12.8 The Atiyah-Patodi-Singer index theorem

So far we have been concerned with index theorems defined on a compact manifold without a boundary. In practical situations in physics, we often need to find an index of an operator defined over a base space $M$ with a boundary. The extensions of the AS index theorem to these cases are discussed here. Our argument is restricted to the spin bundle over $M$ since this is the only situation we shall be concerned with in chapter 13.

12.8.1 $\eta$ -invariant and spectral flow

Let $\mathrm{i}\overline{\mathcal{V}}$ be a Hermitian Dirac operator defined on an odd-dimensional manifold $M$ , $\dim M=2l+1$ . Since $\mathrm{i}\overline{\mathcal{V}}$ is Hermitian, the eigenvalues ${\lambda }_k$ are real. We define the $\eta$ -invariant of $\mathrm{i}\overline{\mathcal{V}}$ by the spectral asymmetry of $\mathrm{i}\overline{\mathcal{V}}$ ,

$$\eta \equiv \sum _{{\lambda }_k>0}1-\sum _{{\lambda }_k<0}1.\text{\hspace{1em}(12.118)}$$

This is not well defined and requires a proper regularization. For example, we may define $\eta$ by ${\lim }_{s\to 0}\eta (s)$ where

$$\eta (s)\equiv \sum _k^{\prime }\mathrm{sgn}\left({\lambda }_k\right){\left|{\lambda }_k\right|}^{-2s}\text{R e}s>0.\text{\hspace{1em}(12.119)}$$

It can be shown that, under proper boundary conditions, $\eta (s)$ has no pole at $s=0$ .

Exercise 12.7

Use the Mellin transformation

$$\frac{1}{2}\Gamma \left(\frac{s+1}{2}\right)a^{-(s+1)/2}={\int }_0^{\infty }dx{x}^s{e}^{-a{x}^2}a>0$$

to verify that

$$\eta (s)=\frac{2}{\Gamma \left(\frac{1}{2}(s+1)\right)}{\int }_0^{\infty }dx{x}^s\mathrm{tr}i\bar{V}{e}^{-x^2(i\bar{V}{)}^2}.\text{\hspace{1em}(12.120)}$$

Suppose a Dirac field is interacting with an external gauge potential ${\mathcal{A}}_t$ , $t\in [0,1]$ . The Dirac operator $\mathrm{i}\tilde{\nabla }({\mathcal{A}}_t)$ has a $t$ -dependent eigenvalue problem. If an eigenvalue of $\mathrm{i}\tilde{\nabla }({\mathcal{A}}_t)$ crosses zero, the $\eta$ -invariant jumps by $\pm 2$ . This jump

denotes the spectral flow from $\lambda ⩾0$ modes to $\lambda ⩽0$ modes; if $\eta$ jumps by $+2(-2)$ , there is a flow of a state from $\lambda <0$ to $\lambda >0$ ( $\lambda >0$ to $\lambda <0$ ), see figure 12.1. In addition to the discontinuous change associated with the spectral flow, i $\mathbb{W}$ also has a continuous variation ${\eta }_c$ . We have

(a)

(b)

Figure 12.1. Whenever an eigenvalue $\lambda$ crosses zero $(a)$ , the $\eta$ -invariant jumps by $\pm 2(b)$ . The sign depends on the way in which $\lambda$ crosses zero.

$$\eta (t=1)-\eta (t=0)={\int }_0^1\mathrm{d}t\frac{\mathrm{d}{\eta }_c}{\mathrm{d}t}+2\times (\text{s p e c t r a l f l o w}).\text{\hspace{1em}(12.121)}$$

12.8.2 The Atiyah-Patodi-Singer (APS) index theorem

Let us consider a $(2l+2)$ -dimensional Dirac operator

$$\mathrm{i}{\hat{D}}_{2l+2}=\mathrm{i}{\sigma }_1\frac{\partial }{\partial t}+{\sigma }_2\otimes \mathrm{i}\bar{\mathbb{V}}({\mathcal{A}}_t)=\left(\begin{array}{cc}0& D\\ D^{†}& 0\end{array}\right)\text{\hspace{1em}(12.122a)}$$

where

$$D=\mathrm{i}{\partial }_t-\nabla ({\mathcal{A}}_t)D^{†}=\mathrm{i}{\partial }_t+\nabla ({\mathcal{A}}_t).\text{\hspace{1em}(12.122b)}$$

[Remark: The positions of $D$ and $D^{†}$ are reversed since

$${\gamma }^{2l+3}=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$$

for our choice of $\gamma$ -matrices; cf (12.79).]

Theorem 12.3 · Atiyah-Patodi-Singer theorem

Let $M$ be an odd-dimensional manifold and $\mathrm{i}\vec{\mathcal{V}}({\mathcal{A}}_t)$ a Dirac operator on $M$ interacting with an external gauge field ${\mathcal{A}}_t$ . Then,

$$\begin{array}{rl}\mathrm{ind}D& =\dim \ker D-\dim \ker D^{†}\\ & ={\int }_{M\times I}\hat{A}(\mathcal{R})\mathrm{ch}(\mathcal{F}){|}_{\text{v o l}}-\frac{1}{2}[\eta (\mathrm{i}\nabla ({\mathcal{A}}_1))-\eta (\mathrm{i}\nabla ({\mathcal{A}}_0))].\end{array}\text{\hspace{1em}(12.123)}$$

The general argument shows that the continuous part ${\eta }_c$ of the $\eta$ -invariant satisfies

$${\int }_0^1\mathrm{d}t\frac{\mathrm{d}{\eta }_c}{\mathrm{d}t}=2{\int }_{M\times I}\hat{A}(\mathcal{R})\mathrm{ch}(\mathcal{F}){|}_{\text{v o l}}.\text{\hspace{1em}(12.124)}$$

Then the RHS of (12.123) is simply the spectral flow

$$-\frac{1}{2}[\eta (t=1)-\eta (t=0)]+\frac{1}{2}{\int }_0^1\mathrm{d}t\frac{\mathrm{d}{\eta }_c}{\mathrm{d}t}=-\text{s p e c t r a l f l o w}.$$

Thus, we find another expression for the APS index theorem,

$$\mathrm{id}{\hat{D}}_{2l+2}=-\text{s p e c t r a l f o w}.\text{\hspace{1em}(12.125)}$$

The proof of the APS index theorem in its most general form is found in Atiyah et al (1975a, b, 1976). The physicists’ proof is found in Alvarez-Gaume et al (1985). We use the APS index theorem to study the odd-dimensional parity anomaly in section 13.6.

Example 12.8

To see why the spectral flow appears in the index theorem, we consider an example taken from Atiyah (1985). Let $M=S^1$ and $\theta$ be its coordinate. Consider a Hermitian operator

$$\mathrm{i}{\nabla }_t\equiv \mathrm{i}\left(\frac{\partial }{\partial \theta }-\mathrm{i}t\right)=\mathrm{i}{\partial }_{\theta }+tt\in \mathbb{R}.\text{\hspace{1em}(12.126)}$$

The term $-\mathrm{i}t$ is thought of as a U(1) gauge potential. The eigenvector and the eigenvalue of $\mathrm{i}{\nabla }_t$ are

$${\psi }_{n,t}(\theta )=\frac{1}{\sqrt{2\pi }}{\mathrm{e}}^{-\mathrm{i}n\theta }(n\in \mathbb{Z}){\lambda }_n(t)=n+t.$$

Since $\mathrm{Spec}\mathrm{i}{\nabla }_t=\mathrm{Spec}\mathrm{i}{\nabla }_{t+1}$ , the family of operators $\mathrm{i}{\nabla }_t$ is periodic in $t$ with the period 1, see figure 12.2. This periodicity manifests itself in the gauge equivalence of $\mathrm{i}{\nabla }_t$ and $\mathrm{i}{\nabla }_{t+1}$ :

Figure 12.2. Time evolution of the eigenvalues of $\mathrm{i}{\nabla }_t$ . Spec $\mathrm{i}{\nabla }_t$ has period 1. The $i$ th eigenvalue crosses zero at $t=0$ and, hence, there is a unit spectral flow.

$$\mathrm{i}{\nabla }_{t+1}={\mathrm{e}}^{\mathrm{i}\theta }\mathrm{i}{\nabla }_t{\mathrm{e}}^{-\mathrm{i}\theta }.$$

There is precisely unit spectral flow from $\lambda <0$ to $\lambda >0$ at $t=0$ while $t$ changes from $-\epsilon$ to $1-\epsilon$ , $\epsilon$ being a small positive number. From $\mathrm{i}{\nabla }_t$ , we construct a two-dimensional Dirac operator

$$\mathrm{i}/D_2\equiv \mathrm{i}{\sigma }_1\otimes \frac{\partial }{\partial t}+{\sigma }_2\otimes \mathrm{i}{\nabla }_t=\left(\begin{array}{cc}0& D\\ D^{†}& 0\end{array}\right)\text{\hspace{1em}(12.127a)}$$

where

$$D\equiv \mathrm{i}{\partial }_t+{\partial }_{\theta }-\mathrm{i}t{D}^{†}\equiv \mathrm{i}{\partial }_t-{\partial }_{\theta }+\mathrm{i}t.\text{\hspace{1em}(12.127b)}$$

These operators act on functions which satisfy the boundary conditions

$$\varphi (\theta +2\pi ,t)=\varphi (\theta ,t)\varphi (\theta ,t+1)={\mathrm{e}}^{\mathrm{i}\theta }\varphi (\theta ,t).\text{\hspace{1em}(12.128)}$$

Let ${\varphi }_0\in \ker D^{†}$ . We have a Fourier expansion

$${\varphi }_0(\theta ,t)=\sum a_n(t){\mathrm{e}}^{-\mathrm{i}n\theta }.$$

It follows from $D^{†}{\varphi }_0=0$ that

$$a_n^{\prime }(t)+(n+t)a_n(t)=0$$

which is easily solved to yield

$$a_n(t)=c_n\exp \left(-\frac{(n+t{)}^2}{2}\right).$$

The boundary conditions (12.128) require that

$$\sum _n{c}_n\exp \left(-\frac{(n+t+1{)}^2}{2}\right)e^{-in\theta }=\sum _n{c}_n\exp \left(-\frac{(n+t{)}^2}{2}\right)e^{-i(n-1)\theta }$$

from which we find that $c_n$ is independent of $n$ . Thus, $\ker D^{†}$ is one dimensional and is spanned by the theta function,

$${\varphi }_0(\theta ,t)=\sum \exp \left(-\frac{(n+t{)}^2}{2}-\mathrm{i}n\theta \right).\text{\hspace{1em}(12.129)}$$

Suppose ${\tilde{\varphi }}_0(\theta ,t)\in \ker D$ . If we put ${\tilde{\varphi }}_0(\theta ,t)=\sum b_n(t){\mathrm{e}}^{-\mathrm{i}n\theta }$ , $b_n(t)$ satisfies

$$b_n^{\prime }(t)-(n+t)b_n(t)=0.$$

The solution of this equation is

$$b_n(t)=b_n(0)\exp \frac{(n+t{)}^2}{2}$$

and, hence, ${\tilde{\varphi }}_0$ cannot be normalized. This shows that

$$\mathrm{ind}D=\dim \ker D-\dim \ker D^{†}=-1$$

which agrees with $-$ (spectral flow).

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