I am trying to have a go at the Atiyah-Singer Index theorem and my question is very basic. Any help is appreciated!
In Nakahara's book Geometry,Topology and Physics the theorem is stated as follows :
$$ IND(E,D) = (-1)^{m(m+1)/2} \int_{M} ch \left( \oplus (-1)^r E_r \right) \frac{Td(TM^\mathbb{C})}{E(M)}\Big|_{Vol}$$
The r.h.s contains objects like Chern,Todd and Euler classes each of which are sub-classes of the DeRham Cohomology group of TM and are hence (Lie-algebra valued) differential forms of various degrees.
So essentially the right hand side tells me to divide by differential forms?? I have no idea how one would perform such an operation. The splitting principle eases things somewhat but one is still dividing by differential forms. How do I make sense out of this? In addition my guess is that the notation $\Big|_{Vol}$ picks the volume forms out of the many terms of various degrees that appear on the r.h.s. Is that correct? If yes could there be more than one term?
many thanks!