Anomalies and Their Occurrence in the Study of Topology in Yang-Mills Gauge Theories

Abstract

When an exact symmetry of the classical action is not preserved as a symmetry under quantization, such as path integral quantization, a gauge theory such as Yang-Mills is said to be anomalous. It is the intention to introduce the subject of anomalies and study briefly some of the topological implications that are associated with the presence of non-abelian gauge anomalies in quantum field theories.

1. Introduction

The objective is to study the topological and cohomological properties of abelian and non-abelian chiral anomalies, which arise in the context of Yang-Mills gauge theories [1, 2, 3]. The occurrence of anomalies can be traced back to the nontrivial topology of the configuration or Yang-Mills orbit space. The abelian chiral anomaly in $d=2p$ dimensions and the non-abelian gauge anomalies when $d=2$ can be calculated. These results may be interpreted in terms of suitable index theorems on spaces of dimension $d=2p$ and $d+2$, respectively. There are consistency conditions for the anomalies and the Schwinger terms can be interpreted in terms of cohomology such that the cocycles take values in the space of functionals of the gauge fields. The cohomological descent procedure which starts from the Chern character forms provides a method for obtaining nontrivial candidates for both the non-abelian anomalies and the Schwinger terms.

The requirement that a field theory be invariant under rigid transformations $\left\{g_i\right\}$ of a gauge group G as well as be invariant under local $\left\{g_i=g_i\left(x\right)\right\}$ transformations is called the gauge invariance principle. This symmetry principle is dynamical because besides including rigid symmetry, it also yields information about the way the interaction with the gauge fields has to be expressed. This gauge invariance is obtained by introducing gauge or Yang-Mills potentials $A_{\mu }^j\left(x\right),$ such that j is a group index. These transform under the adjoint representation of a compact Lie group G and $x=\left(x^{\mu }\right)$ is the spacetime manifold coordinate. The coordinate derivative ${\partial }_{\mu }$ is gauged by the covariant derivative $D_{\mu }$. The Yang-Mills potentials are subject to gauge transformations $A\to A^g$, so independent degrees of freedom have to be identified to avoid overcounting. These symmetries express redundancy in the description of the Yang-Mills fields. Let $\mathcal{A}$ be the infinite-dimensional space of all Yang-Mills potentials A. The action of the gauge group of $\mathcal{A}$ determines orbits, which contain the Yang-Mills field connected to it by a gauge transformation. These extra degrees of freedom can be eliminated by fixing a gauge. This means finding a solution $A_{\mu }^{g_0}$ to a gauge fixing constraint $f\left(A^g\right)=0$. The $A_{\mu }^{g_0}$ of a given orbit is the potential satisfying the same condition as f. In non-abelian theories the solution to this equation is not unique. For example, the Coulomb gauge condition does not determine A uniquely, since $\nabla {\mathbf{A}}^{\prime }=\nabla \mathbf{A}$ allows these potentials to be related, that is, when $\mathrm{\Delta }\tau =0,{\mathbf{A}}^{\prime }=\mathbf{A}+\nabla \tau .$ Thus, there is no unique intersection between the surface $f\left(A_{\mu }^g\right)=0$ and the orbits $A_{\mu }^g$. The phenomenon known as the Gribov ambiguity implies no unique intersection between the surface $f\left(A^g\right)=0$ and the orbits [4]. However, as the intersections of an orbit with the surface $f\left(A^g\right)=0$ are separated by finite gauge transformations, this ambiguity does not disturb the perturbative representation of the theory, which is based on an expansion of a classical configuration. The geometrical nature behind the ambiguity is that of a nontrivial principle bundle P, the bundle of Yang-Mills potentials for which there is no global section. The structure group of this bundle is the group of gauge transformations. The fibers are the orbits of the potentials and the base is determined by independent gauge degrees of freedom. Selecting a gauge would correspond to choosing a vector potential in each orbit in a continuous way. No global gauge fixing is possible on compactified spacetimes. In trying to extend the local chart to the whole manifold, the effect is that, beyond a certain distance, the gauge condition does not fix the gauge uniquely. There are better ways of thinking about a gauge transformation. For example, it is possible to look at a gauge transformation as the change produced in $A_{\mu }$ by a vertical bundle automorphism f of P, where f is trivial on the base space $f_b={\mathbf{1}}_M.$ If $f:P\to P$ is an automorphism $f^{\ast }\left(\omega \right)$ is the gauge transformed connection. So, when $A={\sigma }^{\ast }\left(\omega \right)$ on U, then the new $A^{\prime }$ is given by ${\sigma }^{\ast }\left(f^{\ast }\omega \right)={\left(f\circ \sigma \right)}^{\ast }\left(\omega \right)=A^{\prime }$, where in analogy we denote $f\circ g$ by ${\sigma }^{\prime }$ and both are equivalent.

2. Geometry of gauge theories

Let us provide more information about gauge theories that are required along the way. The group of gauge transformations on spacetime M is the group $\mathcal{C}={\mathrm{Aut}}_V\left(P\right)$ of vertical automorphisms of P(G, M), or ${\mathrm{Aut}}_{V}P=\left\{f\in {\text{Diff}}_M\left(P\right)|f_b=I_M\right\}$, so f is trivial on the base. A Yang-Mills potential comes from a connection $\omega$ on a principle bundle P(G, M) over a spacetime M such that gauge transformation g(x) relates the local representatives ${\sigma }^{\ast }\left(\omega \right)=A$ and $A^{\prime }={\left(f\circ \sigma \right)}^{\ast }\left(\omega \right)$ determined by the pullback of $\omega$ by two local sections $\sigma$ and ${\sigma }^{\prime }$ over $U\subset M.$ If ${\sigma }_{\alpha }^{\ast }\left(\omega \right)=A_{\alpha }$ and ${{\sigma }_{\alpha }}^{\text{'}}\left(x\right)=f\left({\sigma }_{\alpha }\left(x\right)\right)={\sigma }_{\alpha }\left(x\right)\phi \left({\sigma }_{\alpha }\left(x\right)\right)={\sigma }_{\alpha }{\phi }_{\alpha }\left(x\right)$ the gauge transformation is

$$A_{\alpha }^{\prime }\left(x\right)=\mathrm{Ad}\left({\phi }_{\alpha }^{-1}\left(x\right)\right)A_{\alpha }\left(x\right)+{\phi }_{\alpha }^{-1}\left(x\right)d{\phi }_{\alpha }\left(x\right).$$E1

If two local sections are given ${\sigma }_{\beta }\left(x\right)={\sigma }_{\alpha }\left(x\right)g_{\mathit{\alpha \beta }}\left(x\right)$, (then)

$${\sigma }_{\beta }^{\prime }\left(x\right)=f\left({\sigma }_{\beta }\left(x\right)\right)={\sigma }_{\beta }\left(x\right){\phi }_{\beta }\left(x\right)=f\left({\sigma }_{\alpha }\left(x\right)g_{\mathit{\alpha \beta }}\left(x\right)\right)={\sigma }_{\alpha }\left(x\right){\phi }_{\alpha }\left(x\right)g_{\mathit{\alpha \beta }}\left(x\right).{}^{‘}$$E2

Consequently, we express ${\phi }_{\beta }=g_{\mathit{\alpha \beta }}^{-1}{\phi }_{\alpha }\left(x\right)g_{\mathit{\alpha \beta }}\left(x\right)$ for $x\in U_{\alpha }\cap U_{\beta }$. The elements of the gauge group $\mathcal{C}$ of gauge transformations are sections in $\mathrm{\Gamma }\left(\tilde{\mathrm{A}}\mathrm{d}\left(P\right),M\right).$ It can be shown that $\mathcal{C}$ is a Baker-Campbell-Hausdorff group. The Lie algebra is given by sections of the associated bundle of Lie algebras $\mathrm{Ad}\left(P\right)=P{\times }_{\mathrm{Ad}G}\mathcal{C}$. Thus, a gauge transformation can be viewed as a section of $\tilde{\mathrm{A}}\mathrm{d}\left(P\right),$ and an infinitesimal gauge transformation is a section of Ad(P) [5, 6, 7].

Gauge-unrelated Yang-Mills potentials are projected onto different points of the quotient $\mathcal{A}/\mathcal{C}$, the orbit space of the Yang-Mills connections on P(M, G), In general, the action of $\mathcal{C}$ on $\mathcal{A}$ is not free but with certain technical restrictions, it is possible to obtain a free action. This may be done, for example, by restricting $\mathcal{C}$ to the group of automorphisms, preserving a point $p\in P$ or to the group of based gauge transformations, that is, leaving infinity or a point of $M^m=S^m$ fixed. It is not hard to see the bundle of connections is not trivial in general. If it were, it would be permissible to write $\mathcal{A}=\mathcal{C}\times \mathcal{A}/\mathcal{C}$. The functional space of connections $\mathcal{A}$ is an affine space, and hence, it is contractable and it has no topological invariants. However, $\mathcal{A}=\mathcal{C}\times \mathcal{A}/\mathcal{C}$ implies that $0={\pi }_j\left(\mathcal{A}\right)={\pi }_j\left(\mathcal{C}\right)+{\pi }_j\left(\mathcal{A}/\mathcal{C}\right)$ this cannot be fulfilled in general since $\mathcal{C}$ possesses nonzero homotopy groups. The bundle may be nontrivial and no continuous gauge fixing is possible. Since $\mathcal{A}$ is topologically trivial, the topology of $\mathcal{A}/\mathcal{C}$ comes entirely from $\mathcal{C}$.

Consider briefly the topology of the orbit space $\mathcal{A}/\mathcal{C}$, the space of measurable, physical fields, that is, configuration space of the Yang-Mills theory. This depends on the topology of the manifold M. The usual asymptotic conditions allow us to replace the m-dimensional space M by its conformal compactification $S^m$, which is useful for topological considerations. To obtain a finite Yang-Mills action, assume A(x) become pure gauge at infinity $A\left(x\right)\to g^{-1}\left(x\right)dg\left(x\right),$ where $g:S^{m-1}\to G$ and $S^{m-1}$ is the sphere at infinity. These mappings fall into homotopy classes ${\pi }_{m-1}\left(G\right)$, which classify the bundle $P_k\left(G,S^m\right)$ over $S^m$. If m is even, the class k is obtained by computing the Chern class $c_{\left(m/2\right)}$ over $S^m$. This k-dependence is reflected in other constructions. Thus, $\mathcal{A}$ splits into spaces ${\mathcal{A}}_k$ of connections on the bundle $P_k\left(G,S^m\right)$ with gauge group ${\mathcal{C}}_k$. For a given k, ${\mathcal{A}}_k\left({\mathcal{C}}_k,{\mathcal{A}}_k/{\mathcal{C}}_k\right)$ is a principle bundle with structure group ${\mathcal{C}}_k.$ As pointed out by Atiyah and Jones [8], the homotopy type of ${\mathcal{A}}_k/{\mathcal{C}}_k$ does not make reference to k, so ${\pi }_i\left({\mathcal{A}}_k/{\mathcal{C}}_k\right)={\pi }_i\left(\mathcal{A}/\mathcal{C}\right)$ so we may set $k=0$. As far as homotopy properties are concerned, the trivial bundle may be taken as $P_0\left(G,S^m\right)$ for which $P_0=G\times S^m$, so that ${\mathcal{C}}_0=G\left(S^m\right)$ and k may be ignored. From the homotopy sequence induced by $\mathcal{A}\left(\mathcal{C},\mathcal{A}/\mathcal{C}\right)$, we find that

$$0={\pi }_n\left(\mathcal{A}\right)={\pi }_n\left(\mathcal{A}/\mathcal{C}\right)\to {\pi }_{n-1}\left(\mathcal{C}\right)\to {\pi }_{n-1}\left(\mathcal{A}\right)=0,$$E3

from which we infer,

$${\pi }_j\left(\mathcal{A}/\mathcal{C}\right)={\pi }_j\left(\mathcal{C}\right).$$E4

The non-triviality of the orbit space is tied to the non-triviality of the homotopy groups of G. For an abelian gauge theory in four dimensions ${\pi }_j\left(\mathcal{A}/\mathcal{C}\right)={\pi }_{j+1}\left(U\left(1\right)\right)=0.$. This means U(1) -gauge bundle of quantum electrodynamics is trivial and so no Gribov ambiguity in quantum electrodynamics. The non-abelian anomaly of chiral gauge theories in four dimensions may be exposed topologically by looking for non-contractible two-spheres in orbit spaces for which ${\pi }_2\left(\mathcal{A}/\mathcal{C}\right)={\pi }_5\left(G\right)=Z$. In general, in even dimension, a sufficient condition for the existence of a perturbative non-abelian anomaly is that ${\pi }_2\left(\mathcal{A}/\mathcal{C}\right)={\pi }_1\left(\mathcal{C}\right)={\pi }_{2p+1}\left(G\right)=Z$.

3. Quantization of gauge theories and the appearance of anomalies

In general, a quantum theory is called anomalous if an exact symmetry of the classical action is not preserved as a symmetry as a result of path integral quantization of the theory. It is then said there is an obstruction to the lifting of the classical symmetry in the quantum theory [9, 10].

An example of this is the axial vector Noether current in a theory with massless fermions associated with the rigid chiral symmetry of the classical action that is not conserved in the quantum case. The anomaly contributes to physical processes such as the decay of the ${\pi }^0$ boson to two photons. When anomalies effect the gauge symmetries of the theory, it becomes inconsistent. To have unitarity and renormalizability in $d=4$, gauge invariance turns out to be a crucial property of Yang-Mills theories [11].

A brief discussion of the U(1) anomaly, or chiral anomaly, will be presented in an arbitrary d-dimensional space. Some of the explicit calculations have been suppressed, so we can concentrate on studying topology. The fact d is even is due to the fact that chirality can only be defined in even dimensions. One way to proceed is to use the mathematical approach of Fujikawa [12], which leads ultimately to a field-theoretic illustration of the index theorem. In classical theories, all that matters is the Lagrangian. In the quantum case, it is the nonlocal functional W[A] that is the relevant quantity

$$W\left[A\right]=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{e}^{\mathit{\text{iS}}\left[A,\psi \right]}=e^{\mathit{iZ}\left[A\right]}.$$E5

In (5)$\mathcal{D}\psi \mathcal{D}\overline{\psi }$ is the fermionic functional integration measure and the action S is given as

$$S\left[A,\psi \right]=\int \mathit{dx}\overline{\psi }i/D\psi ,D_{\mu }={\partial }_{\mu }+A_{\mu },A_{\mu }=A_{\mu }^i{T}_i,$$E6

with $/D={\mathrm{\gamma }}^{\mu }{D}_{\mu }$ and $\psi ,\overline{\psi }$ are independent Grassmann variables. The integration measure in d dimensions is written simply as dx. The $T_i$ are the generators of the Lie algebra in matrix form, act on the group representation index in $\psi .$ The Minkowski metric on spacetime is ${\eta }^{\mathit{\mu \nu }}=\left(+,-,\cdots ,-\right)$ and the gamma matrices satisfy $\left\{{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right\}=2{\eta }^{\mathit{\mu \nu }},{\mathrm{\gamma }}^{0†}={\mathrm{\gamma }}^0,{\mathrm{\gamma }}^{i†}=-{\mathrm{\gamma }}^i,$ for $i=1,2,\dots ,\left(d-1\right)$ and ${\mathrm{\gamma }}_{d+1}={\left(i\right)}^{d/2-1}{\mathrm{\gamma }}^0\cdots {\mathrm{\gamma }}^{d-1}$. The exponential in (5) has an oscillatory nature, so the path integral is not well defined because of this. This problem can be overcome by moving it to Euclidean space

$$W_E\left[A\right]=\int \mathit{D\psi D}\overline{\psi }{e}^{-S_E\left(A,\psi \right)}=e^{-Z_E\left[A\right]}.$$E7

Here, $S_E\left(A,\psi \right)$ is the Euclidean action and the Dirac matrices entering $/D$ are all hermitian now: $\left\{{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right\}=2{\delta }^{\mathit{\mu \nu }},{\mathrm{\gamma }}^{\mu †}={\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}_{d+1}=i^{d/2}{\mathrm{\gamma }}^0\cdots {\mathrm{\gamma }}^{d-1},{\mathrm{\gamma }}_{d+1}^2=1,{\mathrm{\gamma }}_{d+1}^{†}={\mathrm{\gamma }}_{d+1},{\mathrm{\gamma }}_{d+1}{\mathrm{\gamma }}^{\mu }=-{\mathrm{\gamma }}^{\mu }{\mathrm{\gamma }}_{d+1}.$ The rotation that changes W[A] into $W_E\left[A\right]$ is given by the substitutions $x_{\mu }^0=-{\mathit{ix}}_E^0,x_M^i\to x_E^i=x_{\mathit{Ei}},$$A_0^M\to {\mathit{iA}}_0^E,A_i^M\to A_i^E=A^{Ei},i=1,\dots ,\left(d-1\right)$ and ${\psi }_M\to {\psi }_E,\overline{\psi }\to i{\overline{\psi }}_E$, and ${\mathrm{\gamma }}_M^i=i{\mathrm{\gamma }}_E^i,{\mathrm{\gamma }}_M^0={\mathrm{\gamma }}_E^0$ are identified.

The system is invariant under rigid chiral transformations of the Dirac fields ${\psi }^{\prime }=e^{\mathit{i\alpha }{\gamma }_{d+1}}\psi ,\mathit{\delta \psi }=\mathit{i\alpha }{\mathrm{\gamma }}_{d+1}\psi .$ Classically, this means there is a conserved Noether chiral current or axial current $j_{d+1}^{\mu }$, which can be deduced by writing the variation of $S\left[A,\psi \right]$ under a nonconstant parameter transformation $\alpha =\alpha \left(x\right)$ as

$$\mathit{\delta S}=\int d\mathbf{x}{\partial }_{\mu }\alpha j_{d+1}^{\mu },j_{d+1}^{\mu }=\overline{\psi }{\mathrm{\gamma }}^{\mu }{\mathrm{\gamma }}_{d+1}\psi .$$E8

Quantum mechanically, the axial vector current is no longer conserved. There is an anomalous divergence

$${\partial }_{\mu }{J}_{d+1}^{\mu }=\frac{1}{W\left[A\right]}\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{\partial }_{\mu }{j}_{d+1}^{\mu }{e}^{\mathit{\text{iS}}\left[A,\psi \right]}.$$E9

The quantity ${\partial }_{\mu }{J}_{d+1}^{\mu }$ is called the abelian or axial anomaly. Abelian refers to the fact that symmetry is an axial U(1) transformation. It has been pointed out that the nonconservation of axial current $j_{d+1}^{\mu }$ may be connected to the transformation properties of the functional fermionic measure in the quantum case.

Suppose we try to derive current conservation by using the rigid chiral transformations of the Dirac fields above with $\alpha =\alpha \left(x\right)$. It is required the path integral retain the same form under this change of variable. Expanding the difference between the two integrals in terms of $\alpha ,$ one before and one after the change using (3.4)

$$0=\int \delta \left(\mathcal{D}\psi \mathcal{D}\overline{\psi }\right)e^{-S_E\left[A,\psi \right]}+\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{e}^{-S_E\left[A,\psi \right]}\cdot \int d\mathbf{x}\left({\partial }_{\mu }\alpha \right)j_{d+1,E}^{\mu }.$$E10

Were the fermionic measure invariant, the second term should go to zero or ${\partial }_{\mu }{J}_{d+1,E}^{\mu }=0$. However, the path integral measure is not invariant. The value of $\delta \left(\mathit{D\psi }D\overline{\psi }\right)$ in (10) is minus twice the trace, in the sense of the space of functions, of $\mathit{i\alpha }{\mathrm{\gamma }}_{d+1}$,

$$D{\psi }^{\prime }D{\overline{\psi }}^{\prime }=exp\left(-2\mathrm{Tr}\left(\mathit{i\alpha }{\mathrm{\gamma }}_{d+1}\right)\right)\mathit{D\psi }D\overline{\psi }.$$E11

What appears to be the inverse of the usual Jacobian occurs because Grassmann odd variables are involved. If Euclidean space is compactified to the sphere $S^d$, both $\psi$ and $\overline{\psi }$ can be expanded in terms of the eigenfunctions of $i/D$ with real eigenvalues ${\lambda }_n$

$$\psi =\sum _n{a}_n{\psi }_n,\overline{\psi }=\sum _n{\overline{b}}_n{\psi }_n,\left({\psi }_i,{\psi }_j\right)=\int d\mathbf{x}{\psi }_i^{†}{\psi }_j={\delta }_{\mathit{ij}}.$$E12

The character of both $\psi$ and $\overline{\psi }$ is carried by the coefficients $a_n,{\overline{b}}_n.$ The operator $i/D,$ which is elliptic in Euclidean space, is a Fredholm operator, which means it has a discrete spectrum. The Jacobian is $2\int d\mathbf{x}{\sum }_n{\mathit{\alpha \psi }}^{†}i{\mathrm{\gamma }}_{d+1}{\psi }_n.$ Therefore, (10) is

$$0=\int \mathit{D\psi }D\overline{\psi }\left[-2\int d\mathbf{x}\alpha \sum _n{\psi }^{†}i{\mathrm{\gamma }}_{d+1}{\psi }_n-\int d\mathbf{x}{\partial }_{\mu }{j}_{d+1,E}^{\mu }\alpha \right]e^{-S_E\left[A,\psi \right]}.$$E13

This holds for any $\alpha \left(x\right)$, so from (9), we can write

$${\partial }_{\mu }{J}_{d+1,E}^{\mu }=\frac{1}{W_E\left[A\right]}\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{\partial }_{\mu }{j}_{d+1,E}^{\mu }{e}^{-S_E\left[A,\psi \right]}=-2i\sum _n{\psi }_n^{†}{\mathrm{\gamma }}_{d+1}{\psi }_n.$$E14

Therefore, the anomalous divergence of the current is a consequence of the nontrivial Jacobian associated with the transformation.

Expression (14) is divergent but can be regularized using the standard heat kernel regularization procedure. Therefore, calling (14)$Q_E$, we have

$$Q_E=\underset{M\to \infty }{lim}\left(-2i\right)\sum _n{e}^{-{\lambda }_n^2/M^2}{\psi }_n^{†}{\mathrm{\gamma }}_{d+1}{\psi }_n.$$E15

Suppose the integral $\int d\mathbf{x}{Q}_E$ is considered now. Then, the ${\psi }_n$ corresponding to zero modes makes a contribution to the integral. If $i/D{\psi }_n={\lambda }_n{\psi }_n,$ then $i/D{\mathrm{\gamma }}_{d+1}{\psi }_n=-{\lambda }_n\left({\mathrm{\gamma }}_{d+1}{\psi }_n\right).$ For ${\lambda }_n\ne 0,{\psi }_n$ and $\left({\mathrm{\gamma }}_{d+1}{\psi }_n\right)$ are eigenvectors that correspond to different eigenvalues of a hermitian operator, which means they are orthogonal. Introducing $P_{+}=\left(1+{\mathrm{\gamma }}_{d+1}\right)/2$ and $P_{-}=\left(1-{\mathrm{\gamma }}_{d+1}\right)/2,{\mathrm{\gamma }}_{d+1}=P_{+}-P_{-}.$ This is just the difference between the positive and negative chirality zero modes. Hence, they are related to the index of the operator in this way

$$\int d\mathbf{x}{Q}_E=-2i\left[dimker\left(i/D_{+}\right)-dimker\left(i/D_{-}\right)\right]=-2i\mathrm{ind}\left(i/D_{+}\right).$$E16

The elliptic non-self adjoint ${\left(i/D\right)}^{†}=i/D_{-}$ operator for each chirality are

$$i/D_{\pm }=i/D{P}_{\pm },i/D=i/D_{+}+i/D_{-}.$$E17

Thus, the integration of the abelian anomaly gives the index for the twisted spin complex associated with the operator $i/D_{+}$. It is possible to compute (15) directly by writing it as

$$\begin{array}{l}{Q}_E=-2i\underset{M\to \infty }{lim}\underset{x\to y}{lim}\sum _n{\psi }_n^{†}\left(y\right){\mathrm{\gamma }}_{d+1}{e}^{-{\left(i\cancel{D}\right)}^2\left(x\right)/M^2}{\psi }_n\left(x\right)\\ =-2i\underset{M\to \infty }{lim}\underset{x\to y}{lim}\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}{e}^{-{\left(i\cancel{D}\right)}^2\left(x\right)/M^2}\sum _n{\psi }_n\left(x\right){\psi }_n^{†}\left(y\right)\right)\\ =2i\underset{M\to \infty }{lim}\underset{x\to y}{lim}\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}{e}^{-{\left(i\cancel{D}\right)}^2\left(x\right)/M^2}\right){\delta }^d\left(x-y\right).\end{array}$$E18

In the last step, the completeness relation ${\sum }_n{\psi }_n\left(x\right){\psi }_n^{†}\left(y\right)={\delta }^d\left(x-y\right)$ for ${\psi }_n$ was used. The trace involves both the group and spinorial indices. Expressing ${\delta }^d\left(x-y\right)$ with respect to a plane wave basis,

$${\delta }^d\left(x-y\right)=\frac{1}{{\left(2\pi \right)}^d}\int d\mathbf{k}{e}^{\mathit{ik}\cdot \left(x-y\right)}.$$E19

Thus, for $Q_E$, we obtain

$$Q_E=-2i\underset{M\to \infty }{lim}\frac{1}{{\left(2\pi \right)}^d}\int d\mathbf{k}{e}^{-\mathit{ik}x}\mathrm{Tr}\left({\gamma }_{d+1}{e}^{-{\left(i\cancel{D}\right)}^2/M^2}\right)e^{\mathit{ik}\cdot x}.$$E20

It is now required to deal with the limit $M\to \infty$. It suffices to know the terms that contribute to this limit. Since

$$\begin{array}{l}/D^2=D_{\mu }{D}_{\nu }{\mathrm{\gamma }}^{\mu }{\mathrm{\gamma }}^{\nu }=\frac{1}{2}{D}_{\mu }{D}_{\nu }\left\{{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right\}+\frac{1}{2}{D}_{\mu }{\mathcal{D}}_{\nu }\left[{\gamma }^{\mu },{\gamma }^{\nu }\right]=D^{\mu }{D}_{\mu }+\frac{1}{4}\left[D_{\mu },D_{\nu }\right]\left[{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right]\\ =D^{\mu }{D}_{\mu }+\frac{1}{4}{F}_{\mathit{\mu \nu }}\left[{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right].\end{array}$$E21

Introduce a new variable s defined in terms of k as $s^{\mu }=k^{\mu }/M$ in which case (15) becomes

$$Q_E=-2i\underset{M\to \infty }{lim}\frac{1}{{\left(2\pi \right)}^d}\int d\mathbf{x}{M}^d\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}exp\left(-{\left(\frac{i}{M}/D-/s\right)}^2\right)\right).$$E22

The exponential term can be reexpressed using (21),

$$exp\left(-{\left(\frac{i}{M}/D-/s\right)}^2\right)=exp\left(\frac{D^2}{M^2}+\frac{1}{4M^2}{F}_{\mathit{\mu \nu }}\left[{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right]+\frac{2i}{M}{s}^{\mu }{D}_{\mu }\right)e^{-s^2}.$$E23

It is the case that all potentially divergent terms (22) vanish since $\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}{\mathrm{\gamma }}^{{\mu }_1}\cdots {\mathrm{\gamma }}^{{\mu }_k}\right)$ for $k<d$. The result of the limit is the contribution to the exponential (23) proportional to $1/M^d$ and at the same time has enough gamma matrices. It is one that arises entirely from the middle term in (23), $F_{\mathit{\mu \nu }}\left[{\mathrm{\gamma }}^{\mu },{\mathrm{\gamma }}^{\nu }\right]$, so $Q_E$ is calculated

$$\begin{array}{l}{Q}_E=\frac{1}{\left(d/2\right)!}\frac{-2i}{{\left(4\pi \right)}^d}\int d\mathbf{x}{e}^{-s^2}\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}{F}_{{\mu }_1{\mu }_2}\cdots F_{{\mu }_{d-1}{\mu }_d}\left[{\mathrm{\gamma }}^{{\mu }_1},{\mathrm{\gamma }}^{{\mu }_2}\right]\cdots \left[{\mathrm{\gamma }}^{{\mu }_{d-1}},{\mathrm{\gamma }}^{{\mu }_d}\right]\right)\\ -\frac{2i}{{\left(4\pi \right)}^d}\frac{1}{\left(d/2\right)!}{\left(2\pi \right)}^{d/2}\mathrm{Tr}\left(F_{{\mu }_1{\mu }_2}\cdots F_{{\mu }_{d-1}{\mu }_d}\right)\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}{\mathrm{\gamma }}^{{\mu }_1}\cdots {\mathrm{\gamma }}^{{\mu }_d}\right)\\ =-2i{\left(\frac{-i}{4\pi }\right)}^{d/2}\frac{1}{\left(d/2\right)!}{\mathrm{ϵ}}^{{\mu }_1\cdots {\mu }_d}\mathrm{Tr}\left(F_{{\mu }_1{\mu }_2}\cdots F_{{\mu }_{d-1}{\mu }_d}\right).\end{array}$$E24

The d-dimensional integral $\int d\mathbf{s}{e}^{-s^2}={\pi }^{d/2}$ has been substituted as well as the trace equation $\mathrm{Tr}\left({\mathrm{\gamma }}_{d+1}{\mathrm{\gamma }}^{{\mu }_1}\cdots {\mathrm{\gamma }}^{{\mu }_d}\right)={\left(-2i\right)}^{d/2}{\mathrm{ϵ}}^{{\mu }_1\cdots {\mu }_d}$.

To obtain the anomaly in d-dimensional Minkowski spacetime, a rotation from Euclidean space back is carried out. There is a factor (−i) coming from the trace part $\left({\partial }_0^E\to -i{\partial }_0^M\right),\left(A_0^E\to -{\mathit{iA}}_0^M\right),$ a factor ${\left(-1\right)}^{d/2}$ since ${\left({\partial }_{\mu }{J}_{d+1}^{\mu }\right)}_E$ goes into ${\left(-1\right)}^{d/2}{\partial }_{\mu }{J}_{d+1}^{\mu }$ and a minus sign, since in Minkowski spacetime ${\mathrm{ϵ}}^{0\cdots d-1}={\left(-1\right)}^{d-1}.$ The anomaly in Minkowski spacetime is found to be

$$Q={\left(\frac{i}{4\pi }\right)}^{d/2}\frac{2}{\left(d/2\right)!}{\mathrm{ϵ}}^{{\mu }_1\cdots {\mu }_d}\mathrm{Tr}\left(F_{{\mu }_1{\mu }_2}\cdots F_{{\mu }_{d-1}{\mu }_d}\right).$$E25

This is what results from the Feynman triangle diagram calculation when $d=4$ for the anomaly. The abelian anomaly is gauge invariant. The last thing to do is get the index of the operator $i/D_{+}$. Using (15), (24), and the definition of the Chern characteristic at the end, the index is calculated as

$$\begin{array}{l}\mathrm{ind}i/D_{+}=\frac{i}{2}\int d\mathbf{x}{Q}_E={\left(-\frac{i}{2\pi }\right)}^{d/2}\frac{2}{\left(d/2\right)!}\frac{1}{2^{d/2}}\int d\mathbf{x}{\mathrm{ϵ}}^{{\mu }_1\cdots {\mu }_d}\mathrm{Tr}\left(F_{{\mu }_1{\mu }_2}\cdots F_{{\mu }_{d-1}{\mu }_d}\right)\\ ={\left(-\frac{i}{2\pi }\right)}^{d/2}\frac{1}{\left(d/2\right)!}\frac{1}{2^{d/2}}{\int }_{S^d}{\mathit{dx}}^{{\mu }_1}\wedge \cdots \wedge {\mathit{dx}}^{{\mu }_d}\mathrm{Tr}\left(F_{{\mu }_1{\mu }_2}\cdots F_{{\mu }_{d-1}{\mu }_d}\right)\\ ={\left(-1\right)}^{d/2}{\int }_{S^d}{\left(\frac{i}{2\pi }\right)}^{d/2}\frac{1}{\left(d/2\right)!}\mathrm{Tr}\left(F^{d/2}\right)={\left(-1\right)}^{d/2}{\int }_{S^d}{\mathit{ch}}_{\left(d/2\right)}\left(F\right).\end{array}$$E26

This shows the expectation value ${\left({\partial }_{\mu }{J}_{d+1}^{\mu }\right)}_E$ is given by the index of the Dirac operator. As $M\sim D^d,$ the spin connection does not enter and the index density for the Dirac operator reduces to the Chern characteristic [12, 13, 14].

4. Nontrivial topology and non-abelian case

Non-abelian gauge anomalies admit path integral representation and appear in theories with gauge fields that are coupled to chiral fermions called Weyl fermions. They come up in even dimensions like the abelian anomaly. The generating function of the theory with dynamical gauge fields is made up of a kinetic term, the functional measure $\mathcal{D}{A}_{\mu }$ and so forth but there are not important for the problem of gauge invariance of the theory to be analized as it is concentrated in (22), since the other contributions are gauge invariant. Since gauge anomalies still make the theory inconsistent, quantize the fermion degrees of freedom first retaining $A_{\mu }$ as an external field to investigate the gauge transformation behavior of the resulting theory [15].

The functional is (5), but now $\psi$ is a Weyl spinor that has positive chirality, ${\mathrm{\gamma }}_{d+1}\psi =\psi ,P_{+}\psi =\psi ,$ so $\cancel{D}$ can be replaced by ${\cancel{D}}_{+}$ now. The variation of Z[A] under a gauge transformation ${\zeta }^i\left(x\right)$ with ${\zeta }^i\cdot Y$ as generator is given by

$$\left({\zeta }^i\cdot Y\right)Z\left[A\right]=\int d\mathbf{x}{\zeta }^i\left(x\right)Y_{\left(i\right)}\left(x\right)Z\left[A\right].$$E27

The functional derivative is given by

$$\frac{\delta Z\left[A\right]}{\delta A_{\mu }^i\left(x\right)}=\frac{1}{W\left[A\right]}\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{j}_{\left(i\right)}^{\mu }\left(x\right)exp\left(\mathit{\text{iS}}\left[A,\psi \right]\right)=J_{\left(i\right)}^{\mu }\left(x\right),$$E28

where $j_{\left(i\right)}^{\mu }=\overline{\psi }{\mathrm{\gamma }}^{\mu }\left({\mathit{iT}}_{\left(i\right)}\right)\psi ,{\left({\mathit{iT}}_{\left(i\right)}\right)}^{†}=\left({\mathit{iT}}_{\left(i\right)}\right)$ is the current coupled to $A_{\mu }^i\left(x\right)$ and $J_{\left(i\right)}^{\mu }$ is defined by (28). Consequently, it is found that

$$\left(\zeta \cdot Y\right)Z\left[A\right]=\int d\mathbf{x}{\zeta }^i\left(x\right)D_{\mu }{J}_{\left(i\right)}^{\mu }\left(x\right)=\int d\mathbf{x}{\zeta }^i\left(x\right){\mathcal{V}}^i\left[A\right]\left(x\right).$$E29

If current $J_{\left(i\right)}^{\mu }$ is not covariantly conserved, a breakdown of a conservation law. The functional will not be gauge invariant and there appears an anomaly. Thus, $D_{\mu }{j}_{\left(i\right)}^{\mu }=0$ does not continue to be maintained in the quantum case.

As with the abelian case, the non-invariance of Z[A] may be tied to the fact that, although the exponential in (5) is gauge invariant, given by the classical action, the fermionic functional measure $\mathcal{D}\psi \mathcal{D}\overline{\psi }$ is not. This implies a Jacobian determinant appears that formally is,

$$\frac{\mathcal{D}\left(g^{-1}\psi \right)\mathcal{D}\left(\overline{\psi }g\right)}{\mathcal{D}\psi \mathcal{D}\overline{\psi }}=e^{i{\alpha }_1\left[A,g\right]}.$$E30

If $W\left[A^g\right]=e^{{\alpha }_1\left[A,g\right]}W\left[A\right],W\left[A\right]=e^{\mathit{iZ}\left[A\right]},$ it follows that $Z\left[A^g\right]=Z\left[A\right]+{\alpha }_1\left[A,g\right].$ Then, $Z\left[A^g\right]-Z\left[A\right]\doteq \left(\zeta \cdot Y\right)Z\left[A\right]=\int d\mathbf{x}{\zeta }^i{\mathcal{V}}_{\left(i\right)}\left[A\right]\left(x\right)$. This means ${\alpha }_1$ is given by

$${\alpha }_1\left[A,g\right]=\int d\mathbf{x}\left({\zeta }^i{\mathcal{V}}_{\left(i\right)}\left[A\right]\left(x\right)\right)$$E31

The euclidean generating functional requires the calculation of the determinant of the matrix $i\cancel{D}$ for its evaluation. The problem is $i\cancel{D}$ alters the chirality here, so there is no well-defined eigenvalue problem for chiral fermion fields, hence no basis of eigenfunctions. The equation $\cancel{D}\mathit{\psi }=\psi$ does not make sense for a Weyl spinor. One way to overcome this is to consider complex Dirac fermions instead of chiral Weyl fermions and replace $i\cancel{D}$ with $i{\cancel{D}}^{\ast }$ where

$${\cancel{D}}^{\ast }=\cancel{D}{P}_{+}+\cancel{\partial }{P}_{-}=\cancel{\partial }+{\cancel{A}}_{\mathit{P}}.$$E32

The new generating functional becomes $\int \mathcal{D}\psi \mathcal{D}\overline{\psi }exp(-\int d\mathbf{x}\overline{\psi }i{\cancel{D}}^{\ast }\psi =det\left(i{\cancel{D}}^{\ast }\right)$. With such an operator, the negative chirality fermions do not couple to A. They just give a proportionality factor in (7). Thus, the chiral gauge theory defined by ${\cancel{D}}^{\ast }$ is the same as the one based on $\cancel{D}$ up to this factor. This operator has a well-defined eigenvalue problem associated with it, but its eigenvalues are not real since $i\overline{\cancel{D}}$ is not a self-adjoint operator. The new action is

$$S_E^{\ast }\left(A,\psi \right)=\int d\mathbf{x}\overline{\psi }i{\cancel{D}}^{\ast }\psi .$$E33

is invariant under gauge transformations ${\delta }_{\zeta }=-\zeta P_{+}\psi ,{\delta }_{\zeta }\overline{\psi }=\overline{\psi }{P}_{-}\zeta ,{\delta }_{\zeta }{A}_{\mu }={\partial }_{\mu }\zeta +\left[A_{\mu },\zeta \right]$ with $\zeta \left(x\right)={\zeta }^i\left(x\right)T_i$. These are ordinary gauge transformations for the positive chirality components. Under these transformations, however, the measure is not invariant and an anomaly results, If ${\psi }^{\prime }$ and $A^{\prime }$ are the transformed fields

$$\begin{array}{l}{W}_E^{\ast }\left[A^{\prime }\right]=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{e}^{-I_E^{\ast }\left(A^{\prime },{\psi }^{\prime }\right)}=\int \mathcal{D}{\psi }^{\prime }\mathcal{D}{\overline{\psi }}^{\prime }{e}^{-I_E^{\ast }\left(A^{\prime },{\psi }^{\prime }\right)}\\ =\int \mathcal{D}{\psi }^{\prime }\mathcal{D}{\psi }^{\prime }{e}^{-I_E^{\ast }\left(A,\psi \right)}=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }J{e}^{-I_E^{\ast }\left(A,\psi \right)}.\end{array}$$E34

Integration variables have simply been relabelled in the second and invariance of the action used in the third equality and J is the Jacobian determinant. Since the operator $i{\cancel{D}}^{\ast }$ is elliptic, it has a discrete spectrum on a compact manifold. As $i{\cancel{D}}^{\ast }$ is not self-adjoint, it has eigenvalues that are complex. This means that the left and right eigenfunctions have to be introduced

$$i{\cancel{D}}^{\ast }{\varphi }_n={\lambda }_n{\varphi }_n,{\chi }_n^{†}i{\cancel{D}}^{\ast }={\lambda }_n{\chi }^{†},\int d\mathbf{x}{\chi }_m^{†}{\varphi }_m={\delta }_{\mathit{mn}}.$$E35

These can be used to form expansions for $\psi$ and $\overline{\psi }$ as before $\psi ={\sum }_n{a}_n{\varphi }_n,\overline{\psi }={\sum }_n{\overline{b}}_n{\chi }_n^{†}.$ It is necessary to regularize the Jacobian, and so as usual, a cutoff M is introduced

$$J=exp\left(\underset{M\to \infty }{lim}\int d\mathbf{x}\sum _n\left({\chi }_n^{†}\left(x\right){\mathrm{\gamma }}_{d+1}{\mathit{\zeta \varphi }}_n\left(x\right)\right)e^{-{\lambda }_n^2/M^2}\right).$$E36

To evaluate (36), it is only necessary to retain the term independent of the cutoff M. Omitting considerable calculation, the result is

$$J=exp\left\{\underset{M\to \infty }{lim}\int d\mathbf{x}\underset{x\to y}{lim}\mathrm{Tr}\left[{\mathrm{\gamma }}_{d+1}\zeta \left(x\right)e^{-{\left(i\cancel{D}\right)}^2\left(x\right)/M^2}\delta \left(x-y\right)\right]\right\}.$$E37

and Tr indicates a trace on the spacial as well as on the gauge group indices and ${\sum }_n{\varphi }_n\left(x\right){\chi }_n^{†}\left(y\right)={\delta }^d\left(x-y\right)$. It can also be shown that the expression for the anomaly when $d=4$ is

$$-\frac{i}{24{\pi }^2}\int \mathrm{Tr}\left[\zeta \right(d\left(A\mathit{dA}+\frac{1}{2}{A}^2\right)].$$E38

This implies that

$${\left(D_{\mu }{J}^{\mu }\right)}_{\left(j\right)}=-\frac{i}{24{\pi }^2}{\mathrm{ϵ}}^{\mathit{\text{μνρσ}}}\mathrm{Tr}\left[T_{\left(j\right)}{\partial }_{\mu }\left(A_{\nu }{\partial }_{\mu }{A}_{\sigma }+\frac{1}{2}{A}_{\nu }{A}_{\rho }{A}_{\sigma }\right)\right].$$E39

We can now investigate the non-abelian anomaly as a probe for nontrivial topology. There is a similarity between the expression for the U(1) anomaly when $d=4$ and the non-abelian gauge anomaly. They are given by the divergence and covariant divergence of the currents $J_{d+1}^{\mu }$ and $J_{\left(i\right)}^{\mu }$. Apart from the numerical factor of the $A^3$ term, they are similar.

5. The anomaly as a way to the topology of gauge theories

It has been shown by Atiyah and Singer that the expression for the non-abelian anomaly in $d=2p$ dimensions may be obtained from that of the abelian anomaly in a $\left(d+2\right)$-dimensional space by using an appropriate $\left(d+2\right)$-dimensional index theorem. An outline as to how this may be done is given. In fact, the non-abelian anomaly can be related to the nontrivial topology of the orbit space [16, 17].

Suppose we consider again the action of a Yang-Mills theory with a chiral fermion on an even-dimensional Euclidean space compactified to $S^{2p}$. Let G be a simple, simply connected group such as SU(n). The effective action Z[A] is

$$e^{-Z\left[A\right]}=\int \mathit{D\psi }D\overline{\psi }{e}^{-\int d^{2p}x\overline{\psi }i\cancel{D}{P}_{+}\psi }.$$E40

The identification of (40) with $det\left(i/{\mathit{DP}}_{+}\right)$ is not possible since apart from the regularization problems, the operator $i/{\mathit{DP}}_{+}$ does not have a well-defined eigenvalue problem. This can be avoided by using the operator $/D^{\ast }$ with a Dirac spinor instead of a Weyl spinor. The new action allows us to express (40) as a determinant

$$e^{-Z\left[A\right]}=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{e}^{-d^{2p}x\overline{\psi }i{\cancel{D}}^{\ast }\psi }=det\left(i/D^{\ast }\right).$$E41

As we know, the functional integral is not gauge invariant

$$e^{-Z\left[A^g\right]}=e^{i{\alpha }_1\left[A,g\right]}-e^{-Z\left[A\right]}.$$E42

This lack of invariance has the effect that (41) cannot be expressed over the orbit space $\mathcal{A}/\mathcal{G}$.

The fermion determinant may itself be used to study the nontrivial topology of the orbit space. The variation has to be imaginary since its modulus is gauge invariant. Define now $/D_{\ast }=/D_{+}+/D_{-}=/\partial +/A{P}_{+}$. It is the case that when a Weyl realization for the gamma matrices is used

$$\begin{array}{l}(det\left(i/D^{\ast }\right)(det\left(i/D^{\ast †}\right)=det(i/D^{\ast }i/D^{\ast †}=det\left[\left(i/D_{+}+i/{\partial }_{-}\right)\left(i/D_{-}+i/{\partial }_{+}\right)\right]\\ =det\left(i/D_{+}i/D_{-}+i/{\partial }_{-}i/{\partial }_{+}\right)=det\left(\begin{array}{cc}i/{\partial }_{-}i/{\partial }_{+}& 0\\ 0& i/D_{+}/D_{-}\end{array}\right)\end{array}$$E43

The factor $det\left(i/{\partial }_{-}i/{\partial }_{+}\right)$ is ignored here, so it is found that ${\left|det\left(i/D^{\ast }\right)\right|}^2$ is proportional to $det\left(i/D_{+}i/D_{-}\right)$. Since $det{\left(i/D_{+}i/D_{-}\right)}^2=det{\left(i/D\right)}^2$, where $/D$ is the usual Dirac operator, up to a constant, we obtain $\mid det\left(i{\mathcal{D}}^{\ast }\right)\mid ={\left(det{\left(i/D\right)}^{1/2}\right)}^{1/2}$. This can be regulated in a gauge-invariant manner. Thus, only the imaginary part of $det\left(i/D^{\ast }\right)$ can change as implied by (42). As the real part always admits a gauge-invariant definition on general grounds, it can be said only the imaginary part may change. The real part has a gauge-invariant definition.

Let us try to look more deeply into the topological nature of the non-abelian gauge anomaly. It is important to examine the connection between the d-dimensional non-abelian anomaly and the $d+2$ index theorem which describes the abelian anomaly in $d+2$ dimensions. Let $g\left(\theta ,x\right)$ be a family of gauge transformations that depend on a parameter $\theta \in S^1$ satisfying the boundary conditions $g\left(0,x\right)=g\left(2\pi ,x\right)=e$, These define mappings $g:S^1\times S^{2p}\to G$ classified by the homotopy classes ${\pi }_{2p+1}\left(G\right)$. Notice with these conditions on $g\left(\theta ,x\right)$, the product $S^1$ and $S^{2p}$ is topologically equivalent to $S^{2p+1}$. Let A(x) be a gauge field such that it corresponds to a connection on a trivial bundle and let

$$A^{\theta }=g^{-1}\left(\theta ,x\right)\left(d+A\right)g\left(\theta ,x\right),$$E44

where $d={\sum }_{\mu =0}^{d-1}{\mathit{dx}}^{\mu }\left(\partial /\partial x^{\mu }\right)$ is the resulting one-parameter family of group-transformed Yang-Mills configurations. Suppose the operator $i{\mathcal{D}}_{\ast }\left(A\right)$ has no zero modes. Since just the phase can pick up an anomalous change under a gauge transformation, the operator $i/D^{\ast }\left(A\right)$ does not have them either for all $\theta \in U\left(1\right)$ as the modulus of the determinant is gauge invariant, so $\mid det\left(i/D^{\ast }\left(A^{\theta }\right)\right)\mid =\mid det\left(i/D^{\ast }\left(A\right)\right)\mid$, and we have

$$e^{-Z\left(A^{\theta }\right)}=det\left(i/D^{\ast }\left(A^{\theta }\right)\right)=det\left({\mathit{iD}}^{\ast }\left(A\right)\right)e^{\mathit{i\alpha }\left(A,\theta \right)}.$$E45

Of course, $\theta$ enters in $\alpha$ and in $A^{\theta }$ by means of $g\left(\theta ,x\right)$, so the functional $det\left(i/D_{\ast }\left(A^{\theta }\right)\right)$ defines a particular mapping, $\theta \to e^{\mathit{i\alpha }\left[A,\theta \right]}\in U\left(1\right).$ These mappings can be characterized by means of an integral winding number,

$$k=\frac{1}{2\pi }{\int }_0^{2\pi }\mathit{d\theta }\frac{\partial \alpha \left[A,\theta \right]}{\partial \theta }.$$E46

The integrand is somewhat formal since it is not exact, otherwise $k=0$. To actually get this winding number, it is necessary to extend $A^g$ to a two-parameter family $A^{t,\theta }$ of gauge fields. If $A^{\theta }$ describes a circle $S^1$ in the space of gauge fields $\mathcal{A},$ then $A^{t,\theta }$ is defined on a two-dimensional disk $W^2$ with $\partial W^2=S^1$

$$A^{t,\theta }={\mathit{tA}}^{\theta }=t{g}^{-1}\left(\theta ,x\right)\left(d+A\right)g\left(\theta ,x\right),$$E47

with $d$ as below (44), $\left(t,\theta \right)$ are polar coordinates of the disk $W^2$, with $0\le t\le 1$. On the boundary, $A^{t,\theta }$ becomes the one-parameter family of gauge-related configurations of (44). If $A^{\theta }$ describes a circle in $\mathcal{A}$, then $A^{t,\theta }$ describes a two-dimensional disk in the space of all gauge fields on $S^{2p}$ that belongs to the trivial topological class in $\mathcal{A}$. This triviality is not important as far as the topology of the orbit space $\mathcal{A}/\mathcal{C}$. The determinant $det\left(i/D^{\ast }\left(A^{t,\theta }\right)\right)$ turns into a complex function of the gauge fields on the circle of gauge fields at its boundary. Omitting details of the calculation of the winding number, it may be shown it is equal to the difference between the positive and negative chirality modes of the ordinary Dirac operator $i/D_{2p+2}$ in $d+2$ dimensions

$$\mathit{ind}\left(i/D_{2p+2}\right)=n_{+}-n_{-}=k={\int }_0^{2\pi }{d}_{\theta }\alpha \left[A,\theta \right],$$E48

where $/D_{2p+2}={\sum }_{\mu =0}^{2p+1}{\gamma }^{\mu }{D}_{\mu }$. The $\gamma$‘s are now those of a $\left(d+2\right)$-dimensional space whose coordinates are $t,\theta ,x^{\mu }$ and $\mathit{d\theta }\left(\partial /\partial \theta \right)\equiv d_{\theta }$. This is executed by relating the zero modes $i/D^{\ast }\left(A^{t,\theta }\right)$ to those of $/D_{2p+2}$. In fact, the winding number (46) of the phase of the d-dimensional Weyl determinant is measured by the homotopy class in ${\pi }_{2p+1}\left(G\right)$ of mapping $g\left(\theta ,x\right)$.

The next thing to do is evaluate $\mathrm{ind}\left({\left(i/D\right)}_{2p+2}\right)$. Gauge field $A^{t,\theta }\left(x\right)$ is defined on the manifold $W^2\times S^{2p}$ with a boundary $S^1\times S^{2p}$, and so is the operator $i/D_{2p+2}$. To use the index theorem for manifolds without boundary, so no boundary corrections arise, consider the manifold $S^2\times S^{2p}$. The Yang-Mills fields $A$ on this larger manifold $S^2\times S^{2p}$ are the pullbacks of connections on a principal bundle over $S^2\times S^{2p}$ with a structure group $G$. So as far as the two sphere is concerned, we can use two local charts $D_{\pm }^2\times S^{2p}$, where the disks $D_{+}$ and $D_{-}$ are parametrized by $\left(t,\theta \right)$, $\left(\left(s,\theta \right)\right)$ with $W_{+}$ being the previous $W^2$. This should avoid singular parametrization. The region of overlap along the equator corresponds to $t=1=s$; the north (south) poles of $S^2$ are given by $t=0$, $s=0$. Two local gauge fields $\left(0,0,A_{\pm }\right)$ on the two charts $W_{+}\left(0\le t\le 1\right)$ and $W_{-}\left(0\le s\le 1\right)$ can be taken as

$$A_{+}\left(t,\theta ,x\right)=t\left[g^{-1}\left(A+d+d_{\theta }\right)g\right]=A^{t,\theta }+{\mathit{tg}}^{-1}{d}_{\theta }g,A_{-}\left(s,0,x\right)=A.$$E49

The lower disk is trivial. Let us define an operator $\overline{d}$ (by)

$$\overline{d}=d+d_{\theta }+d_t+d_s.$$E50

It can be seen that $A_{\pm }$ at the equator $t=1=s$ are gauge related

$$A_{+}=g^{-1}\left(\overline{d}+A_{-}\right)g$$E51

since $d_{t}g\left(\theta ,x\right)=0=d_{s}g\left(\theta ,x\right)$. Hence, $A_{\pm }$ define a connection on the principal bundle over $S^2\times S^{2p}$ with group G. So $g\left(\cdot ,x\right):S^1\to \mathcal{C}$ is just the transition function between the local expressions $A_{\pm }$ at the boundary $S^1\times S^{2p}$ of the two patches. By the previous considerations, this also defines a mapping $g\left(\theta ,x\right)$ homotopically equivalent to a mapping $g:S^{2p+1}\to G$. These transition functions define loops in $\mathcal{C}$ that are classified by ${\pi }_1\left(\mathcal{C}\right)$.

The expression of the index that needs to be computed for $p=d/2$ is taken to be

$$\mathrm{ind}\left(i/D_{2p+2}\left(A\right)\right)={\int }_{S^2\times S^{2p}}{\mathrm{ch}}_{p+1}\left(F\right)={\int }_{W_{+}\times S^{2p}}{\mathrm{ch}}_{p+1}\left(F_{+}\right)+{\int }_{W\times S^{2p}}{\mathrm{ch}}_{p+1}\left(F_{-}\right).$$E52

The Chern character $\left(2p+2\right)$-form ${\mathit{ch}}_{p+1}\left(F\right)$ is closed and the local potential $\left(2p+1\right)$-forms are given by the Chern-Simons forms $Q^{2p+1}$ for the gauge fields on the corresponding charts. In fact, (51) is given by

$$Q^{\left(2p+1\right)}\left(A_{+},F_{+}\right){\left|{}_{t=1}-Q^{\left(2p+1\right)}(A_{-},F_{-})\right|}_{s=1}.$$E53

However, on account of (5.12), this expression is the variation of $Q^{\left(2p+1\right)}$ under a gauge transformation and given by $Q^{\left(2l-1\right)}\left(A^g,F^g\right)-Q^{\left(2l-1\right)}\left(A,F\right)=d{\alpha }^{2p-2}\left(A,F,a\right)+Q^{\left(2l-1\right)}\left(g^{-1}\mathit{dg},0\right)$ upon replacing l by $p+1$,

$$Q_{+}^{\left(2p+1\right)}-Q_{-}^{\left(2p+1\right)}=Q^{\left(2p+1\right)}\left(g^{-1}\overline{d}g,0\right)+\overline{d}{\alpha }^{\left(2p\right)},$$E54

where $\overline{d}$ appears in (54) as the exterior derivative $\overline{d}=d+d_{\theta }$. The second term in this is an exact form and it does not contribute,

$$\mathit{ind}\left(i/D_{2p+2}\right)={\left(-1\right)}^p{\left(\frac{i}{2\pi }\right)}^{\left(p+1\right)}\frac{p!}{\left(2p+1\right)!}{\int }_{S^1\times S^{2p}}\mathit{Tr}{\left(g^{-1}\overline{d}g\right)}^{2p+1}.$$E55

In fact, this is the number of times $g\left(\theta ,x\right)$ wraps around $S^{2p+1}$ and is in ${\pi }_{2p+1}\left(G\right)$.

The local density ${\mathit{id}}_{\theta }\alpha \left[A,\theta \right]$, which describes the non-abelian anomaly in a d-dimensional spacetime can be identified and both sides of (48) have to be related. In terms of $Q^{\left(2p+1\right)}$, the index $i/{\mathcal{D}}_{2p+1}$ reduces to

$$\mathit{ind}\left(i/{\mathcal{D}}_{2p+2}\right)={\int }_{S^1\times S^{2p}}{Q}^{\left(2p+1\right)}\left(A^{\theta }+\widehat{v},F^{\theta }\right),$$E56

where $\widehat{v}\equiv g^{-1}{d}_{\theta }g=v\mathit{d\theta }$ and $F^{\theta }=\overline{d}{\left(A^{\theta }+v\right)}^2=g^{-1}\mathit{Fg}$, since the term in Q evaluated at $s=1$ does not have a $\mathit{d\theta }$ component and cannot contribute to the integral $S^1\times S^{2p}$. The only term which contributes to (56) is the term linear in $\widehat{v}$. By (48), the local density for the anomaly is the local density for $\mathrm{ind}\left(i/{\mathcal{D}}_{2p+2}\right)$. Thus, what is needed is the first-order variation

$$Q_1^{\left(2p\right)}\left(v,A^{\theta },F^{\theta }\right)=Q^{\left(2p+1\right)}\left(A^{\theta }+v,F^{\theta }\right)-Q^{\left(2p+1\right)}\left(A^{\theta },F^{\theta }\right).$$E57

As $Q^{\left(2p+1\right)}$ is known, this is not difficult to compute. For $d=2p=4$, using (48) this gives

$$\frac{1}{2\pi }{d}_{\theta }\alpha ={\int }_{S^4}{Q}_1^4\left(v,A^{\theta },F^{\theta }\right)=-\frac{i}{48{\pi }^3}{\int }_{S^4}\mathit{Tr}(vd\left(A^{\theta }{\mathit{dA}}^{\theta }+\frac{1}{2}{\left(A^{\theta }\right)}^3\right).$$E58

Factoring $\mathit{d\theta }$ and setting $\theta =0$, we find the expression for the anomaly

$$\frac{\partial \alpha }{\partial \theta }\left[A,g\right]=-\frac{i}{24{\pi }^2}{\int }_{S^4}\mathrm{Tr}\left[vd\left(A\mathit{dA}+\frac{1}{2}{A}^3\right)\right].$$E59

From (29) and (31), it is found that

$$V_{\left(i\right)}\left[A\right]\left(x\right)=D_{\mu }{J}_{\left(i\right)}^{\mu }=-\frac{i}{24{\pi }^2}\mathit{Tr}\left[T_{\left(i\right)}{ϵ}^{\mathit{\text{νρσκ}}}{\partial }_{\nu }\left(A_{\rho }{\partial }_{\sigma }{A}_{\kappa }+\frac{1}{2}{A}_{\rho }{A}_{\sigma }{A}_{\kappa }\right)\right].$$E60

There are topological consequences related to these results. These considerations provide a check of the relation ${\pi }_1\left(\mathcal{C}\right)={\pi }_5\left(G\right)$. What can be said about ${\pi }_2\left(\mathcal{A}/\mathcal{C}\right)$. Recall the set of Yang-Mills potentials $A^{t,\theta }$ in $\mathcal{A}$. The potentials $A^{\theta }$ when $t=1$ on the boundary are gauge related. The projection of this disk on the orbit space is a two-sphere since all $A^{\theta }$ are projected onto a reference potential A.

The t part of $A^{t,\theta }$ is topologically trivial so the two-sphere in $\mathcal{A}/\mathcal{C}$ is always contractible ${\pi }_2\left(\mathcal{A}/\mathcal{C}\right)=0$ if and only if the loop $A^{\theta }$ in $\mathcal{A}$ is trivial so ${\pi }_1\left(\mathcal{C}\right)=0$. The homotopic non-triviality of the gauge group is equivalent to that of the two-sphere. If not the homotopy used to contract the sphere could be used to deform the loop $g\left(\theta ,x\right)$ to a point $g\left(x\right)$ and ${\pi }_1\left(\mathcal{C}\right)={\pi }_2\left(\mathcal{A}/\mathcal{C}\right)$. There is a global topological obstruction to removing the phase factor in the definition of the determinant when the non-triviality is present. This means the determinant cannot be restricted to the orbit space consistently, so there is no definition in a gauge-invariant manner. Instead, if det A is computed using a regularization procedure, it is found that

$$det\left(A^g\right)=e^{i{\alpha }_1\left[A,g\right]}det\left(A\right),$$E61

where ${\alpha }_1$ can be considered a mapping of $\mathcal{A}\times \mathcal{C}$ in the multiplicative group of complex numbers, hence the one. The action of two transformations required that mod $2\pi$,

$${\alpha }_1\left[A^g,g^{\prime }\right]={\alpha }_1\left[A,g{g}^{\prime }\right]+{\alpha }_1\left[A,g\right]=0.$$E62

This is a one-cycle condition. It relates topological properties with cohomology since the variation of the phase of the determinant is an obstruction to projecting gauge orbits in $\mathcal{A}$ onto points in $\mathcal{A}/\mathcal{C}.$ In this picture, a complex line bundle over $\mathcal{A}/\mathcal{C}$ is defined, the determinant bundle, characterized by its first Chern class. This determines an element in ${\pi }_1\left(\mathcal{C}\right)={\pi }_{2p+1}\left(G\right)={\pi }_2\left(\mathcal{A}/\mathcal{C}\right)=Z.$ The determinant bundle over a non-contractible two-sphere in orbit space with a winding number in ${\pi }_1\left(\mathcal{C}\right)$ is identical to the bundle describing a monopole of the same number of units of magnetic charge. Upon writing a representation of the first Chern class in a topologically nontrivial configuration is the same as giving a specific form of the anomaly.

In d = 4 the abelian anomaly is governed by ${\pi }_0\left(\mathcal{A}/\mathcal{C}\right)={\pi }_3\left(G\right)$ and the non-abelian anomaly by ${\pi }_2\left(\mathcal{A}/\mathcal{C}\right)={\pi }_5\left(G\right).$ In fact Witten’s global anomaly depends on ${\pi }_1\left(\mathcal{A}/\mathcal{C}\right)={\pi }_4\left(G\right)$ of course with ${\pi }_5\left(G\right)=Z\left({\pi }_{2p+1}\left(G\right)\right)=Z$ is a sufficient but not necessary condition for the existence of an anomalous variation of Z [A]. The determinant may have a local variation even if the topology does not force it to acquire a nontrivial topological phase.

6. Conclusions

This work has given a practical way of extracting the local variation and with it the associated anomaly, which is not necessarily related to the existence of a nonzero integer winding number.

Hence, even if the variation of the phase functional does not have a global topological meaning for a compactified spacetime $S^d,$ it still provides the physical perturbative expression for the non-abelian anomaly. It may appear as a topological obstruction in less topologically trivial classes in the local cohomology of gauge fields. This way of proceeding though has the consequence that it results in the cohomological descent procedure [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].