FINITE-GAP INTEGRATION OF THE <i>SU</i>(2) BOGOMOLNY EQUATIONS

Braden, H. W.; Enolskii, V. Z.

doi:10.1017/s0017089508004758

Cited by 5 publications

(3 citation statements)

References 7 publications

(15 reference statements)

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“…We are seeking the θ-functional integration of the Weyl equations themselves. In [BE09] we use an ansatz of Nahm [Nah82] which reduces the integration of the 2n-th order system of ODE of the Weyl equations to an n-th order ODE system that is equivalent to the linear spectral problem of the Lax representation for Nahm equation. Therefore the algebro-geometric solution to the Weyl equation is given in terms of a Baker-Akhiezer function of the Nahm equation whose spectral parameter is a function of monopole coordinates.…”

Section: Humbert Varietymentioning

confidence: 99%

On the Existence of Non-Abelian Monopoles: the Algebro-Geometric Approach

Braden¹,

Enolskii²

2010

AIP Conference Proceedings

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We develop the Atiyah-Drinfeld-Manin-Hitchin-Nahm construction to study SU (2) non-abelian charge 3 monopoles within the algebro-geometric method. The method starts with finding an algebraic curve, the monopole spectral curve, subject to Hitchin's constraints. We take as the monopole curve the genus four curve that admits a C 3 symmetry, η 3 + αηζ 2 + βζ 6 + γζ 3 − β = 0, with real parameters α, β and γ. In the case α = 0 we prove that the only suitable values of γ/β are ±5 √ 2 (β is given below) which corresponds to the tetrahedrally symmetric solution. We then extend this result by continuity to non-zero values of the parameter α and find finally a new one-parameter family of monopole curves with C 3 symmetry.2. The Atiyah-Drinfeld-Manin-Hitchin-Nahm construction. Although the Bogomolny equation is a first order partial differential equation in R 3 few explicit solutions 1 arXiv:1009.3837v1 [math-ph]

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Section: Humbert Varietymentioning

confidence: 99%

On the Existence of Non-Abelian Monopoles: the Algebro-Geometric Approach

Braden¹,

Enolskii²

2010

AIP Conference Proceedings

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“…Bringing methods from integrable systems to bear upon the construction of solutions to Nahm's equations for the gauge group SU (2) Ercolani and Sinha [ES89] later showed how one could solve (a gauge transform of) the Nahm equations in terms of a Baker-Akhiezer function for the curve Ĉ. Thus given a curve Ĉ the machinery of integrable systems allows one (in principle) to construct solutions to Nahm's equations and thence monopoles [BE09A].…”

Section: Introductionmentioning

confidence: 99%

On charge-3 cyclic monopoles

2011

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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.

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“…Neither of these constraints is our focus here, 1 and our attention is instead on the data needed for reconstructing the solutions of the Nahm equation. Reconstructing the monopole data from a solution of the Nahm equations was considered in [10].…”

Section: Introductionmentioning

confidence: 99%