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In this paper, we define an analytic index $\textrm{ind}^{a,\Lambda}_{\widehat{K}}$ in differential $K$-theory associated to families ofgeneralized twisted de Rham operators for any given submersion $\pi:X\to B$with closed fibers. We prove that the analytic index $\textrm{ind}^{a,\Lambda}_{\widehat{K}}$ of the generator associated to any given complex flatvector bundle $(F, \nabla^F)$ is equal to the generator associated to thecohomology bundle $(H(Z, F|_Z), \nabla^{H(Z, F|_Z)})$ of $(F, \nabla^F)$.Moreover, we prove the real part of the Riemann-Roch-Grothendieck theorem forcomplex flat vector bundles at the differential form level. This gives a newproof of the real part of the Riemann-Roch-Grothendieck theorem for complexflat vector bundles previously proved by Ma-Zhang, and also generalizes ourprevious work on the $\mathbb{Z}_2$-graded version of the same theorem in theeven dimensional fiber case.

Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles II