On the Tetrahedrally Symmetric Monopole

  title={On the Tetrahedrally Symmetric Monopole},
  author={Harry W. Braden and V. Z. Enolski},
  journal={Communications in Mathematical Physics},
We study SU(2) BPS monopoles with spectral curves of the form η3+χ(ζ6+bζ3−1) = 0. Previous work has established a countable family of solutions to Hitchin’s constraint that L2 was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The… 

Cyclic Monopoles, Affine Toda and Spectral Curves

We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha

On charge-3 cyclic monopoles

We determine the spectral curve of charge-3 BPS su(2) monopoles with C3 cyclic symmetry. The symmetry means that the genus 4 spectral curve covers a (Toda) spectral curve of genus 2. A well adapted

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

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,

On Charge 3 Symmetric Monopoles

Monopoles are solutions of an SU(2) gauge theory in $\mathbb{R}^{3}$ satisfying a lower bound for energy and certain asymptotic conditions, which translate as topological properties encoded in their

The Construction of Monopoles

We show that the Higgs and gauge fields for a BPS monopole may be constructed directly from the spectral curve without having to solve the gauge constraint needed to obtain the Nahm data. The result

The charge $2$ monopole via the ADHMN construction

Recently we have shown how one may use use integrable systems techniques to implement the ADHMN construction and obtain general analytic formulae for the charge n su(2) Euclidean monopole. Here we do

Construction of Nahm data and BPS monopoles with continuous symmetries

. We study solutions to Nahm’s equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data,

Algebro-geometric solutions to the modified Sawada-Kotera hierarchy

Based on solving the Lenard recursion equation and the zero-curvature equation, we derive the modified Sawada-Kotera (SK) hierarchy associated with a 3 × 3 matrix spectral problem. Resorting to the

Klein's curve

Riemann surfaces with symmetries arise in many studies of integrable systems. We illustrate new techniques in investigating such surfaces by means of an example. By giving a homology basis well

Spectral curves are transcendental

  • H. Braden
  • Mathematics
    Letters in Mathematical Physics
  • 2021
Some arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $$n\ge 2$$ n ≥ 2 Euclidean BPS monopole is not defined over $$\overline{\mathbb {Q}}$$ Q ¯ if



Symetric Monopoles

We discuss SU (2) Bogomolny monopoles of arbitrary charge k invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations

Remarks on the complex geometry of the 3-monopole

We develop the Ercolani-Sinha construction of SU(2) monopoles and make this effective for (a five parameter family of centred) charge 3 monopoles. In particular we show how to solve the

Symmetric Monopoles

We discuss the spectral curves and rational maps associated with SU (2) Bogomolny monopoles of arbitrary charge k . We describe the effect on the rational maps of inverting monopoles in the plane with

Monopoles, Curves and Ramanujan

We develop the Ercolani-Sinha construction of SU(2) monopoles and make this effective for (a five parameter family of centred) charge 3 monopoles. In particular we show how to solve the

On the construction of monopoles

We show that any self-dual SU (2) monopole may be constructed either by Ward's twistor method, or Nahm's use of the ADHM construction. The common factor in both approaches is an algebraic curve whose

On the Construction of Monopoles for

Monopoles are solutions of a first order partial differential equation — the Bogomolny equation. They can be thought of as approximated by static, magnetic particles in R. In these notes we will

Monopoles and Baker functions

The work in this paper pertains to the solutions of Nahm's equations, which arise in the Atiyah-Drinfield-Hitchin-Manin-Nahm construction of solutions to the Bogomol'nyi equations for static


Abstract The Atiyah–Drinfeld–Hitchin–Manin–Nahm (ADHMN) construction of magnetic monopoles is given in terms of the (normalizable) solutions of an associated Weyl equation. We focus here on solving

Algebro-geometric approach to nonlinear integrable equations

A brief but self-contained exposition of the basics of Riemann surfaces and theta functions prepares the reader for the main subject of this text, namely the application of these theories to solving