ADHM Polytopes

TL;DR

This paper develops a framework for constructing ADHM data for Yang-Mills instantons with four-dimensional polytope symmetries, extending previous methods and illustrating with explicit examples.

Contribution

It introduces a novel framework linking ADHM data construction to extended Dynkin diagrams and McKay correspondence for polytope symmetries, including new explicit instanton solutions.

Findings

Constructed ADHM data for instantons with polytope symmetries

Identified lowest charges for given symmetries

Confirmed polytope structures via topological charge density plots

Abstract

We discuss the construction of ADHM data for Yang-Mills instantons with the symmetries of the regular polytopes in four dimensions. We show that the case of the pentatope can be studied using a simple modification of the approach previously developed for platonic data. For the remaining polytopes, we describe a framework in which the building blocks of the ADHM data correspond to the edges in the extended Dynkin diagram that arises via the McKay correspondence. These building blocks are then assembled into ADHM data through the identification of pairs of commuting representations of the associated binary polyhedral group. We illustrate our procedure by the construction of ADHM data associated with the pentatope, the hyperoctahedron and the 24-cell, with instanton charges 4, 7 and 23, respectively. Furthermore, we show that within our framework these are the lowest possible charges with these symmetries. Plots of topological charge densities are presented that confirm the polytope structure and the relation to JNR instanton data is clarified.