Reduction of relative multisymplectic manifolds

Abstract

We extend the multisymplectic reduction theory of Blacker -- itself the extension of Marsden--Weinstein--Meyer reduction to $k$-plectic manifolds -- to the setting of \emph{relative} multisymplectic geometry, in which a smooth map $F:M\to N$ carries a closed nondegenerate relative $(k+1)$-form $\varpi =(\ddot{I}‰,\hat{I}\cdot )$ in the mapping-cone complex of $F$. We introduce the Leibniz algebra of relative Hamiltonian pairs and the associated relative moment maps $\hat{I}¼\in \hat{I}{\text{c◯}}^{k-1}(F,{\mathfrak{g}}^{\ast })$, and prove a relative multisymplectic reduction theorem: for a closed equivariant level $\ddot{I}•\in \hat{I}{\text{c◯}}^{k-1}(F,{\mathfrak{g}}^{\ast })$, the level pair of $\hat{I}¼$ descends to a reduced smooth map $F_{\ddot{I}}•:M_{\ddot{I}}•\to N_{\ddot{I}}•$ carrying a unique reduced closed relative form ${\varpi }_{\ddot{I}}•$; remarkably, the horizontality of the trivializing component is forced by the relative closedness of the level. We further prove relative analogues of the reduction of dynamics, of the structure theory of split moment maps $\hat{I}¼=\hat{I}½\cdot \hat{I}º$ -- for which the splitting datum is a one-step cocycle in the relative Cartan model, so that split relative Hamiltonian $G$-spaces carry canonical relative homotopy moment maps -- and of the Duistermaat--Heckman-type variation formula: with respect to suitable conjugate distributions, the reduced relative class satisfies ${\partial }_{\hat{I}}»[{\varpi }_{\ddot{I}}ˆ]=⟨c,\hat{I}»⟩\cdot [\hat{I}{º}_{\ddot{I}}ˆ]$, where $c$ is the Chern form of the target model bundle acting through the natural $\hat{I}\text{c◯}(N)$-module structure of the mapping cone. Finally, we transport the exact stationary phase approximation to the target component and prove a genuinely relative localization constraint: the fixed-point contributions of the pulled-back split package on the source cancel identically.

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