Relative homotopy moment maps

Abstract: Associated to any smooth map $F\colon M\to N$ equipped with a closed, nondegenerate relative $(n+1)$-form $\varpi$ -- a \emph{relative $n$-plectic structure} -- is an $L_\infty$-algebra of relative observables $L_{\infty}(F,\varpi)$, constructed by the author in earlier work. In this article we develop the corresponding theory of moment maps: for a Lie group $G$ acting compatibly on $M$ and $N$ and preserving $\varpi$, we define a \emph{relative homotopy moment map} as an $L_\infty$-morphism from $\mathfrak{g}$ into $L_{\infty}(F,\varpi)$ lifting the infinitesimal action, thereby providing a full relative generalization of the homotopy moment maps of Callies, Frégier, Rogers and Zambon. We characterize such morphisms by explicit component equations, show that a relative homotopy moment map is equivalent to a homotopy moment map on the target $N$ together with a coherent trivialization of its pullback to $M$, and relate relative moment maps to a relative Cartan model computing relative equivariant de Rham cohomology. Every cocycle in the relative Cartan model extending $\varpi$ induces a relative homotopy moment map via explicit formulas, and we prove the one-step case in full detail. In the existence theory a new phenomenon appears: under a mild connectivity hypothesis the Lie-algebra-cohomology obstruction present in the absolute theory vanishes identically in the relative setting. Finally, we show that quasi-Hamiltonian $G$-spaces with group-valued moment map $μ\colon M\to G$ fit into this framework: the pair $(η,ω)$ built from the Cartan $3$-form is a relative $2$-plectic structure whose Alekseev--Malkin--Meinrenken axioms amount precisely to a canonical one-step cocycle in the relative Cartan model, and hence every quasi-Hamiltonian $G$-space carries a canonical relative homotopy moment map, which we compute explicitly.