The adiabatic limits of signature operators for \(\text{spin}^q\) manifolds (Q5949467)

Let \(M\) be a closed oriented \(n\)-dimensional Riemannian manifold. A spin\(^q\) structure on \(M\) (also called spin\(^h\) structure, see \textit{Ch. Bär} [Math. Nachr. 201, 7-35 (1999; Zbl 0946.58020)]) is the quaternionic analog of a spin\(^c\) structure. More specifically, \(M\) carries an \(\text{ SO}(3)\)-principal bundle \(P_{\text{ SO}}(3)\) and a \(\text{ Spin}^q(n)\)-principal bundle \(P_{\text{ Spin}}^q(n)\) together with a \(\text{ Spin}^q(n)\)-equivariant bundle map \(P_{\text{ Spin}}^q(n) \to P_{\text{ SO}}(n) \times_\pi P_{\text{ SO}}(3)\) where \(\text{ Spin}^q(n) = {\text{ Spin}}(n) \times_{Z_2} {\text{ Sp}}(1)\), \(P_{\text{ SO}(n)}\) is the bundle of oriented tangent frames, and \(\times_\pi\) denotes the fiber product.NEWLINENEWLINENEWLINESince \(P_{\text{ Spin}}^q(n)\) acts canonically on \(\mathbb{C}\text{P}^1 = {\text{ Spin}}^q(n)/{\text{ Spin}}^c(n)\) one can form the \textit{twistor space} of a spin\(^q\) manifold by setting \(Z := P_{\text{ Spin}}^q(n) \times_{\text{ Spin}}^q(n)\mathbb{C}\text{P}^1\). The Levi-Civita connection on \(P_{\text{ SO}(n)}\) and a chosen connection on \(P_{\text{ SO}}(3)\) induce a connection on \(P_{\text{ Spin}^q(n)}\) and hence a splitting \(TZ = H \oplus V\) of the tangent bundle of the twistor space into a horizontal and a vertical subbundle.NEWLINENEWLINENEWLINEThe Riemannian metric on \(M\) induces one on \(H\) and the Fubini-Study metric on \(\mathbb{C}\text{P}^1\) yields one on \(V\). Making \(H\) and \(V\) perpendicular we have thus obtained a Riemannian metric \(g_Z\) on \(Z\). Given \(\varepsilon > 0\) we can rescale the metric on \(H\) by the factor \(\varepsilon^{-1}\) and leave the one on \(V\) unchanged. This gives us a one-parameter family of metrics \(g_\varepsilon\) on \(Z\) with \(g_1 = g_Z\).NEWLINENEWLINENEWLINENow let \(n\) be odd and let \(A^Z_\varepsilon\) denote the signature operator of the Riemannian manifold \((Z,g_\varepsilon)\). Let \(\eta(A^Z_\varepsilon)\) be its eta-invariant. The main result of the paper says that the \textit{adiabatic limit} \(\lim_{\varepsilon \to 0} \eta(A^Z_\varepsilon)\) exists and gives a formula for it. Certain twisted signature operators are also considered. The proofs depend on methods developed by \textit{X. Dai} [J. Am. Math. Soc. 4, No. 2, 265-321 (1991; Zbl 0736.58039)].