Brane Potentials and Moduli Spaces

It is shown that the supergravity moduli spaces of D1-D5 and D2-D6 brane systems coincide with those of the Coulomb branches of the associated non-abelian gauge theories. We further discuss situations in which worldvolume brane actions include a potential term generated by probing certain supergravity backgrounds. We find that in many cases, the appearance of the potential is due to the application of the Scherk-Schwarz mechanism. We give some examples and discuss the existence of novel supersymmetric brane configurations.Comment: 26 pages, phyzzx.te

Mirror Symmetry and Toric Geometry in Three-Dimensional Gauge Theories

We study three dimensional gauge theories with N=2 supersymmetry. We show that the Coulomb branches of such theories may be rendered compact by the dynamical generation of Chern-Simons terms and present a new class of mirror symmetric theories in which both Coulomb and Higgs branches have a natural description in terms of toric geometry.Comment: 16 pages. LaTeX file+two figures. Final version to appear in JHE

Non-Abelian Berry Phases and BPS Monopoles

We study a simple quantum mechanical model of a spinning particle moving on a sphere in the presence of a magnetic field. The system has two ground states. As the magnetic field is varied, the ground states mix through a non-Abelian Berry phase. We show that this Berry phase is the path ordered exponential of the smooth SU(2) 't Hooft-Polyakov monopole. We further show that, by adjusting a potential on the sphere, the monopole becomes BPS and obeys the Bogomolnyi equations. For this choice of potential, it turns out that there is a hidden supersymmetry underlying the system and the Bogomolnyi equations are analogous to the tt* equations of Cecotti and Vafa. We conjecture that the Bogomolnyi equations also govern the Berry phase of N=(2,2) supersymmetric sigma models with other target spaces.Comment: 15 pages. v2: footnotes added to point the reader towards later developments where conjectures made in this paper were subsequently proven. A shortened version of this paper was published in PRL under the title "Scheme for Building a 't Hooft-Polyakov Monopole

Vortices and Impurities

We describe the BPS dynamics of vortices in the presence of impurities. We argue that a moduli space of solitons survives the addition of both electric and magnetic impurities. However, dynamics on the moduli space is altered. In the case of electric impurities, the metric remains unchanged but the dynamics is accompanied by a connection term, acting as an effective magnetic field over the moduli space. We give an expression for this connection and compute the vortex-impurity bound states in simple cases. In contrast, magnetic impurities distort the metric on the moduli space. We show that magnetic impurities can be viewed as vortices associated to a second, frozen, gauge group. We provide a D-brane description of the dynamics of vortices in product gauge groups and show how one can take the limit such that a subset of the vortices freeze.Comment: 19 pages, 2 figures. v2: version to appear in JHE

Monopoles, Vortices, Domain Walls and D-Branes: The Rules of Interaction

Non-abelian gauge theories in the Higgs phase admit a startling variety of BPS solitons. These include domain walls, vortex strings, confined monopoles threaded on vortex strings, vortex strings ending on domain walls, monopoles threaded on strings ending on domain walls, and more. After presenting a self-contained review of these objects, including several new results on the dynamics of domain walls, we go on to examine the possible interactions of solitons of various types. We point out the existence of a classical binding energy when the string ends on the domain wall which can be thought of as a BPS boojum with negative mass. We present an index theorem for domain walls in non-abelian gauge theories. We also answer questions such as: Which strings can end on which walls? What happens when monopoles pass through domain walls? What happens when domain walls pass through each other?Comment: 46 Pages (35 pages of body + appendices). 12 Figures. v2: References added. Minor correction to index theorem in appendix