Construction of cyclic calorons

Abstract

Abstract. Analytic Nahm data of calorons, i.e., Yang-Mills instantons on R 3 × S 1 , with spatial CN -symmetries are considered. The Nahm equations for the bulk data are reduced to the periodic Toda equation by applying the CN -symmetric ansatz for monopoles by Sutcliffe. It is found that the bulk data are solved by elliptic theta functions which enjoy the hermiticity and the reality conditions. The defining relation to the boundary data are given and the C3-symmetric Nahm data are found as an example. IntroductionThe Atiyah-Drinfeld-Hitchin-Manin (ADHM) [1] and the Nahm [2] constructions are powerful tool for getting the information of the moduli space to the anti-selfdual (ASD) Yang-Mills solitons, such as the instantons and the Bogomoln'yi-Prasad-Sommerfield (BPS) monopoles [3]. The principal ingredients of the ADHM/Nahm construction are the ADHM/Nahm data, respectively. One can obtain the gauge fields through the so called Nahm transform once the ADHM/Nahm data are determined. Although the ADHM/Nahm data are significant objects in the analysis on the ASD Yang-Mills solitons, the construction of the exact forms to the ADHM/Nahm data is unexplored field.Among the ASD Yang-Mills solitons, calorons, or periodic instantons, are quite interesting object [4]. They are ASD instantons on R 3 × S 1 , so that if we take the circumference of S 1 be infinitely large, the calorons are reduced to instantons. On the other hand, it is naturally expected to be the BPS monopoles, if we take the ratio of the size of the calorons to the circumference be infinity. Hence, calorons are expected to give the connection between instantons and BPS monopoles. There are articles demonstrating this interpolation in analytic [5,6,7,8,9].In this paper, we give an outline for the construction of the Nahm data of calorons with instanton charge N associated to spatial C N -symmetries around an axis. It is known that the Nahm data of calorons are composed of the bulk data and the boundary data, respectively. We will apply the C N -symmetric ansatz for monopole Nahm data given by Sutcliffe [10,11] as the bulk Nahm data of calorons, and find that the defining equations for the bulk data are the wellknown periodic Toda equations. As we will see, it is not appropriate to fix the boundary data in the basis on which the Toda equations are solved. Hence we will make a unitary transformation into another basis, in which the reality conditions are manifest. We will give a conjecture on the unitary transformations between these basis. We restrict ourselves to consider the calorons of the SU (2) gauge theory, and of trivial holonomy cases.

Aspects ofC3-symmetric calorons from numerical Nahm transform

“…In Fig.1, we present the action isosurface plots for values of µ 0 ∈ M 0 = (0.0, 0.5), M 1 = (0.5, 1.0). For the sector M 0 , we use ϑ 3 , while for the sector M 1 we use ϑ 0 , according to (19). One can manifestly see that all the densities exhibit C 3 -symmetry.…”

“…However, it is known that the calorons from Corrigan-Fairlie-t'Hooft ansatz [16], i.e., the Harrington-Shepard type, of higher instanton charges do not have the monopole limits [17]. In addition, it has been shown that there is a one-parameter family of 3-calorons Nahm data with C 3 -symmetry which do not have monopole limits [18,19]. From these facts, it will be interesting to ask that under which conditions the calorons have monopole limits.…”