12.1 Elliptic operators and Fredholm operators

In the following, we will be concerned with differential operators defined on vector bundles over a compact manifold $M$ without a boundary. We exclusively deal with a nice class of differential operators called the Fredholm operators.

12.1.1 Elliptic operators

Let $E$ and $F$ be complex vector bundles over a manifold $M$ . A differential operator $D$ is a linear map

$$D:\Gamma (M,E)\to \Gamma (M,F).\text{\hspace{1em}(12.1)}$$

Take a chart $U$ of $M$ over which $E$ and $F$ are trivial. We denote the local coordinates of $U$ as $x^{\mu }$ . We introduce the following multi-index notation,

$$M\equiv ({\mu }_1,{\mu }_2,\dots ,{\mu }_m){\mu }_j\in \mathbb{Z},{\mu }_j\ge 0$$

$$\left|M\right|\equiv {\mu }_1+{\mu }_2+\cdots +{\mu }_m$$

$$D_M=\frac{{\partial }^{|M|}}{\partial x^M}\equiv \frac{{\partial }^{{\mu }_1+\cdots +{\mu }_m}}{\partial (x^1{)}^{{\mu }_1}\dots \partial (x^m{)}^{{\mu }_m}}.$$

If $\dim E=k$ and $\dim F=k^{\prime }$ , the most general form of $D$ is

$$[Ds(x){]}^{\alpha }=\sum _{\begin{array}{c}|M|\le N\\ 1\le a\le k\end{array}}{A}_a^{M\alpha }(x)D_M{s}^a(x)1\le \alpha \le k^{\prime }\text{\hspace{1em}(12.2)}$$

where $s(x)$ is a section of $E$ . Note that $x$ denotes a point whose coordinates are $x^{\mu }$ . This slight abuse simplifies the notation. $A^M\equiv (A^M{)}_a^{\alpha }$ is a $k\times k^{\prime }$ matrix which may depend on the position $x$ . The positive integer $N$ in (12.2) is called the order of $D$ . We are interested in the case in which $N=1$ (the Dirac operator) and $N=2$ (the Laplacian). For example, if $F$ is a spin bundle over $M$ , the Dirac operator $D\equiv \mathrm{i}{\gamma }^{\mu }{\partial }_{\mu }+m:\Gamma (M,E)\to \Gamma (M,E)$ acts on a section $\psi (x)$ of $E$ as

$$[D\psi (x){]}^{\alpha }=\mathrm{i}({\gamma }^{\mu }{)}_{\beta }^{\alpha }{\partial }_{\mu }{\psi }^{\beta }(x)+m{\psi }^{\alpha }(x).$$

The symbol of $D$ is a $k\times k^{\prime }$ matrix

$$\sigma (D,\xi )\equiv \sum _{|M|=N}{A}_a^{M\alpha }(x){\xi }_M\text{\hspace{1em}(12.3)}$$

where $\xi$ is a real $m$ -tuple $\xi =({\xi }_1,\dots ,{\xi }_m)$ . The symbol is also defined independently of the coordinates as follows. Let $E\stackrel{\pi }{\to }M$ be a vector bundle and let $p\in M$ , $\xi \in T_p^{\ast }M$ and $s\in {\pi }_E^{-1}(p)$ . Take a section $\tilde{s}\in \Gamma (M,E)$ such that $\tilde{s}(p)=s$ and a function $f\in \mathcal{F}(M)$ such that $f(p)=0$ and $\mathrm{d}f(p)=\xi \in T_p^{\ast }M$ . Then the symbol may be defined by

$$\sigma (D,\xi )s=\frac{1}{N!}D\left(f^N\tilde{s}\right){|}_p.\text{\hspace{1em}(12.4)}$$

The factor $f^N$ automatically picks up the $N$ th-order term due to the condition $f(p)=0$ . Equation (12.4) yields the same symbol as (12.3).

If the matrix $\sigma (D,\xi )$ is invertible for each $x\in M$ and each $\xi \in {\mathbb{R}}^m-\{0\}$ , the operator $D$ is said to be elliptic. Clearly this definition makes sense only when $k=k^{\prime }$ . It should be noted that the symbol for a composite operator $D=D_1\circ D_2$ is a composite of the symbols, namely $\sigma (D,\xi )=\sigma (D_1,\xi )\sigma (D_2,\xi )$ . This shows that composites of elliptic operators are also elliptic. In general, powers and roots of elliptic operators are elliptic.

Example 12.1

Let $x^{\mu }$ be the natural coordinates in ${\mathbb{R}}^m$ . If $E$ and $F$ are real line bundles over ${\mathbb{R}}^m$ , the Laplacian $\Delta :\Gamma ({\mathbb{R}}^m,E)\to \Gamma ({\mathbb{R}}^m,F)$ is defined by

$$\Delta \equiv \frac{{\partial }^2}{\partial {\left(x^1\right)}^2}+\cdots +\frac{{\partial }^2}{\partial {\left(x^m\right)}^2}.\text{\hspace{1em}(12.5)}$$

According to (12.3), the symbol is

$$\sigma (\Delta ,\xi )=\sum _{\mu }({\xi }_{\mu }{)}^2.$$

This is in agreement with the result obtained from (12.4),

$$\begin{array}{rl}\sigma (\Delta ,\xi )s& =\frac{1}{2}\Delta (f^2\tilde{s}){|}_p=\frac{1}{2}\sum \frac{{\partial }^2}{\partial (x^{\mu }{)}^2}(f^2\tilde{s}){|}_p\\ & =\frac{1}{2}\left(f^2\Delta \tilde{s}+2f\Delta f\tilde{s}+2f\sum \frac{\partial f}{\partial x^{\mu }}\frac{\partial \tilde{s}}{\partial x^{\mu }}+2\sum \frac{\partial f}{\partial x^{\mu }}\frac{\partial f}{\partial x^{\mu }}\tilde{s}\right){|}_p\\ & =\sum ({\xi }_{\mu }{)}^{2}s.\end{array}$$

This symbol is clearly invertible for $\xi \ne 0$ , and hence $\Delta$ is elliptic.

However, the d’Alembertian

$$\square \equiv \frac{{\partial }^2}{\partial {\left(x^1\right)}^2}+\cdots +\frac{{\partial }^2}{\partial {\left(x^{m-1}\right)}^2}-\frac{{\partial }^2}{\partial {\left(x^m\right)}^2}\text{\hspace{1em}(12.6)}$$

is not elliptic since the symbol

$$\sigma (\square ,\xi )=({\xi }^1{)}^2+\cdots +({\xi }^{m-1}{)}^2-({\xi }^m{)}^2$$

vanishes everywhere on the light cone,

$$({\xi }^m{)}^2=({\xi }^1{)}^2+\cdots +({\xi }^{m-1}{)}^2.$$

Exercise 12.1

Let $M={\mathbb{R}}^2$ and consider a differential operator $D$ of order two. The symbol of $D$ is of the form

$$\sigma (D,\xi )=A_{11}{\xi }^1{\xi }^1+2A_{12}{\xi }^1{\xi }^2+A_{22}{\xi }^2{\xi }^2.$$

Show that $D$ is elliptic if and only if $\sigma (D,\xi )=1$ is an ellipse in $\xi$ -space.

12.1.2 Fredholm operators

Let $D:\Gamma (M,E)\to \Gamma (M,F)$ be an elliptic operator. The kernel of $D$ is the set of null eigenvectors

$$\ker D\equiv \{s\in \Gamma (M,E)|Ds=0\}.\text{\hspace{1em}(12.7)}$$

Suppose $E$ and $F$ are endowed with fibre metrics, which will be denoted $⟨,{⟩}_E$ and $⟨,{⟩}_F$ , respectively. The adjoint $D^{†}:\Gamma (M,F)\to \Gamma (M,E)$ of $D$ is defined by

$$⟨s^{\prime },Ds{⟩}_F\equiv ⟨D^{†}{s}^{\prime },s{⟩}_E\text{\hspace{1em}(12.8)}$$

where $s\in \Gamma (M,E)$ and $s^{\prime }\in \Gamma (M,F)$ . We define the cokernel of $D$ by

$$\mathrm{coker}D\equiv \Gamma (M,F)/\mathrm{im}D.\text{\hspace{1em}(12.9)}$$

Among elliptic operators we are interested in a class of operators whose kernels and cokernels are finite dimensional. An elliptic operator $D$ which satisfies this condition is called a Fredholm operator. The analytical index

$$\mathrm{ind}D\equiv \dim \ker D-\dim \mathrm{coker}D\text{\hspace{1em}(12.10)}$$

is well defined for a Fredholm operator. Henceforth, we will be concerned only with Fredholm operators. It is known from the general theory of operators that elliptic operators on a compact manifold are Fredholm operators. Theorem 12.1 shows that $\mathrm{ind}D$ is also expressed as

$$\mathrm{ind}D=\dim \ker D-\dim \ker D^{†}.\text{\hspace{1em}(12.11)}$$

Theorem 12.1

Let $D:\Gamma (M,E)\to \Gamma (M,F)$ be a Fredholm operator. Then

$$\mathrm{coker}D\cong \ker D^{†}\equiv \{s\in \Gamma (M,F)|D^{†}s=0\}.\text{\hspace{1em}(12.12)}$$

Proof

Let $[s]\in \mathrm{coker}D$ be given by

$$[s]=\{s^{\prime }\in \Gamma (M,F)|s^{\prime }=s+Du,u\in \Gamma (M,E)\}.$$

We show that there is a surjection $\ker D^{†}\to \mathrm{coker}D$ , namely any $[s]\in \mathrm{coker}D$ has a representative $s_0\in \ker D^{†}$ . Define $s_0$ by

$$s_0\equiv s-D\frac{1}{D^{†}D}{D}^{†}s.\text{\hspace{1em}(12.13)}$$

We find $s_0\in \ker D^{†}$ since $D^{†}{s}_0=D^{†}s-D^{†}D(D^{†}D{)}^{-1}{D}^{†}s=D^{†}s-D^{†}s=0$ . Next, let $s_0,s_0^{\prime }\in \ker D^{†}$ and $s_0\ne s_0^{\prime }$ . We show that $[s_0]\ne [s_0^{\prime }]$ in $\Gamma (M,F)/\mathrm{im}D$ . If $[s_0]=[s_0^{\prime }]$ , there is an element $u\in \Gamma (M,E)$ such that $s_0-s_0^{\prime }=Du$ . Then $0=⟨u,D^{†}(s_0-s_0^{\prime }){⟩}_E=⟨u,D^{†}Du{⟩}_E=⟨Du,Du{⟩}_F\ge 0$ , hence $Du=0$ , which contradicts our assumption $s_0\ne s_0^{\prime }$ . Thus, the map $s_0\mapsto [s]$ is a bijection and we have established that $\mathrm{coker}D\cong \ker D^{†}$ .

12.1.3 Elliptic complexes

Consider a sequence of Fredholm operators,

$$\cdots \to \Gamma (M,E_{i-1})\stackrel{D_{i-1}}{\to }\Gamma (M,E_i)\stackrel{D_i}{\to }\Gamma (M,E_{i+1})\stackrel{D_{i+1}}{\to }\dots \text{\hspace{1em}(12.14)}$$

where $\{E_i\}$ is a sequence of vector bundles over a compact manifold $M$ . The sequence $(E_i,D_i)$ is called an elliptic complex if $D_i$ is nilpotent (that is $D_i\circ D_{i-1}=0$ ) for any $i$ . The reader may refer to $\Gamma (M,E_i)={\Omega }_i(M)$ and $D_i=\mathrm{d}$ (exterior derivative) for example. The adjoint of $D_i:\Gamma (M,E_i)\to \Gamma (M,E_{i+1})$ is denoted by

$$D_i^{†}:\Gamma (M,E_{i+1})\to \Gamma (M,E_i).$$

The Laplacian ${\Delta }_i:\Gamma (M,E_i)\to \Gamma (M,E_i)$ is

$${\Delta }_i\equiv D_{i-1}{D}_{i-1}^{†}+D_i^{†}{D}_i.\text{\hspace{1em}(12.15)}$$

The Hodge decomposition also applies to the present case,

$$s_i=D_{i-1}{s}_{i-1}+D_i^{†}{s}_{i+1}+h_i\text{\hspace{1em}(12.16)}$$

where $s_{i\pm 1}\in \Gamma (M,E_{i\pm 1})$ and $h_i$ is in the kernel of ${\Delta }_i$ , ${\Delta }_i{h}_i=0$ .

Analogously to the de Rham cohomology groups, we define

$$H^i(E,D)\equiv \ker D_i/\mathrm{im}{D}_{i-1}.\text{\hspace{1em}(12.17)}$$

As in the case of the de Rham theory, it can be shown that $H^i(E,D)$ is isomorphic to the kernel of ${\Delta }_i$ . Accordingly, we have

$$\dim H^i(E,D)=\dim {\mathrm{Harm}}^i(E,D)\text{\hspace{1em}(12.18)}$$

where ${\mathrm{Harm}}^i(E,D)$ is a vector space spanned by $\{h_i\}$ . The index of this elliptic complex is defined by

$$\mathrm{ind}D\equiv \sum _{i=0}^m(-1{)}^i\dim H^i(E,D)=\sum _{i=0}^m(-1{)}^i\dim \ker {\Delta }_i.\text{\hspace{1em}(12.19)}$$

The index thus defined generalizes the Euler characteristic, see example 12.2.

How is this related to (12.10)? Consider the complex $\Gamma (M,E)\stackrel{D}{\to }\Gamma (M,F)$ . We may formally add zero on both sides,

$$0\stackrel{i}{\to }\Gamma (M,E)\stackrel{D}{\to }\Gamma (M,F)\stackrel{\phi }{\to }0\text{\hspace{1em}(12.20)}$$

where $i$ is the inclusion. The index according to (12.19) is

dim ker $D-\{\dim \Gamma (M,F)-\dim \mathrm{im}D\}=\dim \ker D-\dim \mathrm{coker}D$

where we have noted that $\dim \mathrm{im}i=0$ , $\ker \phi =\Gamma (M,F)$ and $\mathrm{coker}D=\ker \phi /\mathrm{im}D$ . Thus, (12.19) yields the same index as (12.10).

It is often convenient to work with a two-term elliptic complex which has the same index as the original elliptic complex $(E,D)$ . This rolling up is carried out by defining

$$E_{+}\equiv \underset{r}{⨁}{E}_{2r},E_{-}\equiv \underset{r}{⨁}{E}_{2r+1}\text{\hspace{1em}(12.21)}$$

which are called the even bundle and the odd bundle, respectively. Correspondingly we consider the operators

$$A\equiv \underset{r}{⨁}\left(D_{2r}+D_{2r-1}^{†}\right),A^{†}\equiv \underset{r}{⨁}\left(D_{2r+1}+D_{2r}^{†}\right).\text{\hspace{1em}(12.22)}$$

We readily verify that $A:\Gamma (M,E_{+})\to \Gamma (M,E_{-})$ and $A^{†}:\Gamma (M,E_{-})\to \Gamma (M,E_{+})$ . From $A$ and $A^{†}$ , we construct the two Laplacians

$$\begin{array}{rl}{\Delta }_{+}\equiv A^{†}A& =\underset{r,s}{⨁}(D_{2r+1}+D_{2r}^{†})(D_{2s}+D_{2s-1}^{†})\\ & ={\oplus }_r\left(D_{2r-1}{D}_{2r-1}^{†}+D_{2r}^{†}{D}_{2r}\right)={\oplus }_r{\Delta }_{2r}\end{array}\text{\hspace{1em}(12.23a)}$$

$${\Delta }_{-}\equiv A{A}^{†}={\oplus }_r{\Delta }_{2r+1}.\text{\hspace{1em}(12.23b)}$$

Then we have

$$\begin{array}{rl}\mathrm{ind}\left(E_{\pm },A\right)& =\dim \ker {\Delta }_{+}-\dim \ker {\Delta }_{-}\\ & =\sum (-1{)}^r\dim \ker {\Delta }_r=\mathrm{ind}(E,D).\end{array}\text{\hspace{1em}(12.24)}$$

Example 12.2

Let us consider the de Rham complex $\Omega (M)$ over a compact manifold $M$ without a boundary,

$$0\stackrel{i}{\to }{\Omega }^0(M)\stackrel{\mathrm{d}}{\to }{\Omega }^1(M)\stackrel{\mathrm{d}}{\to }\cdots \stackrel{\mathrm{d}}{\to }{\Omega }^m(M)\stackrel{\mathrm{d}}{\to }0\text{\hspace{1em}(12.25)}$$

where $m=\dim M$ and $\mathrm{d}$ stands for ${\mathrm{d}}_r:{\Omega }^r(M)\to {\Omega }^{r+1}(M)$ . $H^r(E,D)$ defined by (12.25) agrees with the de Rham cohomology group $H_r(M,\mathbb{R})$ . The index is identified with the Euler characteristic,

$$\mathrm{ind}\left({\Omega }^{\ast }(M),\mathrm{d}\right)=\sum _{r=0}^m(-1{)}^r\dim H^r(M;\mathbb{R})=\chi (M).\text{\hspace{1em}(12.26)}$$

We found in chapter 7 that $b^r\equiv \dim H^r(M,\mathbb{R})$ agrees with the number of linearly independent harmonic $r$ -forms: $\dim H^r(M,\mathbb{R})=\dim {\mathrm{Harm}}^r(M)=\dim \ker {\Delta }_r$ , where ${\Delta }_r$ is the Laplacian

$${\Delta }_r={\left(\mathrm{d}+{\mathrm{d}}^{†}\right)}^2={\mathrm{d}}_{r-1}{\mathrm{d}}_{r-1}^{†}+{\mathrm{d}}_r^{†}{\mathrm{d}}_r\text{\hspace{1em}(12.27)}$$

${\mathrm{d}}_r^{†}:{\Omega }^{r+1}(M)\to {\Omega }^r(M)$ being the adjoint of ${\mathrm{d}}_r$ . Now we find that

$$\chi (M)=\sum _{r=0}^m(-1{)}^r\dim \ker {\Delta }_r.\text{\hspace{1em}(12.28)}$$

This relation is very interesting since the LHS is a purely topological quantity which can be computed by triangulating $M$ , for example, while the RHS is given by the solution of an analytic equation ${\Delta }_{r}u=0$ . We noted in example 11.6 that $\chi (M)$ is given by integrating the Euler class over $M$ : $\chi (M)={\int }_{M}e(TM)$ . Now (12.28) reads

$$\sum _{r=1}^m(-1{)}^r\dim \ker {\Delta }_r={\int }_{M}e(TM).\text{\hspace{1em}(12.29)}$$

This is a typical form of the index theorem. The RHS is an analytic index while the LHS is a topological index given by the integral of certain characteristic classes. In section 12.3, we derive (12.29) from the Atiyah-Singer index theorem.

The two-term complex is given by

$${\Omega }^{+}(M)\equiv \underset{r}{⨁}{\Omega }^{2r}(M){\Omega }^{-}(M)\equiv \underset{r}{⨁}{\Omega }^{2r+1}(M).\text{\hspace{1em}(12.30)}$$

The corresponding operators are

$$A\equiv \underset{r}{⨁}\left({\mathrm{d}}_{2r}+{\mathrm{d}}_{2r-1}^{†}\right)A^{†}\equiv \underset{r}{⨁}\left({\mathrm{d}}_{2r-1}+{\mathrm{d}}_{2r}^{†}\right).\text{\hspace{1em}(12.31)}$$

It is left as an exercise to the reader to show that

$$\mathrm{ind}\left({\Omega }^{\pm }(M),A\right)=\dim \ker A_{+}-\dim \ker A_{-}=\chi (M).\text{\hspace{1em}(12.32)}$$

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