Cyclic Calorons

TL;DR

This paper studies cyclic calorons with $C_N$-symmetry, revealing their connection to the periodic Toda lattice, expressing solutions via elliptic theta functions, and analyzing their limits and scale parameters.

Contribution

It introduces a novel analysis of $C_N$-symmetric calorons using Sutcliffe's ansatz, linking their bulk data to the periodic Toda lattice and providing explicit elliptic function solutions.

Findings

Calorons' bulk data exhibit periodic Toda lattice structure.

Solutions are expressed in terms of elliptic theta functions.

Scale parameters of $C_3$ calorons have upper bounds, preventing large monopole limits.

Abstract

The Nahm data of periodic instantons, often called calorons, with spatial $C_N$-symmetries are considered, by applying Sutcliffe's ansatz for the monopoles with $C_N$-symmetries. The bulk data of calorons are shown to enjoy the periodic Toda lattice, and the solutions are given in terms of elliptic theta functions. The case of N=3 calorons are investigated in detail. It is found that the "scale parameters" of these calorons have upper bounds in their values, so that they do not have the large scale, or monopole, limits. The instanton limit of the $C_3$-symmetric caloron is obtained.