Doubly Periodic Instantons and their Constituents

TL;DR

This paper uses the Nahm transform to analyze doubly periodic SU(2) instantons, revealing their core constituents and how they merge or split depending on the torus geometry, with implications for understanding instanton structure.

Contribution

It introduces a detailed analysis of doubly periodic instantons with radial symmetry, identifying constituent locations and their behavior on different torus geometries using the Nahm transform.

Findings

Identified instanton core constituents at special Nahm zero modes

Showed cores merge into monopole-like objects on elongated tori

Demonstrated splitting into twisted charge 1/2 instantons on half-tori

Abstract

Using the Nahm transform we investigate doubly periodic charge one SU(2) instantons with radial symmetry. Two special points where the Nahm zero modes have softer singularities are identified as the locations of instanton core constituents. For a square torus this constituent picture is closely reflected in the action density. In rectangular tori with large aspect ratios the cores merge to form monopole-like objects. For particular values of the parameters the torus can be cut in half yielding two copies of a twisted charge 1/2 instanton. These findings are illustrated with plots of the action density within a two-dimensional slice containing the constituents.