Integration of Dirac–Jacobi structures

Abstract

We study precontact groupoids whose infinitesimal counterparts are Dirac-Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.

On the geometry of prequantization spaces

“…2.7 of [8]; in the general case (but assuming different conventions for the multiplicativity of θ Γ and for which of the source and target is a Jacobi-Dirac map) this is Prop. 3.3 in [17]. We will prove only the first isomorphism above (the one for ker s * | Q ); the other one follows by composing the first isomorphism with i * .…”

“…The following two definitions are adapted from [3,17] respectively to match up the conventions of [8,27].…”

“…In particular the precontact groupoid (Γ c (Q), θ, f ) of a Jacobi-Dirac manifold Q can be constructed via the A-path space P a (L), with θ and f coming from a corresponding 1-form and function on the path space. We refer the reader to [8,6,17] and summarize the results in Theorem 4.5 below. The advantage of this method is that it can be used to generalize Theorem 4.2 to the setting of Dirac manifolds (see Theorems 4.9 and 4.11) and that it can be applied to a general group G action as in [10].…”

“…Here ψ is the 1-cocycle onL given by (X, f ) ⊕ (ξ, g) → f ;L × ψ R is the Lie algebroid on Q × R obtained from the Lie algebroid L and the 1-cocycle ψ, and it is isomorphic to the Lie algebroid given by the Dirac structure on Q × R obtained from the "Diracization" of (Q,L) (see Section 2.3 in [17]). …”

“…Consider the groupoid Γ × R over Q × R with target mapt(g, t) = (t(g), t − r Γ (g)) and the obvious sourcẽ s and multiplication. (Γ × R, d(e t θ Γ )) is then a presymplectic groupoid with the property thats is a forward Dirac map onto (Q × R,L), wherẽ L (q,t) = {(X, f ) ⊕ e t (ξ, g) : (X, f ) ⊕ (ξ, g) ∈ L q } is the "Diracization" ( [25,17]) of the Jacobi-Dirac structureL and t is the coordinate on R. In the special case thatL corresponds to a Jacobi structure this is just Prop. 2.7 of [8]; in the general case (but assuming different conventions for the multiplicativity of θ Γ and for which of the source and target is a Jacobi-Dirac map) this is Prop.…”