Multiplicative Dirac structures

TL;DR

This paper introduces multiplicative Dirac structures on Lie groupoids, establishing a correspondence with Dirac structures on Lie algebroids, thus unifying and extending the integration of various geometric structures.

Contribution

It provides a novel framework linking multiplicative Dirac structures on Lie groupoids with Dirac structures on Lie algebroids, extending previous integration theories.

Findings

One-to-one correspondence between multiplicative Dirac structures on groupoids and Dirac structures on algebroids.

Extension of Lie bialgebroid integration to Dirac structures.

Connection established with higher geometric structures like $\ ext{LA}$-groupoids.

Abstract

In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic groupoids). We prove that for every source simply connected Lie groupoid $G$ with Lie algebroid $AG$, there exists a one-to-one correspondence between multiplicative Dirac structures on $G$ and Dirac structures on $AG$, which are compatible with both the linear and algebroid structures of $AG$. We explain in what sense this extends the integration of Lie bialgebroids to Poisson groupoids carried out in \cite{MX2} and the integration of Dirac manifolds of \cite{BCWZ}. We also explain the connection between multiplicative Dirac structures and higher geometric structures such as $\mathcal{LA}$-groupoids and $\mathcal{CA}$-groupoids.