13.6 The parity anomaly in odd-dimensional spaces

So far, we have been working in even-dimensional spaces. One of the reasons for this is that $\mathrm{SO}(2l+1)$ has real or pseudo-real spinor representations but no complex representations, hence no gauge anomaly is expected. However, we can show that gauge theories in odd-dimensional spaces have a different kind of anomaly called the ‘parity anomaly’, in which the parity symmetry of the classical action is not maintained through quantization. It should be noted that the parity anomaly in $2l+1$ dimensions is related to the Abelian anomaly in $2l+2$ dimensions as was pointed out by Alvarez-Gaume et al (1985).

13.6.1 The parity anomaly

Let $M$ be a $(2l+1)$ -dimensional Riemannian manifold. We distinguish one dimension from the others; that is we assume that $M$ is of the form $\mathbb{R}\times \mathcal{M}$ or $S^1\times \mathcal{M}$ , where $\mathcal{M}$ is a $2l$ -dimensional compact manifold without a boundary. We denote the coordinate of $\mathbb{R}$ or $S^1$ by $t$ while that of $\mathcal{M}$ is denoted by $x$ . The index 0 denotes the component in $t$ -space while $\mu$ denotes that in $x$ -space. For example, the components of the $\gamma$ -matrices are $\{{\gamma }^0,{\gamma }^{\mu }(1\le \mu \le 2l)\}$ .

Define the ‘parity’ operation $\mathrm{P}$ by

$${\mathcal{A}}_0(t,x)\to {\mathcal{A}}_0^{\mathrm{P}}(t,x)=-{\mathcal{A}}_0(-t,x)$$

$${\mathcal{A}}_{\mu }(t,x)\to {\mathcal{A}}_{\mu }^{\mathrm{P}}(t,x)={\mathcal{A}}_{\mu }(-t,x)$$

$$\psi (t,x)\to {\psi }^{\mathrm{P}}(t,x)=\mathrm{i}{\gamma }_0\psi (-t,x)$$

$$\bar{\psi }(t,x)\to {\bar{\psi }}^p(t,x)=i\bar{\psi }(-t,x){\gamma }_0.$$

The classical action is invariant under the parity operation,

$$\begin{array}{rl}& \int \mathrm{d}t\mathrm{d}x\bar{\psi }\mathrm{i}\bar{\nabla }\psi \to -\int \mathrm{d}t\mathrm{d}x\bar{\psi }(-t,x){\gamma }^0\mathrm{i}[{\gamma }^0({\partial }_0-{\mathcal{A}}_0(-t,x))\\ & +{\gamma }^{\mu }\left({\partial }_{\mu }+{\mathcal{A}}_{\mu }(-t,x)\right)]{\gamma }^0\psi (-t,x)\\ & =\int \mathrm{d}t\mathrm{d}x\bar{\psi }(t,x)\mathrm{i}[{\gamma }^0\left({\partial }_0+{\mathcal{A}}_0(t,x)\right)\\ & +{\gamma }^{\mu }({\partial }_{\mu }+{\mathcal{A}}_{\mu }(t,x))]\psi (t,x)\end{array}$$

where we put $t\to -t$ in the final line. Let us see whether this invariance is observed by the effective action. The effective action is given by the regularized product of the eigenvalues of $\mathrm{i}\overline{\nabla }$ . We employ the Pauli-Villars regularization to regulate the product, that is

$${\mathcal{L}}_{\text{r e g}}\equiv \bar{\chi }\mathrm{i}\bar{\nabla }\chi +\mathrm{i}M\bar{\chi }\chi \text{\hspace{1em}(13.106)}$$

is added to the original Lagrangian. The Pauli-Villars regulator $\chi$ is a spinor which obeys bosonic statistics and the limit $M\to \infty$ is understood. The regularized determinant is

$${\mathrm{e}}^{-W[\mathcal{A}]}=\frac{\det \mathrm{i}\bar{\mathcal{V}}}{\det (\mathrm{i}\bar{\mathcal{V}}+\mathrm{i}M)}=\prod _i\frac{{\lambda }_i}{{\lambda }_i+\mathrm{i}M}\text{\hspace{1em}(13.107)}$$

where we noted that $\chi$ is bosonic. Here ${\lambda }_i$ is the $i$ th eigenvalue of $\mathrm{i}\vec{\mathbb{V}}$ ; $\mathrm{i}\vec{\mathbb{V}}{\psi }_i={\lambda }_i{\psi }_i$ . Under the parity operation, eigenvalues change sign,

$$\begin{array}{rl}& \mathrm{i}\left[{\gamma }^0\left({\partial }_0-{\mathcal{A}}_0(-t,x)\right)+{\gamma }^i\left({\partial }_i+{\mathcal{A}}_i(-t,x)\right)\right]\mathrm{i}{\gamma }^0{\psi }_i(-t,x)\\ & =\mathrm{i}{\gamma }^0[{\gamma }^0(-{\partial }_{\tau }-{\mathcal{A}}_0(\tau ,x))-{\gamma }^i({\partial }_i+{\mathcal{A}}_i(\tau ,x))]\mathrm{i}\psi (\tau ,x)\\ & =-{\lambda }_i\mathrm{i}{\gamma }^0{\psi }_i(\tau ,x)\end{array}$$

where $\tau =-t$ . This shows that the effective action $W[\mathcal{A}]$ transforms under the parity operation $\mathrm{P}$ as

$$W[\mathcal{A}]\to W[{\mathcal{A}}^{\mathrm{P}}]=-\ln \prod \frac{-{\lambda }_i}{-{\lambda }_i+\mathrm{i}M}=\overline{W[\mathcal{A}]}\text{\hspace{1em}(13.108)}$$

where the bar denotes complex conjugation. (13.108) shows that the imaginary part of $W$ is identified with the parity-violating part

$$W[\mathcal{A}]-W[{\mathcal{A}}^{\mathrm{P}}]=2\mathrm{Im}W[\mathcal{A}].\text{\hspace{1em}(13.109)}$$

Im $W[\mathcal{A}]$ is given by the $\eta$ -invariant defined in section 12.8. In fact,

$$\begin{array}{rl}\mathrm{Im}W[\mathcal{A}]& =\underset{M\to \infty }{\lim }\mathrm{Im}\left(-\sum _i\ln \frac{{\lambda }_i}{{\lambda }_i+\mathrm{i}M}\right)=\underset{M\to \infty }{\lim }\sum _i{\tan }^{-1}(M/{\lambda }_i)\\ & =\frac{\pi }{2}\left(\sum _{\lambda >0}1-\sum _{\lambda <0}1\right)=\frac{\pi }{2}\eta .\end{array}\text{\hspace{1em}(13.110)}$$

Thus, the Pauli-Villars regulator gives a regularized form for the $\eta$ -invariant. We finally have

$$\mathrm{Im}W[\mathcal{A}]=\frac{\pi }{2}\eta =\frac{\pi }{2}\underset{s\to 0}{\lim }\sum _i^{\prime }\mathrm{sgn}{\lambda }_i|{\lambda }_i{|}^{-2s}\text{\hspace{1em}(13.111)}$$

where the prime indicates the omission of zero modes.

13.6.2 The dimensional ladder: 4-3-2

It is remarkable that the parity anomaly (13.110) is closely related to the chiral anomaly in a $(2l+2)$ -dimensional space (Alvarez-Gaume et al 1985). Following Forte (1987), we look at the dimensional ladder,

$$\begin{array}{c}\text{f o u r - d i m e n s i o n a l A b e l i a n a n o m a l y}\\ \downarrow \\ \text{t h r e e - d i m e n s i o n a l p a r i t y a n o m a l y}\\ \downarrow \\ \text{v o - d i m e n s i o n a l n o n - A b e l i a n a n o m a l y .}\end{array}\text{\hspace{1em}(13.112)}$$

We take $M_4=S^2\times S^2$ as a four-dimensional space. The Abelian anomaly is given by the index

$$\mathrm{ind}\mathrm{i}{\overline{\mathcal{V}}}_4={\mathcal{N}}_{+}-{\mathcal{N}}_{-}={\int }_{S^2\times S^2}{\partial }_{\mu }{j}_5^{\mu }={\int }_{S^2\times S^2}{\mathrm{ch}}_2(\mathbb{F}).\text{\hspace{1em}(13.113)}$$

As before, ${\mathcal{N}}_{+}$ $({\mathcal{N}}_{-})$ is the number of positive (negative) chirality zero modes. Let $Q_3$ be the Chern-Simons form of ${\mathrm{ch}}_2(\mathbb{F})$ ; ${\mathrm{ch}}_2(\mathbb{F})=\mathrm{d}{Q}_3(\mathbb{A},\mathbb{F})$ . Then

$\mathcal{N}\equiv {\mathcal{N}}_{+}-{\mathcal{N}}_{-}$ is given by

$$\begin{array}{rl}\mathcal{N}& ={\int }_{S^2\times S^2}{\mathrm{ch}}_2(\mathbb{F})={\int }_{U_{\mathrm{N}}\times S^2}\mathrm{d}{Q}_3\left({\mathbb{A}}_{\mathrm{N}},{\mathbb{F}}_{\mathrm{N}}\right)+{\int }_{U_{\mathrm{S}}\times S^2}\mathrm{d}{Q}_3\left({\mathbb{A}}_{\mathrm{S}},{\mathbb{F}}_{\mathrm{S}}\right)\\ & ={\int }_{S^1\times S^2}\left[Q_3\left({\mathbb{A}}_{\mathrm{N}},{\mathbb{F}}_{\mathrm{N}}\right)-\mathrm{d}{Q}_3\left({\mathbb{A}}_{\mathrm{S}},{\mathbb{F}}_{\mathrm{S}}\right)\right]\\ & =\frac{1}{24{\pi }^2}{\int }_{S^1\times S^2}\mathrm{tr}{\left(g^{-1}\mathrm{d}g\right)}^3\end{array}\text{\hspace{1em}(13.114)}$$

where $g$ is the gauge transformation connecting ${\mathbb{A}}_{\mathrm{N}}$ and ${\mathbb{A}}_{\mathrm{S}}$ ; ${\mathbb{A}}_{\mathrm{N}}=g^{-1}({\mathbb{A}}_{\mathrm{S}}+\mathrm{d}+{\mathrm{d}}_{\theta })g$ . In the previous section, we have shown that $\mathcal{N}$ also represents the non-Abelian anomaly

$$\mathcal{N}=\frac{1}{2\pi }{\int }_0^{2\pi }\mathrm{d}\theta \frac{\partial w(\mathcal{A},\theta )}{\partial \theta }\text{\hspace{1em}(13.115a)}$$

where $w$ is defined by

$$\det \mathrm{i}\hat{D}({\mathcal{A}}^{g(\theta )})={\mathrm{e}}^{\mathrm{i}w(\mathcal{A},\theta )}\det \mathrm{i}\hat{D}(\mathcal{A}).\text{\hspace{1em}(13.115b)}$$

Here $\mathcal{A}$ is the reference gauge potential and

$${\mathcal{A}}^{g(\theta )}=g^{-1}(x,\theta )(\mathcal{A}+\mathrm{d})g(x,\theta )\mathrm{i}\hat{D}=/\partial +{\mathcal{A}\mathcal{P}}_{+}.$$

Next, we show that $\mathcal{N}$ is also related to the parity anomaly in three-dimensional space. Let $\mathrm{i}{\overline{\nabla }}_3$ be a three-dimensional Dirac operator and define a four-dimensional Dirac operator by

$$\mathrm{i}/D_4[\mathcal{A}]\equiv \mathrm{i}{\sigma }_1\otimes I\frac{\partial }{\partial t}+{\sigma }_2\otimes \mathrm{i}/{\nabla }_3[{\mathcal{A}}_t]\text{\hspace{1em}(13.116)}$$

where ${\mathcal{A}}_t$ is a one-parameter family of gauge potentials interpolating ${\mathcal{A}}_0={\mathcal{A}}_{t=0}$ and ${\mathcal{A}}_1={\mathcal{A}}_{t=1}$ . The Atiyah-Patodi-Singer index theorem (section 12.8) is

$$\mathrm{ind}\mathrm{i}{D}_4=-{\int }_{S^2\times S^1\times I}{\mathrm{ch}}_2(\mathcal{F})+\frac{1}{2}[\eta (t=1)-\eta (t=0)]\text{\hspace{1em}(13.117)}$$

where we have noted that the Dirac genus $\hat{A}$ is trivial on $S^2\times S^1\times I$ . Suppose ${\mathcal{A}}_0$ and ${\mathcal{A}}_1$ are related by a gauge transformation,

$${\mathcal{A}}_1=g^{-1}\left({\mathcal{A}}_0+\mathrm{d}\right)g\text{\hspace{1em}(13.118a)}$$

and consider an interpolating potential

$${\mathcal{A}}_t\equiv t{\mathcal{A}}_1+(1-t){\mathcal{A}}_0.\text{\hspace{1em}(13.118b)}$$

Since the spectrum of $\mathrm{i}{\overline{\mathcal{V}}}_3$ is gauge invariant, in particular $\mathrm{Spec}\mathrm{i}{\overline{\mathcal{V}}}_3({\mathcal{A}}_0)=\mathrm{Spec}\mathrm{i}{\overline{\mathcal{V}}}_3({\mathcal{A}}_1)$ , the $\eta$ -invariant is also gauge invariant. Then $\eta (t=0)=\eta (t=1)$ and the APS index theorem (13.117) yields

$$\begin{array}{rl}\text{s p e c t r a l}& =\mathrm{ind}\mathrm{i}{D}_4({\mathcal{A}}_t)\\ & ={\int }_{S^2\times S^2}{\mathrm{ch}}_2(\mathcal{F})={\int }_{S^1\times S^2}\left[Q_3\left({\mathcal{A}}_1,{\mathcal{F}}_1\right)-Q_3\left({\mathcal{A}}_0,{\mathcal{F}}_0\right)\right]\\ & ={\int }_{S^1\times S^2}{Q}_3\left(g^{-1}dg,0\right)=\mathbb{N}.\end{array}\text{\hspace{1em}(13.119)}$$

Thus, the spectral flow of the three-dimensional theory is given by the index $\mathcal{N}$ . In summary, the map $g:S^2\times S^1\to G$ is understood in three different ways:

(1) $g$ is a transition function at the boundary of two patches of a $G$ bundle over $S^2\times S^2$ . It yields the index $\mathcal{N}$ of the four-dimensional Abelian anomaly.
(2) Suppose ${\mathcal{A}}_0$ and ${\mathcal{A}}_1=g^{-1}({\mathcal{A}}_0+\mathrm{d})g$ are gauge potentials on $S^2\times S^1$ . The gauge transformation function $g$ measures the spectral flow $\mathcal{N}$ between $\mathrm{Spe}\mathrm{i}{\vec{\nabla }}_3({\mathcal{A}}_0)$ and $\mathrm{Spe}\mathrm{i}{\vec{\nabla }}_3({\mathcal{A}}_1)$ .
(3) $g:S^2\times S^1\to G$ induces a map $S^1\to \mathfrak{G}$ , the winding number $\mathcal{N}$ of which is identified with the non-Abelian anomaly in two-dimensional space.

Thus, we have obtained the ‘dimensional ladder’ 4-3-2. The extension to higher dimensions is obvious.

6 Note that there is no gauge anomaly in odd-dimensional spaces.

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