The Heat KernelIndex and Geometry

# A Index and Geometry

In this appendix, we briefly demonstrate how interesting topological invariants can be expressed as the index of a differential operator.

We recall some basics. Fix a manifold $M$ and bundles $E, F \to M$, together with an elliptic differential operator $D\colon \Gamma (E) \to \Gamma (F)$. Then $\ker D$ and $\operatorname{coker}D$ are finite-dimensional, and the index of $D$ to be

$\operatorname{index}D = \dim \ker D - \dim \operatorname{coker}D.$

$D$ has a formal adjoint $D^*$, and $\operatorname{coker}D \cong \ker D^*$. So

$\operatorname{index}D = \dim \ker D - \dim \ker D^*.$

Usually, the operator $D^*D$ is more recognizable. If $\psi \in \ker D^*D$, then

$0 = (\psi , D^*D\psi ) = (D\psi , D \psi ).$

So $\ker D = \ker D^*D$. Then we can write

$\operatorname{index}D = \dim \ker D^*D - \dim \ker DD^*.$

The Hodge decomposition theorem allows us to relate the kernel of differential operators to something more topological.

Theorem A.1 (Hodge decomposition theorem)

Let $M$ be a Riemannian manifold. Recall that the Laplacian on $p$-forms is defined by

$\Delta = \mathrm{d}\mathrm{d}^* + \mathrm{d}^*\mathrm{d}= (\mathrm{d}+ \mathrm{d}^*)^2.$

If $\psi \in \Omega ^p(M)$ is such that $\Delta \psi = 0$, then, as above, $\mathrm{d}\psi = \mathrm{d}^* \psi = 0$. So $\psi$ is in particular a closed form. This defines a map

$\ker \Delta \to H^*(M).$

The Hodge decomposition theorem states that this map is an isomorphism.

Note that $\mathrm{d}+ \mathrm{d}^*: \Omega \to \Omega$ is self-adjoint, so its index is just zero. However, by varying its domain and codomain, we can get interesting indices.

Example A.2

Write

$\Omega _{\mathrm{even}} = \bigoplus \Omega ^{2k},\quad \Omega _{\mathrm{odd}} = \bigoplus \Omega ^{2k + 1}.$

Then we have a map

$\mathrm{d}+ \mathrm{d}^*: \Omega _{\mathrm{even}} \to \Omega _{\mathrm{odd}}.$

The index is then

$\dim \ker \Delta |_{\Omega _{\mathrm{even}}} - \dim \ker \Delta |_{\Omega _{\mathrm{odd}}} = \chi (M),$

the Euler characteristic.

Example A.3

We can play the same game with the signature. Suppose $\dim M = 4k$. Recall that the Hodge star operator is an endomorphism $\Omega ^* \to \Omega ^*$ that squares to $1$. Write $\Omega _{\pm }$ for the $\pm 1$ eigenspaces. One can show that $\mathrm{d}+ \mathrm{d}^*$ anti-commutes with the Hodge star, so induces a map

$\mathrm{d}+ \mathrm{d}^*: \Omega _+ \to \Omega _-.$

We claim the index of this is exactly the signature of $M$.

We focus on the kernel of this map; the cokernel is similar. The kernel is the subspace of $H^*(M)$ that is invariant under the Hodge star operator. This consists of the $+1$ eigenspace in $H^{2k}(M)$ plus the subspace spanned by $\psi + *\psi$ for $\psi \in H^{2k - \varepsilon }(M)$ with $0 \leq \varepsilon < 2k$.

Similarly, the kernel of $\mathrm{d}+ \mathrm{d}^*: \Omega _- \to \Omega _+$ consists of the $-1$ eigenspace in $H^{2k}(M)$ plus the subspace spanned by $\psi - *\psi$ for $\psi \in H^{2k - \varepsilon }(M)$ with $0 \leq \varepsilon < 2k$.

When we subtract the two, we are left with the difference between the $\pm 1$ eigenspaces of $H^{2k}(M)$, i.e. the signature.