Differential Geometry of Contextuality

Abstract: Contextuality has long been associated with topological properties. In this work, such a relationship is elevated to identification in the broader framework of generalized contextuality. We employ the usual identification of states, effects, and transformations as vectors in a vector space and encode them into a tangent space, rendering the noncontextual conditions the generic condition that discrete closed paths imply null phases in valuations, which are immediately extended to the continuous case. Contextual behavior admits two equivalent interpretations in this formalism. In the geometric or intrinsic-realistic view, termed "Schr\"odinger", flat space is imposed, leading to contextual behavior being expressed as non-trivial holonomy of probabilistic functions, analogous to the electromagnetic tensor. As a modification of the valuation function, we use the equivalent curvature to connect contextuality with interference, noncommutativity, and signed measures. In the topological or participatory-realistic view, termed "Heisenberg", valuation functions must satisfy classical measure axioms, resulting in contextual behavior needing to be expressed in topological defects in events, resulting in non-trivial monodromy. We utilize such defects to connect contextuality with non-embeddability and to construct a generalized Vorob'ev theorem, a result regarding the inevitability of noncontextuality. We identify in this formalism the contextual fraction for models with outcome-determinism, and a pathway to address disturbance in ontological models as non-trivial transition maps. We also discuss how the two views for encoding contextuality relate to interpretations of quantum theory.