2026/07/08 by Djounvouna Dinamo · 1 voice · 1 citation
Mathematics · #math.SG #math.AT
Associated to any smooth map F\colon M→ N equipped with a closed, nondegenerate relative (n+1)-form \varpi -- a relative n-plectic structure -- is an L_∞-algebra of relative observables L∞(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 relative homotopy moment map as an L_∞-morphism from \mathfrakg into L∞(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→ 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.