Index character associated to the projective Dirac operator

We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai–Melrose–Singer.

💡 Research Summary

The paper addresses the problem of understanding the “fractional index” that appears in the projective Dirac operator introduced by Mathai, Melrose, and Singer (MMS). For a Riemannian manifold (M) (assumed even‑dimensional), the authors construct a canonical infinite‑dimensional Clifford module (E) and a Dirac operator (\not!\partial_M) acting on its smooth sections. The construction proceeds as follows. Starting from the orthonormal frame bundle (FSO(n)) one obtains a principal (PU(N))‑bundle (P\to M) via the natural embedding (SO(n)\hookrightarrow PU(N)) where (N=2^{n/2}). The natural representation (V_{\mathrm{nat}}) of (SU(N)) yields the Hilbert space (L^2(P,V_{\mathrm{nat}})). By regarding this space as sections of the associated bundle \