Cyclic Monopoles, Affine Toda and Spectral Curves

Braden, H. W.

doi:10.1007/s00220-011-1347-1

Cited by 11 publications

(12 citation statements)

References 18 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The work of Braden and Sutcliffe on cyclic monopoles [7,38] was very influential in furthering the understanding of monopole moduli spaces, so there is hence much benefit in studying cyclic calorons for similar purposes. For calorons, we may consider euclidean cyclic subgroups of S 1 ×S 1 , that is, ones which act temporally as well as in space.…”

Section: Cyclic Caloronsmentioning

confidence: 99%

Symmetric calorons and the rotation map

Cork

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

We study SU (2) calorons, also known as periodic instantons, and consider invariance under isometries of S 1 × R 3 coupled with a non-spatial isometry called the rotation map. In particular, we investigate the fixed points under various cyclic symmetry groups. Our approach utilises a construction akin to the ADHM construction of instantons -what we call the monad matrix data for calorons -derived from the work of Charbonneau and Hurtubise. To conclude, we present an example of how investigating these symmetry groups can help to construct new calorons by deriving Nahm data in the case of charge 2.

show abstract

Section: Cyclic Caloronsmentioning

confidence: 99%

Symmetric calorons and the rotation map

Cork

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…Now substituting (n, m) = (n 0 , n 1 , n 2 , n 3 , m 0 , m 1 , m 2 , m 3 ) directly into (2.21) and making use of (2.28, 2.30) we may deduce that the Ercolani-Sinha vector takes the form (n, m) = (n 0 , n, n, n, m 0 , m, m, m), and thus es is fixed under the spatial symmetry: σ( es) = es. (This result was obtained more generally via a different argument in [Bra10].) With this simplification we find the remaining equations encoded in the Ercolani-Sinha conditions take the form…”

Section: The Ercolani-sinha Conditionsmentioning

confidence: 64%

“…The consequences of assuming symmetry is that the spectral curve covers another curve, the quotient curve by the symmetry. In the case of cyclic symmetry we have an n-fold unbranched cover π :Ĉ → C of a hyperelliptic curve of genus n − 1 which is the spectral curve of the a n affine Toda system, and [Bra10] shows how both the constraints (H2,3) reduce to become constraints on the reduced curve. In particular…”

mentioning

confidence: 99%

“…Perhaps the nicest feature of this homology basis (and that induced on C) is that it enables us to make use of a remarkable factorisation theorem due to Accola and Fay [Acc71,Fay73] and also observed by Mumford. This allows the theta functions ofĈ to be described in terms of the theta functions of C for the parts of the Jacobian that are relevant for us [Bra10]. By the end of section 2 we have reduced the construction of cyclically invariant monopoles (with γ = 0) to questions about a genus two hyperelliptic curve C. Section 3 then discusses the restrictions the Ercolani-Sinha constraints place on C. The Ercolani-Sinha constraints are shown to reduce to the single constraint c dX/Y = 0 on a scaled form of C, Y 2 = (X 3 + a X + g) 2 + 4 (where (a, g) := (α/β 2/3 , γ/β)), and here c := π( es) is known.…”

mentioning

confidence: 99%

See 1 more Smart Citation

On charge-3 cyclic monopoles

2011

View full text Add to dashboard Cite

Abstract. We determine the spectral curve of charge 3 BPS su(2) monopoles with C 3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted homology basis is presented enabling the theta functions and monopole data of the genus 4 curve to be given in terms of genus 2 data. The Richelot correspondence, a generalization of the arithmetic mean, is used to solve for this genus 2 curve. Results of other approaches are compared.

show abstract

“…Now we determine the bulk Nahm data satisfying the C N -symmetric condition (5). For this purpose, we will apply the ansatz for the monopole Nahm data with C N -symmetries given by Sutcliffe over a decade ago [10,11], whose uniqueness is proved in [13]. The form of the N × N bulk data is given in terms of the functions f j (s) and p j (s) (j = 0, 1, 2 .…”

Section: N -Symmetric Ansatz For the Bulk Datamentioning

confidence: 99%

Construction of cyclic calorons

Muranaka

Nakamula

Sawado

2013

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

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.

show abstract

Cyclic Monopoles, Affine Toda and Spectral Curves

Cited by 11 publications

References 18 publications

Symmetric calorons and the rotation map

Symmetric calorons and the rotation map

On charge-3 cyclic monopoles

Construction of cyclic calorons

Contact Info

Product

Resources

About