Index character associated with the projective Dirac operator (Q2839376)

In [J. Differ. Geom. 74, No. 2, 265--292 (2006; Zbl 1115.58021)], \textit{V. Mathai} et al. introduced the concept of a pseudodifferential operator on a projective vector bundle. Using this framework, for any Riemannian manifold \(M\), they associated a projective Dirac operator \(\partial_M^{\text{pr}}\). Although \(\partial_M^{\text{pr}}\) cannot be represented as a Fredholm operator in general, they defined its analytic index, \(\text{ind}_a\partial_M^{\text{pr}}\in\mathbb Q\), which may not be integral if \(M\) is not spin, and proved a fractional index formula, \(\text{ind}_a\partial_M^{\text{pr}}=\int_M\hat A(R_M)\).NEWLINENEWLINESuppose that \(n=\dim M\) is even, and identify the automorphism group of the Cifford algebra of \({\mathbb R}^n\) with \(\text{PU}(N)\) for a suitable integer \(N\). A natural principal \(\text{PU}(N)\)-bundle \(P\to M\) is defined. Let \(V^{\text{(nat)}}\) denote the natural representation of \(\text{SU}(N)\). The Levi-Civita connection induces a \(1\)-form \(\nabla\) of first order differential operators acting on \(C^\infty(P,V^{\text{(nat)}})\), which also admits a structure of \(\text{Cl}_{\mathbb C}(M)\). Thus we can define a Dirac operator \(\partial_M\) on \(C^\infty(P,V^{\text{(nat)}})\). This operator is equivariant and transversely elliptic on \(P\) as \(\text{SU}(N)\)-manifold. So its index, \(\text{ind}_{\text{SU}(N)}(\gamma,\partial_M)\), is defined as a distribution on \(\text{SU}(N)\). In [J. Differ. Geom. 78, No. 3, 465--473 (2008; Zbl 1147.58018)] \textit{V. Mathai} et al. proved that \(\partial_M\) descends to the projective Dirac operator \(\partial_M^{\text{pr}}\) on the projective spin bundle over \(\text{Cl}_{\mathbb C}(M)\), and observed that \(\langle\text{ind}_{\text{SU}(N)}(\gamma,\partial_M),\phi\rangle=\text{ind}_a\partial_M^{\text{pr}}\) if \(\phi\in C^\infty(\text{SU}(N))\) is equal to \(1\) around the identity element \(e\) and has small enough support.NEWLINENEWLINE In the paper under review, the author gives a description of the whole distribution \(\text{ind}_{\text{SU}(N)}(\gamma,\partial_M)\), showing that \(\text{ind}_{\text{SU}(N)}(\gamma,\partial_M)=\int_M\hat A(R_M)\,\text{ch}_{\text{SU}(N)}(\gamma,{\mathcal E}/{\mathbb S})\), where \(\text{ch}_{\text{SU}(N)}(\gamma,{\mathcal E}/{\mathbb S})\) is the Chern character of the Clifford module \({\mathcal E}=P\times_{\text{PU}(N)}L^2(\text{PU}(N))\otimes V^{\text{(nat)}}\) over \(M=P/\text{PU}(N)\). Since \(\mathcal E\) is of infinite rank, convolution with auxiliary test functions is needed to define \(\text{ch}_{\text{SU}(N)}(\gamma,{\mathcal E}/{\mathbb S})\), which turns out to be a distribution on \(\text{SU}(N)\).