Integrability and Hyperbolic Monopoles


Harry Braden

Spectral Curves, hyperbolic monopoles and the Chiral Potts Model.

The construction of hyperbolic monopoles via spectral curves will be reviewed. Various connections between the spectral curves of hyperbolic monopoles and those of the Chiral Potts model will be described.

Moustafa Gharamti

Geometry and dynamics of hyperbolic monopoles.

First, we construct a supersymmetic Yang Mills Higgs theory on H^3, and we show that half of the supersymmetry is preserved by the supersymmetric hyperbolic monopole solutions. Then, through studying the low energy dynamics of these solutions we obtain a supermultiplet of zero modes that will yield the defining equations of the geometry of the hyperbolic monopoles via the on shell closure of its algebra.

Nuno M Romão

Spectral curves of hyperbolic monopoles.

Twistor theory for monopoles in hyperbolic 3-space was first draughted by Atiyah in his seminal paper of 1984, but some of the details were only much later worked out. One early insight was that twistor data for Euclidean monopoles should be obtainable from hyperbolic twistor data in a zero-curvature limit; thus, in this sense, the latter contain more information. After work by Murray and Singer, it came to light that the properties of a general hyperbolic monopole could be extracted from a spectral curve embedded in the surface P^1xP^1, but in practice this can be very hard to achieve. In this talk, I will consider the problem of computing the vacuum/asymptotic magnitude of the Higgs field (the so-called monopole mass) m>0 from a given spectral curve. I shall discuss two classes of examples where this calculation can be made quite explicit: (i) 2-monopoles; (ii) monopoles with Platonic symmetries. I will also illustrate how Euclidean twistor data can be recovered essentially by differentiating hyperbolic data in the large m limit, and how for rational values of m one can expect a simplification of the hyperbolic spectral curves. (Joint work with Paul Norbury.)

Paul M Sutcliffe

Platonic hyperbolic monopoles.

Atiyah has shown that circle invariant instantons correspond to hyperbolic monopoles. The ADHM construction of instantons is used to obtain a number of explicit examples of hyperbolic monopoles, often with some Platonic symmetry. A key ingredient is the identification of a new set of constraints on ADHM instanton data to ensure the circle invariance.

Robert Weston

An Introduction to the Chiral Potts Model.

I will present a simple summary of the definition and origins of the Chiral Potts model. This will include a description of the Boltzmann weights and the quantum spin chain formulation, and the connection of the former to the representation theory of quantum affine sl_2.