[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 sth 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.?).