A geometric phase approach to quark confinement from stochastic gauge-geometry flows

TL;DR

This work advances a geometric-phase perspective on confinement by coupling a stochastic gauge–geometry flow to Yang–Mills dynamics, where topology changes in hadronic background manifolds dynamically produce center vortices that generate the Wilson-loop area law. Through a three-pronged framework—stochastic geometry and gauge flows, stochastic quantization, and a topology-driven confinement mechanism—the authors connect singular gauge transformations, instanton sectors, and cobordisms to magnetic vortices and the Aharonov–Bohm phase. A key result is an explicit scaling relation for topology changes tied to the Chern–Simons invariant of limiting 3-manifolds, allowing a Planck-scale starting point to evolve toward QCD-relevant scales (∼165 MeV). The proposal culminates in a concrete, topologically flavored pathway to confinement via knotted Wilson loops and linking data, offering a bridge between geometry, topology, and nonperturbative QCD behavior.

Abstract

We apply a stochastic version of the geometric (Ricci) flow, complemented with the stochastic flow of the gauge Yang--Mills sector, in order to seed the chromo-magnetic and chromo-electric vortices that source the area-law for QCD confinement. The area-law is the key signature of quark confinement in Yang--Mills gauge theories with a non-trivial center symmetry. In particular, chromo-magnetic vortices enclosed within the chromo-electric Wilson loops instantiate the area-law asymptotic behaviour of the Wilson loop vacuum expectation values. The stochastic gauge-geometry flow is responsible for the topology changes that induce the appearance of the vortices. When vortices vanish, due to topology changes in the manifolds associated to the hadronic ground states, the evaluation of the Wilson loop yields a dependence on the length of the path, hence reproducing the perimeter law of the hadronic (Higgs) phase of real QCD. Confinement, instead, is naturally achieved within this context as a by-product of the topology change of the manifold over which the dynamics of the Yang--Mills fields is defined. It is then provided by the Aharonov--Bohm effect induced by the concatenation of the compact chromo-electric and chromo-magnetic fluxes originated by the topology changes. The stochastic gauge-geometry flow naturally accomplishes a treatment of the emergence of the vortices and the generation of turbulence effects. Braiding and knotting, resulting from topology changes, namely stochastic fluctuations, stabilize the chromo-magnetic vortices. Finally, we observe that dimensional transmutation for the Yang-Mills fields can be derived from the scaling property of the geometric part of the stochastic flow. Specifically, a relation that involves the infrared equilibrium limit of the Planck constant can be derived that yields the correct order of magnitude for $Λ_{\rm QCD}$.