Manin pairs and moment maps revisited
Eckhard Meinrenken, Selim Tawfik
TL;DR
The paper develops a lifting framework for quasi-Poisson spaces arising from Manin pairs $( rak{d}, rak{g})$, establishing a bijection between $G$-spaces with $D/G$-valued moment maps and $G\times G$-spaces with $D$-valued moment maps via the Lifting Theorem.It then defines fusion and conjugation operations for these spaces by translating Ševera’s constructions through the lifting, and demonstrates how moduli spaces of flat $D$-bundles provide rich new examples within this Dirac-geometric setting.A quasi-symplectic groupoid integration theory is developed to describe moment-map targets and the corresponding reductions, with explicit forms for the multiplicative 2-forms and their dependence on splittings; reductions of targets yield a coherent quotient framework that preserves quasi-Poisson structures.The equivalence between $D$-valued and $D/G$-valued moment maps is established as a central unifying theme, allowing simple fusion/conjugation constructions and enabling the systematic construction of moduli-space examples for surfaces with bi-colored boundaries.
Abstract
The notion of quasi-Poisson $G$-spaces with $D/G$-valued moment maps was introduced by Alekseev and Kosmann-Schwarzbach in 1999. Our main result is a \emph{Lifting Theorem}, establishing a bijective correspondence between the categories of quasi-Poisson $G$-spaces with $D/G$-valued moment maps and of quasi-Poisson $G\times G$-spaces with $D$-valued moment maps. Using this result, we give simple constructions of fusion and conjugation for these spaces, and new examples coming from moduli spaces.