Implementation of an iterative map in the construction of (quasi)periodic instantons: chaotic aspects and discontinuous rotation numbers

TL;DR

This paper introduces an iterative complex map approach to study (quasi)periodic selfdual gauge fields, revealing chaotic behavior and discontinuous rotation numbers, with implications for instantons and monopoles in four-dimensional Euclidean space.

Contribution

It develops a novel iterative map method to analyze (quasi)periodic instantons, linking complex dynamics with gauge field topological properties and exploring chaotic aspects.

Findings

Chaotic features in the iterative map are linked to gauge field behavior.

Discontinuous rotation numbers are observed in quasiperiodic regimes.

The method provides new insights into instanton and monopole limits.

Abstract

An iterative map of the unit disc in the complex plane (Appendix) is used to explore certain aspects of selfdual, four dimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits. The iterations do not correspond directly to a discretized time evolution of the gauge fields. They are implemented in an indirect fashion. First, (t,r,\theta,\phi) being the standard coordinates, the (r,t) half plane is mapped on the unit disc in an appropriate way. This provides an (r,t) parametrization (Sec.1) of Z_0, the starting point of the iterations and makes the iterates increasingly complex functions of r and t. These are then incorporated as building blocks in the generating function of the fields (Sec.2). We explain (starting in Sec.1 and at different stages) in what sense and to what extent some remarkable features of our map (indicated in the title) are thus carried over into the continuous time development of the fields. Special features for quasiperiodicity are studied (Sec.3). Spinor solutions (Sec.4) and propagators (Sec.5) are discussed from the point of view of the mapping. Several possible generalizations are indicated (Sec.6). Some broader topics are discussd in conclusion (Sec.7).