Fractal Fiber Bundles and Exotic Holonomy in Quantum Spacetime Geometry

Abstract

We propose a novel differential geometric framework for modeling the microstructure of spacetime near the Planck scale, where classical smooth manifold assumptions are expected to break down. Motivated by both theoretical limitations in quantum gravity and observational anomalies in cosmology and black hole physics, we introduce the concept of fractal fiber bundles: geometric structures defined over fractal or scale-dependent base spaces, equipped with scale-dependent connections. These generalized bundles allow for the formulation of exotic holonomy, where parallel transport gives rise to groupoid-valued structures that may exhibit non-associativity and encode physically meaningful phase data. We construct a generalized Ricci flow for evolving geometric structures across scales, and prove a convergence theorem showing the emergence of smooth Riemannian geometry as a large-scale limit of fractal data. Additionally, we demonstrate that nontrivial holonomy behavior can persist at arbitrarily small scales in fractal geometries, leading to a departure from classical Lie group-based gauge symmetry. The framework developed here not only deepens the mathematical foundations of quantum geometry, but also offers potential insights into early-universe cosmology, black hole interiors, and observable signatures such as gravitational wave echoes and dimensional flow.

Dhaka Univ. J. Sci. 74(2): 193–200, 2026 (July)