Multiplicative Ehresmann connections for Lie groupoid fibrations

Abstract: We introduce multiplicative Ehresmann connections on surjective submersions of Lie groupoids, extending both the classical notion of Ehresmann connections on fibre bundles and the more recent notion of multiplicative connections on Lie groupoid extensions. We investigate the existence of such connections, showing that, in general, they may fail to exist even for proper Lie groupoids. In contrast, positive results hold for Morita submersions, uniform Lie groupoid fibrations, locally trivial families of Lie groupoids, and proper families of Lie groupoids. Our main results concern completeness. For Lie groupoid fibrations, we prove that the completeness of a multiplicative connection is governed by the induced connection on the kernel bundle and, under connectivity assumptions, by the base connection. For families of source-proper Lie groupoids, we prove the equivalence between local triviality and the existence of complete multiplicative Ehresmann connections.

• The paper extends classical Ehresmann connections to the Lie groupoid setting by defining multiplicative Ehresmann connections that respect groupoid multiplication.
• It establishes existence criteria for these connections using Morita fibrations, proper families, and local triviality while highlighting cohomological obstructions in nontrivial actions.
• The work characterizes completeness of these connections via reduction principles on kernel groupoids, offering insights applicable in Poisson geometry and gauge theory.

Multiplicative Ehresmann Connections for Lie Groupoid Fibrations: A Technical Analysis

Overview

The paper "Multiplicative Ehresmann connections for Lie groupoid fibrations" (2604.22394) presents a comprehensive extension of the theory of Ehresmann connections to the setting of Lie groupoid fibrations. The authors develop the concept of multiplicative Ehresmann connections (MECs) for surjective submersions of Lie groupoids, addressing their existence and completeness, and delineate the precise circumstances under which such structures are available. The work refines and generalizes seminal contributions on multiplicative connections and their applications in differential and Poisson geometry, representation theory, and stack cohomology.

Multiplicative Ehresmann Connections: Foundations and Generalization

The construction synthesizes classical Ehresmann connections from bundle and foliation theory with the notion of multiplicativity intrinsic to the groupoid context. For a surjective Lie groupoid morphism $T: G \to H$ over $M \to N$, a MEC is defined via a VB-groupoid splitting of $TG$, such that the horizontal distribution forms a VB-subgroupoid and, critically, respects the groupoid multiplication. This formalism includes as special cases:

• Classical Ehresmann connections on surjective submersions.
• Linear connections on vector bundles, viewed as abelian groupoids over their bases.
• Multiplicative connections for groupoid extensions, notably those classifying non-abelian differentiable gerbes.
• Principal connections via the gauge groupoid.

A foundational result is the characterization of MECs as those Ehresmann connections for which the space of horizontal paths is a subgroupoid of the path groupoid. The authors provide rigorous geometric and infinitesimal characterizations, with the latter employed to establish a bridge to the theory of Lie algebroids and infinitesimally multiplicative connections.

Existence Results and Obstructions

Unlike the classical setting, where Ehresmann connections always exist for surjective submersions, the existence of MECs in the multiplicative context is profoundly constrained. The key positive and negative results are as follows:

• Morita Fibrations and Uniform Lie Groupoid Fibrations: For these classes, MECs always exist. The proof exploits the essentially local triviality of these fibrations and leverages pullback constructions.
• Proper Families of Lie Groupoids: A properness condition on the total groupoid suffices to guarantee the existence of a MEC.
• Locally Trivial Families: Such families always admit a MEC, a fact established via partition of unity arguments and glueing of local trivializations.
• General Lie Groupoid Fibrations: Existence fails dramatically in certain cases. Explicitly, for action morphisms associated to nontrivial connected Lie group actions, MECs do not exist unless the action is trivial (Corollary 4.3), even if the groupoid involved is proper and compact. This highlights a major divergence from the proper groupoid extension case.

These existence results are directly connected to the underlying cohomological obstructions, extending prior work on bundle of ideals and groupoid extensions.

Completeness of Multiplicative Ehresmann Connections

A principal contribution concerns the structure and completeness of MECs. The authors demonstrate that the completeness property for a MEC (i.e., global path lifting for horizontal paths) is fundamentally a property of the induced connections on the kernel of the groupoid fibration and, in special cases, the base connection.

Main Results on Completeness:

• If the MEC on the kernel groupoid is complete, then the original MEC is complete.
• Under the source-connectedness of the kernel, completeness reduces further to the completeness of the connection on the base.
• For families of source-proper Lie groupoids, the existence of a complete MEC is equivalent to local triviality of the family (Theorem 6.7).

These claims provide powerful reduction principles, linking global geometric properties to fiberwise and basewise considerations. Notably, completeness can fail in locally nontrivial families even when local criteria seem favorable, evidencing the subtleties endemic to the groupoid context.

Implications and Directions for Future Research

Theoretical Consequences

• Stack Cohomology and Differentiable Gerbes: The formalism and results facilitate a unified approach to constructing generalized Cartan models and describing nonabelian gerbes in the Lie groupoid context.
• Morita Invariance and Deformation Theory: The reduction techniques for completeness invite further study of the deformation theory of groupoid fibrations and the moduli of groupoid structures with prescribed connection data.
• Infinitesimal Correspondence: The linkage between multiplicative connections at the groupoid and Lie algebroid levels supports ongoing developments in Lie groupoid/algebroid integration theory and related cohomological frameworks.

Practical Applications

• Poisson and Symplectic Geometry: The construction provides local models for geometric structures encountered in Poisson geometry, facilitating the study of deformation quantization and symplectic groupoids.
• Gauge Theory and Higher Structures: Recent applications in Yang-Mills theory on Lie groupoids and categorified gauge theories are enabled by explicit construction of MECs and resolution of their existence and completeness.

Open Problems and Speculation

• Necessity of Source-Properness: While source-properness appears sufficient for existence of complete MECs in locally trivial families, it remains an open question whether this is also necessary (Question 6.12).
• Higher Stack and Derived Geometry: The extension of these concepts to higher Lie groupoids and their role in derived differential geometry is a natural next step, especially in light of applications to higher gerbes and field theories.

Conclusion

This work delivers a substantial advancement in the study of connections on Lie groupoid fibrations, offering definitive criteria for the existence and completeness of multiplicative Ehresmann connections. By interweaving geometric, cohomological, and infinitesimal perspectives and establishing sharp boundaries for the availability of these structures, the paper sets the stage for further explorations in differential geometry, representation theory, and higher stack theory. The operationalization of these results in Poisson geometry and gauge theory substantiates their practical significance within modern mathematical physics and geometry.