Shifted lagrangian structures in Poisson geometry

2026/05/27 by Daniel Álvarez, Henrique Bursztyn, Miquel Cueca · 1 voice
Mathematics · #math.SG #math.DG

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Abstract

This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such morphisms and Dirac structures in transitive Courant algebroids given by the product of an exact Courant algebroid and a quadratic Lie algebra. As a key application, we identify the global objects integrating quasi-Poisson manifolds, which we call multiplicative D-valued moment maps; this extends the integration of Poisson manifolds to symplectic groupoids and the lifting of Poisson actions to multiplicative hamiltonian actions. We devise systematic constructions of quasi-symplectic groupoids via fibred products of 2-shifted lagrangians, extending classical reduction procedures. This places known constructions, such as the integrations of Poisson homogeneous spaces and Poisson quotients, into a broader, conceptual framework, while yielding new examples.

Discussions

• Daniel \'Alvarez, Henrique Bursztyn, Miquel Cueca: Shifted lagrangian structures in Poisson geometry https://arxiv.org/abs/2605.29117 https://arxiv.org/pdf/2605.29117 https://arxiv.org/html/2605.29117 [bsky, 0 points, 0 comments]

Related

• Dirac structures, moment maps and quasi-Poisson manifolds
• Dirac geometry, quasi-Poisson actions and D/G-valued moment maps
• Dirac geometry, deformation theory and shifted symplectic geometry
• Integrability of Poisson-Lie group actions
• Poisson and Symplectic structures, Hamiltonian action, momentum and reduction