Black magic session of concordance: Regge mass spectrum from Casson's invariant

TL;DR

This paper explores the connection between knot theory, 3-manifold curvature, and hadron mass spectra, proposing a novel topological interpretation of Regge trajectories via Casson's invariant.

Contribution

It introduces a new topological framework linking knot concordance and Casson's surgery formula to the Regge mass spectrum of hadrons, integrating physics and mathematics.

Findings

Regge trajectories are described by Casson's invariant.

Dynamically generated knots correspond to 3-manifolds of non-negative curvature.

The approach aligns with experimental data on hadron spectra.

Abstract

Recently, there had been a great deal of interest in obtaining and describing of all kinds of knots in links in hydrodynamics, electrodynamics, non Abelian gauge field theories and gravity. Although knots and links are observables of the Chern-Simons (C-S) functional, the dynamical conditions for their generation lie outside of the scope of the C-S theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S^3 are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space thus far is attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of non negative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of non negative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/ linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is an adaptation of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson's surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew-Frautschi (C-F) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding C-F plots. The physics behind Casson's surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons