The Riemann-Roch theorem (and HRR, GRR, GRR for stacks, etc), over $\mathbb{C}$, associates topological information on one side $\chi(X,F)$, the Euler characteristic, to analytic information on the other side.  If you want to relate analysis to topology, the biggest, baddest tool in your arsenal is [The Atiyah-Singer Index Theorem][1].  Then it's just a matter of finding an operator that shows up naturally that should apply.  In my answer on that one, I linked to the original paper, where Atiyah and Singer explicitly do HRR as Theorem 3.


  [1]: https://mathoverflow.net/questions/1162/atiyah-singer-index-theorem