This is mentioned on the 5th page (i.e. p 271) of Bloch: Algebraic cycles and higher K-theory. It relates to a conjecture relating orders of zeroes of Hasse-Weil zeta functions to the alternating sum of ranks of higher Chow groups, where the latter can maybe be interpreted as some sort of Euler-Poincare characteristic for Bloch’s complexes.
What could the relation be between this and Kontsevich’s very general index theorem?
http://londonnumbertheory.wordpress.com/2009/10/15/computational-class-field-theory/ mentions that Atiyah-Singer is maybe a form of Riemann-Roch.
Minhyong Kim MO comment: I’m sorry to be encouraging random association, but Vojta’s proof of Faltings’s theorem does use an arithmetic Riemann-Roch theorem, the Archimedean component of which is a refined index theorem. Sorry, couldn’t resist.
nLab page on Arithmetic index theory