Supersymmetry and geometry of hyperbolic monopoles

Abstract

This thesis studies the geometry of hyperbolic monopoles using supersymmetry in four and six dimensions. On the one hand, we show that starting with a four dimensional supersymmetric Yang-Mills theory provides the necessary information to study the geometry of the complex moduli space of hyperbolic monopoles. On the other hand, we require to start with a six dimensional supersymmetric Yang-Mills theory to study the geometry of the real moduli space of hyperbolic monopoles. In chapter two, we construct an off-shell supersymmetric Yang-Mills-Higgs theory with complex fields on three-dimensional hyperbolic space starting from an on-shell supersymmetric Yang-Mills theory on four-dimensional Euclidean space. We, then, show that hyperbolic monopoles coincide precisely with the configurations that preserve one half of the supersymmetry. In chapter three, we explore the geometry of the moduli space of hyperbolic monopoles using the low energy linearization of the field equations. We find that the complexified tangent bundle to the hyperbolic moduli space has a 2-sphere worth of integrable structures that act complex linearly and behave like unit imaginary quaternions. Moreover, we show that these complex structures are parallel with respect to the Obata connection, which implies that the geometry of the complexified moduli space of hyperbolic monopoles is hypercomplex. We also show, as a requirement of analysing the geometry, that there is a one-to-one correspondence between the number of solutions of the linearized Bogomol’nyi equation on hyperbolic space and the number of solutions of the Dirac equation in the presence of hyperbolic monopole. In chapter four and five, we shift the focus to supersymmetric Yang-Mills theories in six dimensional Minkowskian spacetime. Via dimensional reduction we construct a supersymmetric Yang-Mills Higgs theory on R3 with real fields which we then promote to H3. Under certain supersymmetric constraints, we show that hyperbolic monopoles configurations of this theory preserve, again, one half of the supersymmetry. Then, through investigating the geometry of the moduli space we showthat the moduli space is described by real coordinate functions (zero modes), and we construct two sets of 2-sphere of real complex structures that act linearly on the tangent bundle of the moduli space, but don’t behave like unit quaternions. This result coincides with the result of Bielawski and Schwachhöfer, who called this new type of geometry pluricomplex geometry. Finally, we show that in the limiting case, when the radius of curvature H3 is set to infinity, the geometry becomes hyperkähler which is the geometry of the moduli space of Euclidian monopoles.