :\mathfrak{A}\to \mathfrak{A}/\mathfrak{G}$ 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

Figure 13.1. The BRS operator $\mathcal{S}$ is the restriction of $\delta$ along the fibre.

$$\mathcal{A}(x)=g^{-1}(x)\left(a(x)+d\right)g(x)\text{\hspace{1em}(13.69)}$$

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 $\mathrm{d}$ ; see Leinaas and Olaussen (1982). If $\delta$ is applied on (13.69), we find that

$$\begin{array}{rl}\delta \mathcal{A}& =-g^{-1}\delta g\mathcal{A}+g^{-1}\delta ag-g^{-1}a\delta g-g^{-1}d(\delta g)\\ & =g^{-1}\delta ag-\mathrm{d}(g^{-1}\delta g)-g^{-1}\delta g\mathcal{A}-\mathcal{A}{g}^{-1}\delta g\\ & =g^{-1}\delta ag-{\mathcal{D}}_{\mathcal{A}}\left(g^{-1}\delta g\right)\end{array}\text{\hspace{1em}(13.70)}$$

where ${\mathcal{D}}_{\mathcal{A}}=\mathrm{d}+[\mathcal{A},]$ . The first term of (13.70) represents the derivative of $\mathcal{A}$ along $\mathfrak{A}/\mathfrak{G}$ while the second represents that along the fibre; see figure 13.1. The BRS transformation $\mathcal{S}$ is obtained by restricting the variation $\delta$ along the fibre,

$$\mathcal{S}\mathcal{A}\equiv \delta \mathcal{A}{|}_{\text{f i b r e}}=-{\mathcal{D}}_{\mathcal{A}}\omega \text{\hspace{1em}(13.71a)}$$

where the FP ghost $\omega$ is $g^{-1}\mathbb{S}g\equiv g^{-1}\delta g{|}_{\mathrm{fibre}}$ . We also find that

$$\mathbb{S}\omega =\delta \omega {|}_{\text{f i b r e}}=-g^{-1}\mathbb{S}g{g}^{-1}\mathbb{S}g=-{\omega }^2\text{\hspace{1em}(13.71b)}$$

which reproduces (13.63a).

13.4.3 The Wess-Zumino conditions

Exercise 13.1 shows that $\mathcal{S}$ is nilpotent on any polynomial $f$ of $\mathcal{A}$ and $\omega$ ,

$${\mathcal{S}}^{2}f(\omega ,\mathbf{A})=0.\text{\hspace{1em}(13.72)}$$

The nilpotency is required by the interpretation of $\mathcal{S}$ as an exterior derivative operator. In particular, we should have

$$\mathbb{S}G[\omega ,A]={\mathbb{S}}^{2}W[A]=0.\text{\hspace{1em}(13.73)}$$

This condition is called the Wess-Zumino consistency condition (WZ condition) and can be used to determine the non-Abelian anomaly (Wess and Zumino 1971, Stora 1984, Zumino 1985, Zumino et al 1984). 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 $G[\omega ,A]$ is given, the anomaly is completely pinned down.

13.4.4 Descent equations and solutions of WZ conditions

Stora (1984) and Zumino (1985) constructed the solution of WZ conditions as follows. The Abelian anomaly in $(2l+2)$ -dimensional space is given by

$${\mathrm{ch}}_{l+1}(\mathbf{F})=\frac{1}{(l+1)!}\mathrm{tr}{\left(\frac{\mathrm{i}\mathbf{F}}{2\pi }\right)}^{l+1}\text{\hspace{1em}(13.74)}$$

where $F=\mathrm{d}A+A^2$ , $A=g^{-1}(\mathcal{A}+\mathrm{d})g$ as before. Let $Q_{2l+1}(A,F)$ be the Chern-Simons form of ${\mathrm{ch}}_{l+1}(F)$ ,

$${\mathrm{ch}}_{l+1}(\mathbf{F})=\mathrm{d}{Q}_{2l+1}(\mathbf{A},\mathbf{F}).\text{\hspace{1em}(13.75)}$$

Since the algebraic structure of the triplet $(\Delta ,\mathbb{A},\mathbb{F})$ is exactly the same as that of $(\mathrm{d},A,F)$ , we also have

$${\mathrm{ch}}_{l+1}(\mathbb{F})=\Delta Q_{2l+1}(\mathbb{A},\mathbb{F})=\Delta Q_{2l+1}(\mathbf{A}+\omega ,\mathbf{F})\text{\hspace{1em}(13.76)}$$

where we have noted that $\mathbb{A}=A+\omega$ and $\mathbb{F}=F$ . If we expand $Q_{2l+1}(\mathbb{A},\mathbb{F})=Q_{2l+1}(A+\omega ,F)$ in powers of $\omega$ , we have

$$\begin{array}{rl}& Q_{2l+1}(\mathbb{A},\mathbb{F})=Q_{2l+1}^0(\mathbf{A},\mathbf{F})+Q_{2l}^1(\omega ,\mathbf{A},\mathbf{F})+Q_{2l-1}^2(\omega ,\mathbf{A},\mathbf{F})\\ & +\cdots +Q_0^{2l+1}(\omega ,\mathbf{A},\mathbf{F})\end{array}\text{\hspace{1em}(13.77)}$$

where $Q_r^s$ is sth order in $\omega$ and $r+s=2l+1$ .

We now note that ${\mathrm{ch}}_{l+1}(\mathbb{F})={\mathrm{ch}}_{l+1}(F)$ since $\mathbb{F}=F=g^{-1}\mathcal{F}g$ . In terms of the Chern-Simons forms, this can be expressed as

$$\Delta Q_{2l+1}(\mathbb{A},\mathbb{F})=\mathrm{d}{Q}_{2l+1}(\mathbf{A},\mathbf{F}).\text{\hspace{1em}(13.78)}$$

Substituting (13.77) into (13.78), we have

$$\begin{array}{rl}& (\mathrm{d}+\mathbb{S})[Q_{2l+1}^0(\mathbf{A},\mathbf{F})+Q_{2l}^1(\omega ,\mathbf{A},\mathbf{F})\\ & +\cdots +Q_0^{2l+1}(\omega ,\mathbf{A},\mathbf{F})]=\mathrm{d}{Q}_{2l+1}^0(\mathbf{A},\mathbf{F}).\end{array}\text{\hspace{1em}(13.79)}$$

If we collect terms of the same order in $\omega$ , we have the ‘descent equations’

$$\mathbb{S}{Q}_{2l+1}^0(\mathbf{A},\mathbf{F})+\mathrm{d}{Q}_{2l}^1(\omega ,\mathbf{A},\mathbf{F})=0\text{\hspace{1em}(13.80a)}$$

$$\mathbb{S}{Q}_{2l}^1(\omega ,\mathbf{A},\mathbf{F})+\mathrm{d}{Q}_{2l-1}^2(\omega ,\mathbf{A},\mathbf{F})=0\text{\hspace{1em}(13.80b)}$$

$$\mathbb{S}{Q}_1^{2l}(\omega ,\mathbf{A},\mathbf{F})+\mathrm{d}{Q}_0^{2l+1}(\omega ,\mathbf{A},\mathbf{F})=0\text{\hspace{1em}(13.80c)}$$

$$\mathbb{S}{Q}_0^{2l+1}(\omega ,\mathbf{A},\mathbf{F})=0.\text{\hspace{1em}(13.80d)}$$

Note here that $\mathcal{S}$ increases the degree of $\omega$ by one, see (13.63). Let us look at the $2l$ -form $Q_{2l}^1(\omega ,A,F)$ . If we put

$$G[\omega ,\mathbf{A},\mathbf{F}]\equiv {\int }_M{Q}_{2l}^1(\omega ,\mathbf{A},\mathbf{F})\text{\hspace{1em}(13.81)}$$

$G[\omega ,A,F]$ satisfies the WZ condition,

$$\begin{array}{rl}\mathbb{S}G[\omega ,\mathbf{A},\mathbf{F}]& ={\int }_M\mathbb{S}{Q}_{2l}^1(\omega ,\mathbf{A},\mathbf{F})=-{\int }_M\mathrm{d}{Q}_{2l-1}^2(\omega ,\mathbf{A},\mathbf{F})\\ & =-{\int }_{\partial M}{Q}_{2l-1}^2(\omega ,A,F)=0\end{array}$$

where we have assumed that $M$ has no boundary and use has been made of (13.80b). This shows that once $Q_{2l}^1(\omega ,A,F)$ is obtained, the anomaly $G[\omega ,A,F]$ is easily found.

Proposition 13.1

$Q_{2l}^1$ defined here is given by

$$Q_{2l}^1(\omega ,\mathcal{A},\mathfrak{F})={\left(\frac{\mathrm{i}}{2\pi }\right)}^{l+1}\frac{1}{(l-1)!}{\int }_0^1\delta t(1-t)\mathrm{str}\left[\omega \mathrm{d}\left(\mathcal{A}{\mathfrak{F}}_t^{l-1}\right)\right].\text{\hspace{1em}(13.82)}$$

[Note: In the proof, we tentatively drop the normalization factor $(\mathrm{i}/2\pi {)}^{l+1}$ to simplify the expressions. This factor will be recovered at the very end.]

Proof

We start with (11.105),

$$Q_{2l+1}(\mathcal{A}+\omega ,\mathcal{F})=\frac{1}{l!}{\int }_0^1\delta t\mathrm{tr}[(\mathcal{A}+\omega ){\hat{\mathcal{F}}}_t^l]$$

where

$$\begin{array}{rl}& {\hat{\mathcal{F}}}_t\equiv t\mathcal{F}+(t^2-t)(\mathcal{A}+\omega {)}^2\\ & ={\mathcal{F}}_t+(t^2-t)\{\mathcal{A},\omega \}+(t^2-t){\omega }^2\end{array}$$

$${\mathfrak{F}}_t\equiv \mathrm{d}(t\mathcal{A})+(t\mathcal{A}{)}^2.$$

If we substitute ${\hat{\mathcal{F}}}_t$ into $Q_{2l+1}$ and collect terms of first order in $\omega$ , we have:

$$\begin{array}{rl}& \frac{1}{l!}{\int }_0^1\delta t\mathrm{tr}[\omega {\mathcal{F}}_t^l+(t^2-t)(\mathcal{A}[\mathcal{A},\omega ]{\mathcal{F}}_t^{l-1}+\mathcal{A}{\mathcal{F}}_t[\mathcal{A},\omega ]{\mathcal{F}}_t^{l-2}\\ & +\cdots +{\mathcal{A}\mathcal{F}}_t^{l-1}[\mathcal{A},\omega ])]\\ & =\frac{1}{l!}\int \delta t\mathrm{str}[\omega {\mathcal{F}}_t^l+(t^2-t)\mathcal{A}({\mathcal{F}}_t^{l-1}[\mathcal{A},\omega ])\\ & +{\mathcal{F}}_t^{l-2}[\mathcal{A},\omega ]{\mathcal{F}}_t+\dots )]\\ & =\frac{1}{l!}\int \delta t\mathrm{str}[\omega {\mathcal{F}}_t^l+(t^2-t)l\mathcal{A}[\mathcal{A},v]{\mathcal{F}}_t^{l-1}]\\ & =\frac{1}{l!}\int \delta t\mathrm{str}\left[\omega {\mathfrak{F}}_t^l+l\left(t^2-t\right)\left([\mathcal{A},\mathcal{A}]\omega {\mathfrak{F}}_t^{l-1}+\mathcal{A}\omega [\mathcal{A},{\mathfrak{F}}_t^{l-1}]\right)\right]\\ & =\frac{1}{l!}\int \delta t\mathrm{str}[\omega \left\{{\mathcal{F}}_t^l+l(t-1)\left(t[\mathcal{A},\mathcal{A}]{\mathcal{F}}_t^{l-1}-\mathcal{A}\left[{\mathcal{A}}_t,{\mathcal{F}}_t^{l-1}\right]\right)\right]\end{array}$$

where str is the symmetrized trace defined by (11.8). Now we use

$${\mathcal{D}}_t{\mathcal{F}}_t^{l-1}\equiv \mathrm{d}{\mathcal{F}}_t^{l-1}+[{\mathcal{A}}_t,{\mathcal{F}}_t^{l-1}]=0$$

$$\frac{\partial {\mathcal{F}}_t}{\partial t}=\mathrm{d}\mathcal{A}+t[\mathcal{A},\mathcal{A}]$$

to change the final line of the previous equation to

$$\begin{array}{rl}& \frac{1}{l!}\int \delta t\mathrm{str}\left[\omega \left\{{\mathcal{F}}_t^l+l(t-1)\left[\left(\frac{\partial {\mathcal{F}}_t}{\partial t}-\mathrm{d}\mathcal{A}\right){\mathcal{F}}_t^{l-1}+\mathcal{A}\mathrm{d}{\mathcal{F}}_t^{l-1}\right]\right\}\right]\\ & =\frac{1}{l!}\int \delta t\mathrm{str}\left[\omega \left\{{\mathcal{F}}_t^l+l(1-t)\mathrm{d}\left({\mathcal{A}\mathcal{F}}_t^{l-1}\right)+(t-1)\frac{\partial {\mathcal{F}}_t^l}{\partial t}\right\}\right].\end{array}$$

Integrating by parts, we find that

$$Q_{2l}^1(\omega ,\mathcal{A},\mathcal{F})=\frac{1}{(l-1)!}\int \delta t(1-t)\mathrm{str}[\omega \mathrm{d}(\mathcal{A}{\mathcal{F}}_t^{l-1})].$$

If we recover the normalization, we finally have

$$Q_{2l}^1(\omega ,\mathcal{A},\mathcal{F})={\left(\frac{\mathrm{i}}{2\pi }\right)}^{l+1}\frac{1}{(l-1)!}{\int }_0^1\delta t(1-t)\mathrm{str}[\omega \mathrm{d}(\mathcal{A}{\mathcal{F}}_t^{l-1})].$$

$\square$

For $m=2l=2$ and $m=4$ , we have

$$Q_2^1(\omega ,\mathbf{A},\mathbf{F})={\left(\frac{\mathrm{i}}{2\pi }\right)}^2\mathrm{tr}(\omega \mathrm{d}\mathbf{A})\text{\hspace{1em}(13.83a)}$$

$$Q_4^1(\omega ,\mathbf{A},\mathbf{F})=\frac{1}{6}{\left(\frac{\mathrm{i}}{2\pi }\right)}^3\mathrm{str}\left(\omega \mathrm{d}\left(\mathbf{A}\mathrm{d}\mathbf{A}+\frac{1}{2}{\mathbf{A}}^3\right)\right).\text{\hspace{1em}(13.83b)}$$

These results are also verified by direct computations. Up to the normalization factor, (13.83b) yields the non-Abelian anomaly in four-dimensional space; see (13.56).

Sumitani (1984) pointed out that the approach to the non-Abelian anomalies here is ad hoc and does not clarify the following points:

(1) The WZ condition (13.73) does not fix the normalization of the anomaly and, moreover, the uniqueness of the solution is far from trivial.
(2) It is not clear why we should start from the Abelian anomaly in $(m+2)$ -dimensional space.

To answer these questions we need to develop a more elaborate index theorem called the family index theorem; see Atiyah and Singer (1984), Singer (1985) and Sumitani (1984, 1985). In the next section, we outline the physicists’ approach to this problem, closely following the work of Alvarez-Gaume and Ginsparg (1984).

Endnotes

Unplaced source note

We drop the representation index $r$ to simplify the expression.

Unplaced source note

The set ${\Omega }^m(G)$ should not be confused with ${\Omega }^m(M)$ , the set of $m$ -forms on $M$ . The distinction should be clear from the context.

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