The moment map revisited
A Hamiltonian action of a Lie group on a symplectic manifold gives rise to a symplectomorphism group, and one can think of a corresponding moment map as being associated with that symplectomorphism group. In the paper under review a general notion of moment map is described, which allows one to speak about moment maps associated with families (not necessarily groups) of symplectomorphisms. The general setting is provided by a symplectic manifold \((M,\omega)\), a fibration \(\pi\colon Z\to S\), and a smooth map \(G\colon Z\to M\). For any \(s\in S\) one denotes \(Z_s=\pi^{-1}(s)\), \(g_s=G| _{Z_s}: Z_s\to M\), \(\Lambda_s=g_s(Z_s)\), and one assumes that the mapping \(g_s\) is a Lagrangian embedding. Then a moment map is a lifting \(\Phi: Z\to T^*S\) of \(\pi: Z\to S\) such that for all \(s\in S\) and \(z\in Z_s\) the mapping \(\Phi_s=\Phi| _{Z_s}: Z_s\to T_s^*S\) satisfies the following condition: The dual (i.e., transpose) of the differential \((d\Phi_s)_z: T_z(Z_s)\to T_s^*S\) coincides with the linear map \(T_sS\to T_z^*(Z_s)\) induced by \(dG_z: T_zZ\to T_{G(z)}M\) by factoring to a linear map \(T_zZ/T_z(Z_s)\to T_{G(z)}M/T_{G(z)}(\Lambda_s)\) and then using the isomorphisms \(T_zZ/T_z(Z_s)\simeq T_sS\) (defined by the fibration \(\pi\)) and \(T_{G(z)}M/T_{G(z)}(\Lambda_s)\!\simeq T_{G(z)}^*(\Lambda_s)\simeq T_z^*(Z_s)\) (defined by the Lagrangian embedding \(g_s=G| _{Z_s}: Z_s\to\Lambda_s\hookrightarrow M\) along with the symplectic form \(\omega\) of \(M\)). This formalism can be applied in particular to families (indexed by \(S\)) of symplectomorphisms of \(M\), by working with \(S\times M\) in the role of \(Z\) and with \(M\times M^{-}\) in the role of \(M\) (where \(M^{-}\) stands for \(M\) equipped with the symplectic form \(-\omega\)). The main results of the paper include some basic properties of this generalized moment map. Thus the pull-back by \((G,\Phi): Z\to M\times T^*S\) of the symplectic form of \(M\times T^*\) is the pull-back by \(\pi\) of a closed 2-form \(\mu\) on \(S\), whose cohomology class does not depend on the choice of \(\Phi\) (Theorem~7). In addition, if \([\mu]=0\) then there exists a moment map \(\Phi\) such that the aforementioned map \((G,\Phi)\) is a Lagrangian embedding (Theorem~8). Conversely, if \(\Phi\colon Z\to T^*S\) is a lifting of \(\pi: Z\to S\) such that the mapping \((G,\Phi): Z\to M\times T^*S\) is a Lagrangian embedding, then \(\Phi\) is a moment map (Theorem~9).
scientific article; zbMATH DE number 5004319