Local formulas for multiplicative forms

TL;DR

This paper introduces explicit formulas for integrating multiplicative forms on local Lie groupoids, enabling concrete local integrations of various geometric structures like Poisson and Dirac structures.

Contribution

It provides explicit formulas for integrating multiplicative forms on local Lie groupoids, advancing the understanding of geometric structure integrations.

Findings

Explicit formulas for integrating multiplicative forms.

Concrete local integrations of Poisson, Dirac, and Jacobi structures.

Construction of local symplectic, presymplectic, and contact groupoids.

Abstract

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis-Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids, respectively.