Magnetically charged calorons with non-trivial holonomy

Abstract

Instantons in pure Yang-Mills theories on partially periodic space R 3 × S 1 are usually called calorons. The background periodicity brings on characteristic features of calorons such as nontrivial holonomy, which plays an essential role for confinement/deconfinement transition in pure Yang-Mills gauge theory. For the case of gauge group S U(2), calorons can be interpreted as composite objects of two constituent "monopoles" with opposite magnetic charges. There are often the cases that the two monopole charges are unbalanced so that the calorons possess net magnetic charge in R 3 . In this paper, we consider several mechanism how such net magnetic charges appear for certain types of calorons through the ADHM/Nahm construction with explicit examples. In particular, we construct analytically the gauge configuration of the (2, 1)-caloron with U(1)-symmetry, which has intrinsically magnetic charge. 1

Skyrmions from calorons

“…Since the Skyrme-instanton boundary condition (65) for non-zero µ breaks the gauge symmetry to U (1), there may be a relationship with the U (1)gauged skyrmions found in [42], which also exhibit an axial symmetry, in addition to a non-zero dipole moment, which is similar to the interpretation of (1, 1)-calorons as two oppositely charged magnetic monopoles [26]. Another axially symmetric caloron appears explicitly in [56]. This caloron is possibly more interesting as it has a mixture of instanton and magnetic charge -it is a (2, 1)-caloron.…”

“…Before we prove the existence of the bound (57), we shall first confirm that C(α) given by (58) is real. Indeed, we clearly have κ 0 > 0 by (37), and (56) shows that the numerator of C(α) 2 is positive. For the denominator, we have by (56) that Thus comparing this with (58), we see that C(α) 2 > 0, so C(α) ∈ R. Now consider the quadratic form…”

Symmetric calorons of higher charges and their large period limits

“…The class of calorons with non-trivial holonomy are crucial in the analysis for the confinement/deconfinement phase transition of quarks in QCD [44,45,46,47,48,49,50,51,52]. The higher charge generalization for the non-trivial holonomy calorons is quite restricted except for the cases of instanton charge two [33,34,35,36]. This situation is reflecting the fact that the introduction of non-trivial holonomy in the Nahm data is extremely complicated problem, i.e., it is hard to construct the corresponding Nahm data.…”

“…Here we give an overview to these examples. The simplest case of N = 2 calorons with axial symmetry are the Harrington-Shepard type 2calorons [26,34], whose analytic description of the gauge fields is given in [35]. For the case of non-trivial holonomy, the Nahm data of the charge 2-calorons with axial symmetry are considered in [33,34,36,37], employing the BPS 2-monopole data as their bulk data, which are given in trigonometric functions.…”

A model for gauged skyrmions with low binding energies

“…In this section we shall reproduce the KvBLL calorons from their Nahm data, that is, we shall apply the Nahm transform explicitly. This was also reviewed recently in [45] using a different, but equivalent formulation of the Nahm transform. Our calculation is most similar to that of [33].…”