The modular automorphism group σωt associated with a faithful state ω on a vonNeumann algebra A is canonical, but its “generator data” carries a gauge ambiguity:operators commuting with modular flow (zero modes) can be added without changingthe induced flow on A. We formalize this in the modular Berry connection frameworkintroduced for entangled regions in AdS/CFT and generalized by de Boer et al. usingconditional expectations onto modular fixed-point algebras. We define a state- andalgebra-intrinsic holonomy invariant—the modular Wilson loop Uγ —for loops in pa-rameter space of states or subalgebras. The conjugacy class (hence spectrum) of Uγ isgauge-invariant. We prove a flatness-to-integrability theorem: if the modular holonomyis trivial on a generating set of loops (equivalently, the induced representation of π1 istrivial), then there exists a global gauge in which the zero-mode connection vanishesand modular generator variations satisfy a Frobenius-type integrability condition,yielding globally consistent symmetry generators extracted from modular data. Con-versely, nontrivial holonomy is a rigorous obstruction: any global geometrization builtfrom those modular generators must be path-dependent or impossible. The frame-work interconnects with Connes cocycle comparison, providing an intrinsic transportmechanism, and recovers known results on modular Berry connections in holographywhile extending to general von Neumann algebraic settings.
SIKX HILTON (Tue,) studied this question.