[Paper Review] Lie rackoids integrating Courant algebroids

Camille Laurent-Gengoux, Friedrich Wagemann|arXiv (Cornell University)|Jul 16, 2018

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper introduces an infinite-dimensional Lie rackoid $Y = \mathcal{C}^\infty([0,1], T^*M)$ that integrates the standard Courant algebroid on $\mathbb{T}M = TM \oplus T^*M$ via a rack product derived from automorphisms of the Dorfman bracket. The key contribution is showing that the quotient of $Y$ by a suitable equivalence relation yields the Weinstein Lie groupoid integrating an integrable Dirac structure, with the rackoid product recovering the adjoint action of bisections on paths.

ABSTRACT

We construct an infinite dimensional Lie rackoid Y which hosts an integration of the standard Courant algebroid. As a set, Y = C $\infty$ ([0, 1], T * M) for a compact manifold M. The rackoid product is by automorphisms of the Dorfman bracket. The first part of the article is a study of the Lie rackoid Y and its tangent Leibniz algebroid a quotient of which is the standard Courant algebroid. In a second part, we study the equivalence relation related to the quotient on the rackoid level and restrict then to an integrable Dirac structure. We show how our integrating object contains the corresponding integrating Weinstein Lie groupoid in the case where the Dirac structure is given by a Poisson structure.

Motivation & Objective

• To construct a global differential geometric object—specifically, a Lie rackoid—that integrates the standard Courant algebroid on $\mathbb{T}M = TM \oplus T^*M$ for a compact manifold $M$.
• To provide a framework that naturally recovers the Weinstein Lie groupoid integrating an integrable Dirac structure, overcoming limitations of previous graded differential geometry approaches.
• To clarify the relationship between the rackoid structure on $Y$, the Dorfman bracket automorphisms, and the resulting groupoid structure on the quotient.

Proposed method

• The Lie rackoid $Y$ is defined as the space of smooth paths in the cotangent bundle $T^*M$, with source and target maps given by evaluation at the endpoints of the interval $[0,1]$.
• The rack product on bisections $\Sigma = (\phi_t, \eta_t)$ is defined via the action of diffeomorphisms $\phi_1$ and closed 2-forms $d\beta_\Sigma$, where $\beta_\Sigma = \int_0^1 \phi_s^*\eta_s \, ds$, preserving the Dorfman bracket structure.
• The tangent Leibniz algebroid of $Y$ is shown to admit a quotient isomorphic to the standard Courant algebroid, establishing the infinitesimal compatibility.
• An equivalence relation $\sim$ is defined on $Y$, corresponding to a Fréchet foliation, which identifies paths that are equivalent under the action of bisections preserving the Dirac structure.
• For a Dirac structure $D \subset \mathbb{T}M$, the subobject $Y_D \subset Y$ consists of paths lying in $D$, and the quotient $Y_D / \sim$ is shown to carry a Lie groupoid structure.
• The rackoid product on $Y_D / \sim$ is proven to coincide with the adjoint action of bisections on the Weinstein groupoid, linking the rackoid construction to classical integration theory.

Experimental results

Research questions

• RQ1 Can a global differential geometric object be constructed that integrates the standard Courant algebroid, which is not a Lie algebroid and thus not amenable to classical integration?
• RQ2 How can the rackoid structure on $Y$ be used to recover the Weinstein Lie groupoid for an integrable Dirac structure?
• RQ3 What is the precise relationship between the rackoid product on bisections of $Y$ and the adjoint action of bisections on the Lie groupoid integrating a Dirac structure?
• RQ4 Does the quotient of the Lie rackoid $Y$ by the equivalence relation $\sim$ yield a manifold structure whose tangent space at the identity matches the Courant algebroid?

Key findings

• The infinite-dimensional Lie rackoid $Y = \mathcal{C}^\infty([0,1], T^*M)$ is constructed with a smooth rack product on its bisections, derived from automorphisms of the Dorfman bracket.
• The tangent Leibniz algebroid of $Y$ admits a quotient isomorphic to the standard Courant algebroid on $\mathbb{T}M$, confirming infinitesimal compatibility.
• The equivalence relation $\sim$ on $Y$ induces a Fréchet foliation, and the quotient $Y / \sim$ is shown to carry a manifold structure with the correct tangent space at the identity.
• For an integrable Dirac structure $D$, the subobject $Y_D \subset Y$ of paths in $D$ is coisotropic if and only if $D$ is integrable, and $Y_D / \sim$ is the Weinstein Lie groupoid integrating $D$.
• The rackoid product on $Y_D / \sim$ coincides with the adjoint action of bisections on the Lie groupoid, establishing that the rackoid structure recovers the groupoid structure.