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 [1] (hep-th/0404198) [2] (hep-th/0405194) [3] (hep-th/0412024) [4] (hep-th/0503033) [5] (hep-th/0505136), vortices [6] (hep-th/0511088) [7] (hep-th/0601181), domain wall webs [8] (hep-th/0506135) [9] (hep-th/0508241) [10] (hep-th/0509127), monopolevortex-wall systems [11] (hep-th/0405129) [12] (hep-th/0501207), instanton-vortex systems [13] (hep-th/0412048), effective Lagrangian on walls and vortices [14] (hep-th/0602289), classification of BPS equations [15] (hep-th/0506257), and Skyrmions [16] (hep-th/0508130). † In this paper we keep terminology of "instantons" for Yang-Mills instantons in four Euclidean space. They become particles in 4+1 dimensions.

Magnetic impurity inspired Abelian Higgs vortices

“…This phenomenon, known as the monopole confinement, was proposed by Mandelstam [23,24], Nambu [27], and 't Hooft [40,41], and has served as a profound source of thoughts [33] for the understanding of quark confinement where the underlying Abelian Higgs model platform is extended into various supersymmetric gauge field theory models and the classical Meissner effect is elevated into its supersymmetric versions so that the Nielsen-Olesen magnetic strings become colored ones [3,7,8,10,12,14,15,26,35,36]. See [9,11,21,34,37,38,44,45] for surveys and further literature.…”

Nonintegrability of self-dual Yang–Mills–Higgs system

“…This energy is minimized if the following BPS equations for non-Abelian vortices are satisfied [22,23,24,15]:…”

“…We rewrite the set of domain-wall equations to a second order ODE in the form like the Yang's equation [28,15]. The first of the domain-wall equation (3.25) can be integrated as 28) with an N C × N F matrix H 0 of integration constants, and S = S(y) ∈ GL(N C , C).…”

The Magnetic Monopoles Seventy-five Years Later

“…The matter equation can be solved [59]- [61] by use of the N × N moduli matrix H 0 (z) whose components are holomorphic functions of the complex coordinate z = x 1 + ix 2 ,…”

“…6). These cases have been discussed extensively [61]- [64], within the context of U(N) model of Subsection 6.3.…”