M2-branes and instantons

In the first part of this thesis we discuss some of the issues arising in extending the ABJM action of multiple M2-branes to include couplings to the background 3-form field. These couplings are analogous to the Myers-Chern-Simons terms of the multiple D2-brane action. We review and extend previous results to include terms which are quadratic in the background 3-form. These are fixed by requiring that we recover the correct terms after using the novel Higgs mechanism to reduce the ABJM action to the multiple D2-brane action. We also discuss the problem of constructing a gauge invariant pull-back in the ABJM action. In the second part of this thesis, we begin by exploring the low energy dynamics of charge two instantons in $\SU(2)$ five dimensional Yang-Mills via the moduli space approximation of Manton. We also investigate dyonic instantons which have an excited scalar field and create a potential on the moduli space. In Chapter 5 we explicitly calculate the moduli space metric and potential for charge two (dyonic) instantons. These calculations are performed by using the ADHM construction. In Chapter 6 we perform a numerical study of the low-energy dynamics of instantons and dyonic instantons. We see that instantons undergo right-angled scattering and understand this analytically in terms of symmetries of the underlying ADHM data. We also present a comprehensive study of the scattering behaviour of instantons and dyonic instantons under various initial conditions. Finally we exhibit some examples of closed geodesics on the moduli space of dyonic instantons, and geodesics which hit the moduli space singularities in finite time. In Chapter 7 we investigate instantons with a large amount of symmetry. We first understand how the action of a symmetry on an instanton is lifted to the underlying ADHM data. The transformation of the ADHM data must be undoable by a transformation which leaves the instanton invariant, and we search for symmetric instantons by finding such transformation matrices that are representations of the symmetry group. With this method we are able to find solutions to the ADHM constraints that describe instantons with the symmetries of the 5-cell, 16-cell and 24-cell, with charge 4, 7, and 23 respectively. Finally, we see that these solutions correspond to solutions which can be constructed from the JNR ansatz