Infinitesimal Symmetries of Dixmier-Douady Gerbes

We introduce the infinitesimal symmetries of Dixmier-Douady gerbes over a manifold M, both with and without connective structures and curvings. We explore the algebraic structure possessed by these symmetries, and relate them to equivariant gerbes via a “differentiation functor”. In the case that a gerbe G is equipped with a connective structure A, we give a new construction of the Courant algebroid associated to (G,A) directly in terms of the infinitesimal symmetries of (G,A).

💡 Research Summary

The paper develops a systematic theory of infinitesimal symmetries for Dixmier‑Douady gerbes on a smooth manifold M, both in the bare setting and when equipped with connective structures and curvings. The authors begin by recalling the well‑understood case of principal circle bundles: a 1‑parameter family of diffeomorphisms of M lifts to a 1‑parameter family of bundle automorphisms, and differentiating at the identity yields a vector field on M together with a T‑invariant vertical vector field on the total space. This elementary picture is reformulated in sheaf‑theoretic language, emphasizing that sections of the bundle form a torsor for the sheaf of smooth T‑valued functions and that a lift of a vector field is encoded by a sheaf homomorphism to the sheaf of iℝ‑valued functions.

Motivated by this analogy, the authors define the infinitesimal symmetry of a gerbe G as a sheaf of groupoids 𝓛_G over M. There is a natural projection functor π:𝓛_G→TM (viewed as a sheaf of groupoids with only identity morphisms). For each vector field ξ∈𝔛(M) the fiber 𝓛_G(ξ) is a gerbe banded by iℝ; when ξ=0 this fiber is canonically equivalent to the trivial gerbe B(iℝ_M) of principal iℝ‑bundles. Collecting all fibers yields an exact sequence of 2‑stacks B(iℝ_M) // 𝓛_G // TM, which can be interpreted as a “coherent Lie 2‑algebra” extension.

The paper then introduces a differentiation functor. Given a smooth 1‑parameter subgroup {φ_t}⊂Diff(M) and a corresponding category of lifts 𝓛_G({φ_t}) (objects are gerbe isomorphisms G≃φ_t^*G compatible with the group structure), differentiation at t=0 produces a functor D: 𝓛_G({φ_t}) → 𝓛_G(ξ), where ξ is the infinitesimal generator of {φ_t}. The authors prove that, after restricting to a sufficiently small open set and sufficiently small t, D is an equivalence of categories. This result mirrors the classical statement that any vector field integrates locally to a unique flow, thereby justifying the interpretation of 𝓛_G as the true infinitesimal symmetry object of a gerbe.

When a connective structure A (a 1‑form‑valued data on a good cover) and a curving K (a 2‑form) are present, the infinitesimal symmetry sheaf refines to a sheaf of categories 𝓛(G,A). The projection π:𝓛(G,A)→𝓛_G forgets the connective data, and each fiber over a lift ˆξ∈𝓛_G is a torsor for global 1‑forms on M. The connective structure provides a canonical horizontal lift ξ̂^h∈𝓛_G for any ξ∈𝔛(M). The authors construct a Čech model (g_{ijk},A_{ij}) for the gerbe and its connective data, and show that the corresponding categories L_{gijk} and L_{gijk,Aij} inherit the structure of a 2‑term L_∞‑algebra: they admit addition, scalar multiplication, and a Lie bracket satisfying the higher Jacobi identities. The exact sequence 0 → L_{gijk}(0) → L_{gijk} → C^∞(TM) → 0 is a Lie‑2‑algebra extension, split linearly by the connection A, while the curving K measures the obstruction to splitting at the level of Lie‑2‑algebras (its de Rham class appears as the “twist”).

The central achievement of the paper is a new construction of the Courant algebroid E(G,A) directly from infinitesimal connective symmetries. For each ξ∈𝔛(M) one defines Ê_ξ := π^{-1}(ξ̂^h) ⊂ 𝓛(G,A)(ξ), the set of connective lifts extending the horizontal lift. The disjoint union over all ξ yields a C^∞(M)‑module E(G,A) fitting into the exact sequence 0 → C^∞(T^*M) → E(G,A) → C^∞(TM) → 0. The authors define a bracket