Real secondary index theory (Q945651)

This paper gives a secondary version of the index theory for families of generalized Dirac operators satisfying appropriate conditions. To explain it, recall first some family index theory. Let \(\mathcal{E}\) be the geometric structure needed to define a family of fiberwise generalized Dirac operators on a smooth fiber bundle with base \(B\) and the Bismut super connection. Let \(\operatorname{index}(\mathcal{E})\in K^0(B)\) be the index of the family, and let \(\Omega(\mathcal{E})\) be the closed even form on \(B\) defined by the local index theory, which represents a class \([\Omega(\mathcal{E})]\in H^{\text{ev}}(B,\mathbb{R})\). Then the local index theorem states that \textbf{ch}\(^{\mathbb{R}}(\operatorname{index}(\mathcal{E}))=[\Omega(\mathcal{E})]\). Since the classifying space of \(K^0\) is \(BU\times\mathbb{Z}\), there is a universal class \textbf{ch}\(^{\mathbb{R}}_{\text{univ}}\in H^{\text{ev}}(BU\times\mathbb{Z},\mathbb{R})\) such that \([\Omega(\mathcal{E})]=f^*\) \textbf{ch}\(^{\mathbb{R}}_{\text{univ}}\), where \(f:B\to BU\times\mathbb{Z}\) classifies \(\operatorname{index}(\mathcal{E})\). Moreover \(H^{\text{ev}}(BU\times\mathbb{Z},\mathbb{R})\) is a polynomial algebra with generators \(c^{\mathbb{R}}_2,c^{\mathbb{R}}_4,\dots\) Thus \textbf{ch}\(^{\mathbb{R}}_{\text{univ}}\) has an expression in terms of these generators, obtaining the Chern character. The authors now consider the Atiyah-Hirzebruch filtration of \(K\)-theory, \(K_k^*(B)\), given by the classes with zero restriction to the \((k-1)\)-skeleton of \(B\), and assume that \(\operatorname{index}(\mathcal{E})\in K^*_k(B)\). Then \(c_k(\operatorname{index}(\mathcal{E}))\in H^k(B,\mathbb{Z})\) admits a natural lift to Deligne cohomology, \(\hat{c}_k(\mathcal{E})\in H^k_{\text{Del}}(B)\equiv H^{k-1}(B,\mathbb{R}/\mathbb{Z})\), which depends continuously on the geometry. Thus it has a curvature \(\omega(\hat{c}_k(\mathcal{E}))\in\mathcal{A}^k(B)\), which is given by \(\Omega(\mathcal{E})\). To go further, the authors consider the case of a family of twisted Dirac operators on a family of Spin-manifolds of dimension \(n\), where the twisting bundle is real. In this case, if \(k+n\equiv 2\;(\text{mod}\;4)\), then it is shown that \(\hat{c}_k(\mathcal{E})\) is flat, concluding that this class is a differential-topological invariant. Moreover they show that \(\hat{c}_k(\mathcal{E})\) depends on the real refinement \(\operatorname{index}_{\mathbb{R}}(\mathcal{E})\in KO^{-n}(B)\) and satisfies \(2\hat{c}_k(\mathcal{E})=0\). In fact, they construct a natural transgression \[ d_{B,k-1}^n:U^n_k(B)\to H^{k-1}(B,\mathbb{R}/\mathbb{Z}) \] such that \(d_{B,k-1}^n(\operatorname{index}_{\mathbb{R}}(\mathcal{E}))=\hat{c}_k(\mathcal{E})\) and \(2d_{B,k-1}^n=0\), where \(U^n_k(B)\subset KO^{-n}(B)\) consists of the classes which after complexification belong to \(K^{-n}_k(B)\). Now the universal situation is given as follows. The classifying space of \(KO^{-n}\) is \(\Omega^nBO\), and we have the fibration \[ \Omega^n(U/O)\to\Omega^nBO\to\Omega^nBU\;. \] A class \(x\in KO^{-n}(B)\) is represented by a map \(f:B\to\Omega^nBO\). If \(x\in U^{-n}_\infty(B)\), then \(f\) factors up to homotopy through a map \(g:B\to\Omega^n(U/O)\) because the composition of \(f\) with \(\Omega^nBO\to\Omega^nBU\) is null homotopic. So there must be a universal class \(\bar d_{4k+1-n}\in H^{4k+1-n}(\Omega^n(U/O),\mathbb{R}/\mathbb{Z})\) such that \(g^*\bar d_{4k+1-n}=d^{-n}_{4k+1-n}(x)\). The main result of the paper (Theorem~4.10) computes this universal class in terms of the known generators of \(\Omega^n(U/O)\). In most of the cases, it vanishes or it is a classical characteristic class. But it is not a classical characteristic class when \(n=6\) (it is not a pull back from \(\Omega^6BO=Sp/U\)), and thus it is more exotic and interesting in this case. Some examples with nontrivial classes are given.