Interpolated Topology Change in a Spin Cobordism and the Chiral Weyl Curvature Diagnostic

TL;DR

This paper develops a framework for regulated topology change in Lorentzian quantum gravity using spin cobordisms and introduces a curvature-based diagnostic sensitive to chirality and topological complexity.

Contribution

It introduces the Chiral Weyl Curvature Diagnostic and a smooth spin cobordism framework allowing topology change with controlled curvature and chirality.

Findings

Constructed a spin cobordism between a wormhole and Minkowski spacetime.

The curvature response in the throat is Weyl-dominated.

Parity-odd Weyl term distinguishes chiral configurations.

Abstract

Topology change in Lorentzian quantum gravity demands geometric regulators that control curvature, spin structure, and chirality during nontrivial interpolations. We develop a framework for regulated topology change based on smooth Lorentzian spin cobordisms with interpolating metrics, allowing a transient failure of global hyperbolicity while preserving smoothness, Lorentz signature, and spin compatibility. Within this framework we introduce the Chiral Weyl Curvature Diagnostic, a curvature-based functional that weights topology-changing geometries by conformal curvature, spin admissibility, and topological complexity. The diagnostic functional is built from Weyl curvature invariants and includes a parity-odd dual Weyl term that is sensitive to geometric chirality. Spin consistency is enforced via a Stiefel-Whitney constraint, ensuring that only physically admissible cobordisms contribute. As an example, we construct a smooth spin cobordism between a Morris-Thorne wormhole and asymptotically flat Minkowski spacetime. In the throat region the curvature response is shown to be Weyl-dominated, and the parity-odd Weyl contribution sharply distinguishes chiral knotted embeddings while vanishing for amphichiral configurations. We then show that braids provide a natural language of throat dynamics: evolving wormhole throats trace time-dependent braid movies, with elementary braid generators representing the fundamental topological operations of the cobordism. Replacing crossing number by a braid-based complexity refines the diagnostic functional to operate at the level of these elementary exchanges and extends it naturally to multithroat and networked configurations.