The geometry and dynamics of magnetic monopoles.

*(English)*Zbl 0671.53001
Princeton, NJ: Princeton University Press. 133 p. $ 25.00 (1988).

The Bogomolny equation describes the concept of a static magnetic monopole in non-Abelian gauge theories and is the dimensional reduction to \(R^3\) of the self-dual Yang-Mills equation. The main purpose of the present book is to study the Riemannian geometry of the space of all k- monopoles. The data for a monopole on \(R^3\) consist of a connection \(A\) and a Higgs field \(\Phi\) which take their values in the Lie algebra of the group \(\mathrm{SU}(2)\) and satisfy the equation \(D_A\Phi = {}^*F_A,\) where \(F_A\) is the curvature of the connection and \(^*\) denotes the duality operator on \(R^3\). Every monopole has a magnetic charge \(k\) and a well-defined centre. If we identify gauge-equivalent monopoles of charge \(k\) and fixed centre we obtain the moduli space \(M^0_k.\)

The second important space \(M_k\) depends on a fixed direction \(v\in R^3\) and we allow for the identification only gauge transformations which tend to the identity as \(t.v\to \infty\). There exists a diffeomorphism \(M_k\approx R_k\) between \(M_k\) and the space \(R_k\) of all rational functions \(S(z)\) of degree \(k\) with \(S(\infty)=0\) (Donaldson). The \(L^2\)-norm defines a Riemannian metric and \(M_k\) is a complete Riemannian manifold. Since the space of all pairs \((A,\Phi)\) can be identified with the space of functions taking values in \(\mathfrak{su}(2)\otimes H\) it follows that the tangent space to \(M_k\) is a vector space over \(H\). Furthermore, \(M_k\) is a hyper-Kähler manifold. The \(k\)-fold covering \(\tilde M_k\) of \(M_k\) decomposes as an isometric product of hyper-Kähler manifolds \(\tilde M_k=\tilde M^0_k\times (S^1\times R^3)\) and \(\tilde M^0_k\) is a simply-connected and irreducible Riemannian manifold.

Chapter 5 of the present book contains the twistor construction for an arbitrary hyper-Kähler manifold (Salamon, Hitchin, Karlhede, Lindström, Roček) as well as the description of the twistor space of the hyper-Kähler manifold \(M_k\) (Hitchin, Hurtubise). The remaining chapters concentrate entirely on the case \(k=2\) and the study of the 4-dimensional hyper-Kähler manifold \(M^0_2\). The reader can find here completely new results concerning the Riemannian geometry of \(M^0_2\).

First of all the authors look at the surfaces \[ \Sigma_1=\{S(z)=1/(z^2-u),\quad u\in C\},\quad \Sigma_{23}=\{S(z)=z/(z^ 2-v),\quad v\in C^*\} \] in \(R_2\approx M_2\) and determine in this way the conformal structure of \(M^0_2\). On the other hand, since \(M^0_2\) is a 4-dimensional anti-self-dual Einstein manifold admitting \(\mathrm{SO}(3)\) as a group of isometries, the metric of \(M^0_2\) belongs to the Taub-NUT family and the Einstein equations for such a metric reduce to a system of ordinary differential equations (Gibbons, Pope). Using this information the authors derive the explicit form of the metric on the monopole space \(M^0_2\) (chapter 11). It turns out that the surfaces \(\Sigma_1\), \(\Sigma_{23}\) are totally geodesic, \(\Sigma_1\) has positive curvature and the second surface has negative curvature. The geodesic motion on these surfaces is then studied in chapter 13. The last chapter contains an interpretation of this geodesic motion in terms of the scattering of monopoles, regarded as point-particles.

The second important space \(M_k\) depends on a fixed direction \(v\in R^3\) and we allow for the identification only gauge transformations which tend to the identity as \(t.v\to \infty\). There exists a diffeomorphism \(M_k\approx R_k\) between \(M_k\) and the space \(R_k\) of all rational functions \(S(z)\) of degree \(k\) with \(S(\infty)=0\) (Donaldson). The \(L^2\)-norm defines a Riemannian metric and \(M_k\) is a complete Riemannian manifold. Since the space of all pairs \((A,\Phi)\) can be identified with the space of functions taking values in \(\mathfrak{su}(2)\otimes H\) it follows that the tangent space to \(M_k\) is a vector space over \(H\). Furthermore, \(M_k\) is a hyper-Kähler manifold. The \(k\)-fold covering \(\tilde M_k\) of \(M_k\) decomposes as an isometric product of hyper-Kähler manifolds \(\tilde M_k=\tilde M^0_k\times (S^1\times R^3)\) and \(\tilde M^0_k\) is a simply-connected and irreducible Riemannian manifold.

Chapter 5 of the present book contains the twistor construction for an arbitrary hyper-Kähler manifold (Salamon, Hitchin, Karlhede, Lindström, Roček) as well as the description of the twistor space of the hyper-Kähler manifold \(M_k\) (Hitchin, Hurtubise). The remaining chapters concentrate entirely on the case \(k=2\) and the study of the 4-dimensional hyper-Kähler manifold \(M^0_2\). The reader can find here completely new results concerning the Riemannian geometry of \(M^0_2\).

First of all the authors look at the surfaces \[ \Sigma_1=\{S(z)=1/(z^2-u),\quad u\in C\},\quad \Sigma_{23}=\{S(z)=z/(z^ 2-v),\quad v\in C^*\} \] in \(R_2\approx M_2\) and determine in this way the conformal structure of \(M^0_2\). On the other hand, since \(M^0_2\) is a 4-dimensional anti-self-dual Einstein manifold admitting \(\mathrm{SO}(3)\) as a group of isometries, the metric of \(M^0_2\) belongs to the Taub-NUT family and the Einstein equations for such a metric reduce to a system of ordinary differential equations (Gibbons, Pope). Using this information the authors derive the explicit form of the metric on the monopole space \(M^0_2\) (chapter 11). It turns out that the surfaces \(\Sigma_1\), \(\Sigma_{23}\) are totally geodesic, \(\Sigma_1\) has positive curvature and the second surface has negative curvature. The geodesic motion on these surfaces is then studied in chapter 13. The last chapter contains an interpretation of this geodesic motion in terms of the scattering of monopoles, regarded as point-particles.

Reviewer: Thomas Friedrich (Berlin)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C20 | Global Riemannian geometry, including pinching |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53C80 | Applications of global differential geometry to the sciences |

81T08 | Constructive quantum field theory |

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |