The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. But the K-theoretic proof has the advantage that it proves Atiyah-Singer not only for Dirac-type operators, but much more generally for elliptic, symmetric pseudodifferential operators (PDOs).

I was wondering whether one could do the heat kernel proof also for pseudodifferential operators?

I know that we can conclude Atiyah-Singer for PDOs from the theorem for Dirac operators by using Poincare duality (i.e., by using that on compact spin$^c$-manifolds every K-homology class can be represented by the class of a twisted Dirac operator).

But I'm asking if the proof via the heat kernel directly generalizes to PDOs? One problem that I can see is that it is often used that $e^{itD}$ has finite propagation speed, and I think that this does not hold anymore if D is merely pseudodifferential (right?). But maybe it is possible to work around that? Or what is the reason why I only encounter heat kernel proofs for Dirac type operators but not for PDOs?


The answer is no. Even for the differentil elliptic operator, the heat kernel method can not give the result.

In the heat method proof, we use McKean-Singer formula, $$\mathrm{Ind} D= \mathrm{Tr} \left[e^{-tD^*D}-e^{-tDD^*}\right]=\int_M \mathrm{Tr} [p_t(x,x)-q_t(x,x)]dx,$$ where $p_t(x,y), q_t(x,y)$ are the heat kernel. This is always true, even for PDO.

McKean-Singer conjectured, and proved by a lot of people (Gilkey, Getzler, Bismut...) using different methods, for Dirac operator, when $t\to0$, $$\mathrm{Tr} [p_t(x,x)-q_t(x,x)]\to a(x). \quad (1)$$ Then we can deduce the Atiyah Singer index theorem $$\mathrm{Ind} D=\int_X a(x)dx.$$ When $t\to0$, the limit of $\mathrm{Tr} [p_t(x,x)],\mathrm{Tr} [q_t(x,x)]$ do not exist but the limit of the difference exsit. This is a very deep result relating to the supersymmetry and the structure of Dirac operator.

Your question is to ask if $\mathrm{Tr} [p_t(x,x)-q_t(x,x)]$ has a limit when $t\to0$ for PDO. This is not true even for general elliptic oprator.

An example is that for the Kahler manifold $(M,\omega)$, $D=\sqrt{2}(\bar{\partial}+\bar{\partial}^*)$ is a Dirac operator. the limit in (1) exist. But when $M$ is not Kahler, the limit in (1) maybe does not exist. In Bismut's paper, A local index theorem for non Kähler manifolds http://link.springer.com/article/10.1007%2FBF01443359, he find such limit exist if $\partial \overline{\partial}\omega=0 $, and this is almost a necessary condition.

Let's final remark that the finite propagation speed is not essential, just a trick.

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  • $\begingroup$ This is exactly what I was looking for, thanks! $\endgroup$ – AlexE Nov 30 '13 at 11:38

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