In this talk, we will introduce an interesting index problem naturally associated to the Arveson-Douglas conjecture infunctional analysis. This index problem is a generalization of the classical Toeplitz index theorem, and connects to many different branches of Mathematics. In particular, it can be viewed as ananalyticversion of theGrothendieckRiemannRochtheorem.This is joint work with R. Douglas，M. Jabbari, and G. Yu.