Aspects ofC₃-symmetric calorons from numerical Nahm transform

Abstract

Calorons are finite action solutions to the anti-selfdual Yang-Mills equations on R 3 × S 1 . They are generally constructed by the so called Nahm construction. We perform the numerical Nahm transform for the Nahm data of 3-calorons with C3-symmetry, which do not have the monopole limits. Dissimilar to the cases of having monopole limits, we can trace the zero-circumference limit of S 1 . It is found that the action density of the calorons tends to fade away as S 1 shrinks.

Symmetric calorons and the rotation map

“…for G ∈ GL(k, C). We denote by M j k (ρ 2 ) the set of matrices in M k satisfying (29)- (32), and denote by M j k (ρ) the set of matrices in M j k (ρ 2 ) satisfying (23)- (28). The purpose behind the second consideration is due to the inclusion M j k (ρ) ⊂ M j k (ρ 2 ), which is seen by setting…”

“…A similar argument holds for D p . The statement about eigenvectors follows from equations (31) and (32).…”

“…Since, by lemma 22, G is necessarily of standard form, we may solve (29)- (32) and hence write an ansatz for a representative of M j k (ρ) in terms of these objects:…”

“…For higher charge cases, the success has so far been only with numerical methods. For calorons, this has been done only for very special cases [28,32], but many examples have been considered in the case of monopoles [37]. Recent work by Braden and Enolski [8] has shed light on how to construct monopoles explicitly from Nahm data, yet it is still unknown whether these results apply to calorons.…”

Symmetric calorons of higher charges and their large period limits

“…The cyclic 3-calorons have a particular feature that they do not have a BPS monopole limit, thus the Nahm data are unrelated to that of BPS monopoles. In [40], the numerical Nahm transform of the C 3 -symmetric calorons are performed.…”