On the \(\text{mod } k\) index theorem of Freed and Melrose
The theorem mentioned in the title reads \[ \text{ind}_{(k)}(E) = \int_X \widehat A(TX, \nabla^{TX}) \text{ch} (E,\nabla^E) - k\overline \eta(D_{Y,F})\;\bmod k. \] Here \(X\) is an even dimensional compact smooth \(\text{spin} \mathbb{Z}/k\) manifold with singular set \(Y\). Moreover \(E\) is a \(\mathbb{Z}/k\) complex vector bundle over \(X\) and \(F\) is the associated bundle over \(Y\). Also \(\nabla_{TX}\) is the Levi-Civita connection associated to a suitable \(\mathbb{Z}/k\) metric on \(TX\), and \(\nabla_E\) is a suitable \(\mathbb{Z}/k\) connection on \(E\) preserving a suitable metric \(g^E\) on \(E\). Furthermore \(D_{Y,F}\) is the Dirac operator, and \(\overline\eta\) the reduced \(\eta\) invariant. Finally \(\text{ind}_{(k)}= [i_!E] \in\mathbb{Z}/k\) is the index defined using the \(K\)-theory of \(\mathbb{Z}/k\) vector bundles. In the paper by \textit{D. S. Freed} and \textit{R. B. Melrose} [Invent. Math. 107, 283-299 (1992; Zbl 0760.58039)] this formula was proved as an application of an index theorem of Atiyah-Singer type for \(\mathbb{Z}/k\) manifolds. In this note a new proof is given as an application of the identity involving Chern-Simons currents proved in [\textit{J. M. Bismut} and \textit{W. Zhang}, Math. Ann. 295, 661-684 (1993; Zbl 0795.57010)]. Moreover a mod 2 refinement is proved for real vector bundles.
scientific article; zbMATH DE number 895022