nLab microlocal formulation of index theory

Idea

One may give an abstract definition of an elliptic complex over a complex analytic space: an elliptic pair is the datum of an $\mathbb{R}$-constructible sheaf $F$ and a complex $\mathcal{M}$ of $\mathcal{D}$-modules, such that the intersection

$$char(\mathcal{M})\cap SS(F)=T^*_M M\subset T^*M$$

of the characteristic variety of $\mathcal{M}$ with the microsupport of $F$ is contained in the zero section of the cotangent bundle.

To such an elliptic pair, one may associate a pair of trace kernels $K_{\mathcal{M}}$ and $K_F$ such that the trace kernel $K_{\mathcal{M}}\otimes K_F$ is well defined (because of the microsupport condition). The index of the elliptic pair is given by the Hochshild class $eu(K_{\mathcal{M}}\otimes K_F)$, and the index theorem essentially says that this class may be computed as the product of the classes of the two kernels in play.

Generalization

One may try to generalize the microlocal formulation of index theory to global analytic geometry, using derived microlocalization, to get a global analytic index theory that generalizes Kashiwara and Schapira’s approach to the analytic situation over an arbitrary Banach ring.

Reference

• Masaki Kashiwara and Pierre Schapira Microlocal Euler classes and Hochschild homology arXiv.