\(K\)-theoretic invariants of Hamiltonian fibrations (Q2306969)

If \((M,\omega)\) is a symplectic manifold, Top is a category of paracontact Hausdorff topological spaces, \(\mathrm{Ham}(M,\omega)\) is the group of Hamiltonian diffeomorphisms of \((M,\omega)\), then a Hamiltonian fibration is a topological fiber bundle \(\textbf{M}\) over a base space \(B\in Top\), with fiber given by a symplectic manifold \((M,\omega)\), and structure group reduced to the group \(\mathrm{Ham}(M,\omega)\). One of the most basic examples of a Hamiltonian fibration is given by the projectivization \(P(E)\to B\) of a complex vector bundle \(E\to B\) of rank \(r\) over \(B\). In this paper, the authors introduce new invariants of Hamiltonian fibrations with values in the suitably twisted \(K\)-theory of the base. The first result of the paper states that, if \(E\) is a complex vector bundle on a topological space \(B\), and for each complex line bundle \(U\) on \(B\), \(ch(E\otimes U)\ne ch(\underline{\mathbb{C}}^r)\), then \(\mathrm{P}(E)\) is non-trivial as a Hamiltonian fibration, where \(\underline{\mathbb{C}}^r\) denotes the trivial vector bundle of the same rank as \(E\), and \(ch\) denotes the Chern character. Also, a stronger result states that, if \(E\) is a complex vector bundle on a topological space \(B\), and \(E\otimes U\) is not stably trivial for each complex line bundle \(U\) on \(B\), then \(\mathrm{P}(E)\) is non-trivial as a Hamiltonian fibration. If \(L\) is a complex Hermitian line bundle over a symplectic manifold \((M,\omega)\) with a unitary connection \(\nabla\) having curvature \(R(\nabla)=-i\omega\), then a 4-tuple \(\widehat{M}=(M,L,\nabla,\omega)\) is called a prequantization space. In order for such \((L,\nabla)\) to exist on a symplectic manifold \((M,\omega)\), the cohomology class of the symplectic form must be represented by a class with integer coefficients. Then \((M,\omega)\) is called quantizable. Fiber bundles with prequantization spaces over the base space \(B\) and the structure group \(\mathcal{Q}(\widehat{M})\) of the automorphisms of the prequantization space are called prequantum fibrations. For each compatible almost complex structure \(J\) on \((M,\omega)\) and \(k\in\mathbb{Z}_+\) there is a certain first-order elliptic differential operator on the spaces of \(L^{\otimes k}\)-valued \((0,*)\)-differential forms on \(M\), called the \(\mathrm{Spin}^c\) Dirac operator \(D^k\). The analytic index of the family of \(\mathrm{Spin}^c\) Dirac operators gives a \(K\)-theory class on the base \(B\). The new invariants introduced in this paper arise from the family analytic index of natural \(\mathrm{Spin}^c\) Dirac operators by showing that each \(K\)-theory class generalizes Bott's virtual Hilbert space and does not depend on the choice of almost complex structures. Consequently, it is shown to be an invariant of the isomorphism class of the prequantum fibration. The authors show that, if \(\mathcal{Q}(r)=\mathcal{Q}\left(\widehat{\mathbb{CP}}^{r-1}\right)\) is the group of automorphisms of the natural prequantization \(\widehat{\mathbb{CP}}^{r-1}\) of \(\mathbb{CP}^{r-1}\), \(B\mathcal{Q}(r)\) is the classifying space in the category \(Top\), \(\mathcal{H}\) is a separable infinite-dimensional Hilbert space, and for \(r\in\mathbb{Z}_+\), \(\mathrm{Fred}_r(\mathcal{H})\) is the space of Fredholm operators on \(\mathcal{H}\) of Fredholm index \(r\), then there are natural maps \(e\), \(q\), \(W_r\), with \(e\), \(q\) forming the sequence \(BU(r)\overset{e}{\to}B\mathcal{Q}(r)\overset{q}{\to}\mathrm{Fred}_r(\mathcal{H})\), and a homotopy commutative diagram \[ \begin{tikzcd} BU(r)\arrow[r, "e"] \arrow[d, "i_r"] & B\mathcal{Q}(r) \arrow[d, "q"] \\ BU \arrow[r,"W_r"] & \mathrm{Fred}_r(\mathcal{H}) \end{tikzcd} \] where the map \(e\) is induced by the canonical homomorphism \(U(r)\to\mathcal{Q}(r)\), the map \(i_r:BU(r)\to BU\) is the natural inclusion, and \(W_r\) is a weak equivalence. Moreover, the authors show that the map in complex \(K\)-theory induced by the natural inclusion map \(BU(r)\to B\mathcal{Q}(r)\equiv B\mathcal{Q}\left(\widehat{\mathbb{CP}}^{r-1}\right)\) is surjective on K-theory. Finally, they introduce a twisted version of their invariants and construct a new homotopy equivalence map, with a certain naturality property, from \(BU\) to the space of index-0 Fredholm operators on a Hilbert space, using elements of modern theory of homotopy colimits.