# On the Tetrahedrally Symmetric Monopole

@article{Braden2009OnTT, title={On the Tetrahedrally Symmetric Monopole}, author={Harry W. Braden and V. Z. Enolski}, journal={Communications in Mathematical Physics}, year={2009}, volume={299}, pages={255-282} }

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…

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## References

SHOWING 1-10 OF 30 REFERENCES

### Symetric Monopoles

- Mathematics
- 1995

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

- Mathematics
- 2006

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

- Mathematics
- 1994

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

- Mathematics
- 2007

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

- Mathematics
- 1983

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

- Physics
- 1989

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

- Mathematics
- 1989

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…

### FINITE-GAP INTEGRATION OF THE SU(2) BOGOMOLNY EQUATIONS

- MathematicsGlasgow Mathematical Journal
- 2009

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

- Mathematics, Physics
- 1994

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…