Solitons in the Higgs phase: the moduli matrix approach

Abstract

We review our recent work on solitons in the Higgs phase. We use U ( N C ) gauge theory with N F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices, which are the only elementary solitons in the Higgs phase, are found in terms of the moduli matrix. Stable monopoles and instantons can exist in the Higgs phase if they are attached by vortices to form composite solitons. The moduli spaces of these composite solitons are also worked out in terms of the moduli matrix. Webs of walls can also be formed with characteristic difference between Abelian and non-Abelian gauge theories. Instanton–vortex systems, monopole–vortex–wall systems, and webs of walls in Abelian gauge theories are found to admit negative energy objects with the instanton charge (called intersectons), the monopole charge (called boojums) and the Hitchin charge, respectively. We characterize the total moduli space of these elementary as well as composite solitons. In particular the total moduli space of walls is given by the complex Grassmann manifold SU ( N F )/[ SU ( N C ) × SU ( N F − N C ) × U (1)] and is decomposed into various topological sectors corresponding to boundary condition specified by particular vacua. The moduli space of k vortices is also completely determined and is reformulated as the half ADHM construction. Effective Lagrangians are constructed on walls and vortices in a compact form. We also present several new results on interactions of various solitons, such as monopoles, vortices and walls. Review parts contain our works on domain walls (Isozumi Y et al 2004 Phys. Rev. Lett. 93 161601 ( Preprint hep-th/0404198), Isozumi Y et al 2004 Phys. Rev. D 70 125014 ( Preprint hep-th/0405194), Eto M et al 2005 Phys. Rev. D 71 125006 ( Preprint hep-th/0412024), Eto M et al 2005 Phys. Rev. D 71 105009 ( Preprint hep-th/0503033), Sakai N and Yang Y 2005 Comm. Math. Phys. (in press) ( Preprint hep-th/0505136)), vortices (Eto M et al 2005 Phys. Rev. Lett. 96 161601 ( Preprint hep-th/0511088), Eto M et al 2006 Phys. Rev. D 73 085008 ( Preprint hep-th/0601181)), domain wall webs (Eto M et al 2005 Phys. Rev. D 72 085004 ( Preprint hep-th/0506135), Eto M et al 2006 Phys. Lett. B 632 384 ( Preprint hep-th/0508241), Eto M et al 2005 AIP Conf. Proc. 805 354 ( Preprint hep-th/0509127)), monopole–vortex–wall systems (Isozumi Y et al 2005 Phys. Rev. D 71 065018 ( Preprint hep-th/0405129), Sakai N and Tong D 2005 J. High Energy Phys. JHEP03(2005)019 ( Preprint hep-th/0501207)), instanton–vortex systems (Eto M et al 2005 Phys. Rev. D 72 025011 ( Preprint hep-th/0412048)), effective Lagrangian on walls and vortices (Eto M et al 2006 Phys. Rev. D (in press) ( Preprint hep-th/0602289)), classification of BPS equations (Eto M et al 2005 Preprint hep-th/0506257) and Skyrmions (Eto M et al 2005 Phys. Rev. Lett. 95 252003 ( Preprint hep-th/0508130)).