Spin\(^ q\) structures
The notion of \(\text{Spin}^q\)-structure is introduced. The notion is analogous to that of a Spin-structure for appropriate Riemann manifolds and to that of a \(\text{Spin}^c\)-structure for complex manifolds. It is shown that \(\text{Spin}^q\)-structures are canonically induced by almost quaternionic structures on manifolds. More precisely, the group \(\text{Spin}^q(n)\) is defined by twisting the Spin group \(\text{Spin} (n)\) using the quaternionic unitary group Sp(1) i.e. \(\text{Spin}^q(n) = \text{Spin} (n) \times{_{\mathbb{Z}_2} \text{Sp(1)}}\) and fits appropriately into a short exact sequence \(1\to\mathbb{Z}_2 \to \text{Spin}^q (n)\to \text{SO} (n) \times \text{SO} (3)\to 1\) analogous to \(1\to\mathbb{Z}_2 \to \text{Spin} (n) \to \text{SO} (n) \to 1\) \((\text{Spin} (n)\) is the universal covering group of \(\text{SO} (n))\) in the Spin case. A \(\text{Spin}^q\)-structure on a principal \(\text{SO} (n)\)-bundle \(P_n\) over a manifold \(X\) consists of a principal \(\text{Spin}^q (n)\)-bundle \(P^q_n\) and a principal SO(3)-bundle \(P_3\) over \(X\) together with a \(\text{Spin}^q(n)\)-equivariant bundle map \(P^q_n \to P_n \times P_3\). Using \(P^q_n\) and a real representation of \(\text{Spin}^q (n)\), a \(\text{Spin}^q\)-vector bundle with a Dirac operator on it is constructed and the Bochner-Weitzenboeck type formula for the Dirac Laplacian is obtained. The paper concludes with the calculation of the index of the Dirac operator of a \(\text{Spin}^q\)-manifold of dimension \(n\equiv 0 \bmod 4\).
scientific article; zbMATH DE number 844443