Lie Groupoid Geometry of Cylindrically Consistent Gauge Covariant Flux Derivations in Loop Quantum Gravity

Abstract

We address a kinematical consistency tension in loop quantum gravity. Naive fluxes fail to transform in a graph-compatible gauge-covariant manner, whereas gauge-covariant fluxes obtained by inserting parallel transport typically introduce auxiliary path dependence that can spoil cylindrical consistency under graph refinement. We formulate LQG kinematics in a groupoid-geometric language by treating generalized connections as functors from the analytic path groupoid of [Formula: see text] into [Formula: see text] and by organizing cylindrical functions through the associated projective system. Within this framework, we define a base-pointed, parallel-transport-corrected flux functional [Formula: see text] anchored at [Formula: see text] and controlled by a surface path family [Formula: see text]. Our main result identifies explicit geometric admissibility conditions on [Formula: see text] under which the Poisson action [Formula: see text] defines a well-defined derivation on [Formula: see text] that is gauge covariant at the base point and independent of the chosen representative graph. We also present a counterexample showing that, without [Formula: see text]-adaptation or [Formula: see text]-closure, refinement invariance can fail already at the kinematical level.