The Heat Kernel — Index 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 MM and bundles E,FME, F \to M, together with an elliptic differential operator D:Γ(E)Γ(F)D\colon \Gamma (E) \to \Gamma (F). Then kerD\ker D and cokerD\operatorname{coker}D are finite-dimensional, and the index of DD to be

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

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

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

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

0=(ψ,DDψ)=(Dψ,Dψ). 0 = (\psi , D^*D\psi ) = (D\psi , D \psi ).

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

indexD=dimkerDDdimkerDD. \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 MM be a Riemannian manifold. Recall that the Laplacian on pp-forms is defined by

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

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

kerΔH(M). \ker \Delta \to H^*(M).

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

Note that d+d:ΩΩ\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

Ωeven=Ω2k,Ωodd=Ω2k+1. \Omega _{\mathrm{even}} = \bigoplus \Omega ^{2k},\quad \Omega _{\mathrm{odd}} = \bigoplus \Omega ^{2k + 1}.

Then we have a map

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

The index is then

dimkerΔΩevendimkerΔΩodd=χ(M), \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 dimM=4k\dim M = 4k. Recall that the Hodge star operator is an endomorphism ΩΩ\Omega ^* \to \Omega ^* that squares to 11. Write Ω±\Omega _{\pm } for the ±1\pm 1 eigenspaces. One can show that d+d\mathrm{d}+ \mathrm{d}^* anti-commutes with the Hodge star, so induces a map

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

We claim the index of this is exactly the signature of MM.

We focus on the kernel of this map; the cokernel is similar. The kernel is the subspace of H(M)H^*(M) that is invariant under the Hodge star operator. This consists of the +1+1 eigenspace in H2k(M)H^{2k}(M) plus the subspace spanned by ψ+ψ\psi + *\psi for ψH2kε(M)\psi \in H^{2k - \varepsilon }(M) with 0ε<2k0 \leq \varepsilon < 2k.

Similarly, the kernel of d+d:ΩΩ+\mathrm{d}+ \mathrm{d}^*: \Omega _- \to \Omega _+ consists of the 1-1 eigenspace in H2k(M)H^{2k}(M) plus the subspace spanned by ψψ\psi - *\psi for ψH2kε(M)\psi \in H^{2k - \varepsilon }(M) with 0ε<2k0 \leq \varepsilon < 2k.

When we subtract the two, we are left with the difference between the ±1\pm 1 eigenspaces of H2k(M)H^{2k}(M), i.e. the signature.