Dynamics of periodic monopoles

Abstract

BPS monopoles which are periodic in one of the spatial directions correspond, via a generalized Nahm transform, to solutions of the Hitchin equations on a cylinder. A one-parameter family of solutions of these equations, representing a geodesic in the 2monopole moduli space, is constructed numerically. It corresponds to a slow-motion dynamical evolution, in which two parallel monopole chains collide and scatter at right angles.

Integrable (2k)-Dimensional Hitchin Equations

“…The aim now is to describe some solutions of (10); these will therefore also be solutions of the other systems (8), and (7) in the k = 2 case. The equations (10) or (11) are defined on any k-dimensional complex manifold, and in general one may also allow singularities. For example, in the k = 1 case on a compact Riemann surface of genus g, smooth solutions of (2) exist only when g ≥ 2; on the 2-sphere and the 2-torus, solutions necessarily have singularities [12].…”

Geometry of solutions of Hitchin equations on ${{\mathbb{R}}^{\mathbf{2}}}$

“…The moduli space M is the space of moduli, for fixed values of the parameters. A useful choice of gauge was described in [12,16]; it may be summarized as follows. The gauge field F is taken to be in the σ 3 -direction in su (2), and Φ has the form…”

“…The set R of rotationally-symmetric solutions is a 1-parameter family of fields interpolating between (12) and the non-explicit field S + ∩ R. This family can be obtained as solutions of a boundary-value problem for a single real-valued function, as follows. We work in the gauge (4, 5), with µ + = z 2 and µ − = −1.…”

“…The righthand diagram shows the result of solving the boundary-value problem (13) numerically, and plots π −1 |F | d 2 x as a function of B. The limit B → ∞ is the degenerate solution (12) where F = 0, whereas B = 0 is the solution depicted in the left-hand plot.…”

Dynamics of monopole walls

“…* email address: rafael.maldonado@durham.ac.uk † email address: richard.ward@durham.ac.uk of the centred 2-monopole moduli space is different, and is called ALG [4]. In this case, the generalized Nahm transform has been used to describe some of the geodesics on the moduli space, and their interpretation in terms of periodic monopole dynamics [5,6].This paper focuses on the doubly-periodic case, namely BPS monopoles on T 2 × R, also referred to as monopole walls [7,8]. An N-monopole field which is periodic in the x-and y-directions may be viewed as a set of N monopole walls, each extended in the xy-direction.…”