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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?

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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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