\(\text{Spin}^q\), twistor and \(\text{Spin}^c\)
In an earlier paper [\textit{M. Nagase}, J. Math. Soc. Jap. 47, 93-119 (1995; Zbl 0863.57019)], the author introduced and studied the so-called \(\text{Spin}^q(n)\)-structures: the group \(\text{Spin}^q(n)\) is, by definition, \(\text{Spin}(n)\times_{{\mathbb Z}_2} \text{Sp}(1)\) and \(\text{Spin}^q(n)\)-structures for \(\text{SO}(n)\)-bundles are defined similarly to \(\text{Spin}(n)\)- or \(\text{Spin}^c(n)\)-structures. If \(P\to M\) is the principal \(\text{Spin}^q(n)\)-bundle of such a structure, the \({\mathbb C}P^1\)-bundle \(Z=P\times (\text{Spin}^q(n)/\text{Spin}^c(n))\) is called the (\(\text{Spin}^q(n)\)-style) twistor space associated to the given \(\text{Spin}^q(n)\)-structure. Any almost-quaternionic manifold \(M\) has a canonical \(\text{Spin}^q(n)\)-structure [op.cit.] whose twistor space coincides with the twistor space studied by \textit{S. Salamon} [Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)] provided \(\dim M=8m+4\) and, in contrast to the Salamon twistor space, is trivial if \(\dim M=8m\) (since the canonical \(\text{Spin}^q(n)\)-structure is trivial in this case). In Section 1 of the present paper, the author shows that the twistor space \(Z\) associated to any \(\text{Spin}^q(n)\)-structure admits a natural \(\text{Spin}^c(n+2)\)-structure. Suppose the structure \(\text{SO}(n)\)-bundle of an oriented Riemannian manifold \((M,g)\) of dimension \(n\) admits a \(\text{Spin}^q(n)\)-structure. Any given connection of the principal \(SO(3)\)-bundle of this structure, together with the Levi-Civita connection of \(M\), induces a connection on the \(\text{Spin}^q(n)\)-bundle \(P\) of the structure and so on the associated bundle \(\pi:Z\to M\). The latter connection, as is usual in twistor theory, gives rise to a Riemannian metric \(g^Z= \pi^{*}g+ g^v\) on \(Z\) where \(g^v\) is the Fubini-Study metric on the fibre \({\mathbb C}P^1\) of \(Z\). In Section 2, the author considers the Levi-Civita connection of the metric \(g^Z\) and its relation with the connection \(\pi^{*}\nabla^M+\nabla^v, \nabla\) denoting the Levi-Civita connection. In Section 3 he discusses the representation theory for \(\text{Spin}^q(n)\), defines spinor bundles \({\mathbb S}^q\) over \(M\) and \(S^c\) over \(Z\), and gives Lichnerowicz type formulas for the Dirac operators \(D^q\) on \(S^q\) and \(D^c\) on \(S^c\). If \(M\) is even-dimensional, then the spinor bundles have a natural splitting according to which the Dirac operators \(D^q\) and \(D^c\) are graded into \(D^{q\pm}\), \(D^{c\pm}\) and the author calculates the indices of \(D^{q+}\) and \(D^{c+}\) (for \(M\) compact) in Section 4. In the last section of the paper, the author constructs a \(\text{Spin}(2n+1)\times_{{\mathbb Z}_2} \text{Sp}(1) \times_{{\mathbb Z}_2} U(1)\)-vector bundle \(S^{qc}\) on the twistor space \(Z\) over a compact manifold \(M\) of odd dimension \(2n+1\) and a Dirac operator \(D_{\varepsilon}^{qc}\) for \(S^{qc}\) associated to the metric \(g_{\varepsilon}^Z=\varepsilon^{-1}\pi^{*}g+g^v\). He shows that the limit of the reduced \(\eta\)-invariant \(\widetilde\eta(D_{\varepsilon}^{qc})\) exists in \({\mathbb R}/{\mathbb Z}\) as \(\varepsilon\to 0\). Following an idea originally due to \textit{E. Witten} [Commun. Math. Phys. 100, 197-229 (1985; Zbl 0581.58038)], he also considers the difference \(\lim_{\varepsilon\to 0} \widetilde\eta(D_{\varepsilon}^{qc})- \widetilde\eta(D^q)\). In the main result of the paper, the author gives an integral formula \(\text{mod } {\mathbb Z}\) for this difference involving the twisted \(A^{\wedge}\)-genus.
scientific article; zbMATH DE number 1121190