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.

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When instantons are put into the Higgs phase, vortices are attached to instantons. We construct such composite solitons as 1=4 BPS states in five-dimensional supersymmetric UN C gauge theory with N F N C fundamental hypermultiplets. We solve the hypermultiplet BPS equation and show that all 1=4 BPS solutions are generated by an N C N F matrix which is holomorphic in two complex variables, assuming the vector multiplet BPS equation does not give additional moduli. We determine the total moduli space formed by topological sectors patched together and work out the multi-instanton solution inside a single vortex with complete moduli. Small instanton singularities are interpreted as small sigma-model lump singularities inside the vortex. The relation between monopoles and instantons in the Higgs phase is also clarified as limits of calorons in the Higgs phase. Another type of instantons stuck at an intersection of two vortices and dyonic instantons in the Higgs phase are also discussed.

Dense quantum chromodynamic matter accommodates various kind of topological solitons such as vortices, domain walls, monopoles, kinks, boojums and so on. In this review, we discuss various properties of topological solitons in dense quantum chromodynamics (QCD) and their phenomenological implications. A particular emphasis is placed on the topological solitons in the color-flavor-locked (CFL) phase, which exhibits both superfluidity and superconductivity. The properties of topological solitons are discussed in terms of effective field theories such as the Ginzburg-Landau theory, the chiral Lagrangian, or the Bogoliubov-de Gennes equation. The most fundamental string-like topological excitations in the CFL phase are the non-Abelian vortices, which are 1/3 quantized superfluid vortices and color magnetic flux tubes. These vortices are created at a phase transition by the Kibble-Zurek mechanism or when the CFL phase is realized in compact stars, which rotate rapidly. The interaction between vortices is found to be repulsive and consequently a vortex lattice is formed in rotating CFL matter. Bosonic and fermionic zero-energy modes are trapped in the core of a non-Abelian vortex and propagate along it as gapless excitations. The former consists of translational zero modes (a Kelvin mode) with a quadratic dispersion and CP 2 Nambu-Goldstone gapless modes with a linear dispersion, associated with the CFL symmetry spontaneously broken in the core of a vortex, while the latter is Majorana fermion zero modes belonging to the triplet of the symmetry remaining in the core of a vortex. The low-energy effective theory of the bosonic zero modes is constructed as a non-relativistic free complex scalar field and a relativistic CP 2 model in 1+1 dimensions. The effects of strange quark mass, electromagnetic interactions and non-perturbative quantum corrections are taken into account in the CP 2 effective theory. Various topological objects associated with non-Abelian vortices are studied; colorful boojums at the CFL interface, the quantum color magnetic monopole confined by vortices, which supports the notion of quark-hadron duality, and Yang-Mills instantons inside a non-Abelian vortex as lumps are discussed. The interactions between a non-Abelian vortex and quasi-particles such as phonons, gluons, mesons, and photons are studied. As a consequence of the interaction with photons, a vortex lattice behaves as a cosmic polarizer. As a remarkable consequence of Majorana fermion zero modes, non-Abelian vortices are shown to behave as a novel kind of non-Abelian anyon. In the order parameters of chiral symmetry breaking, we discuss fractional and integer axial domain walls, Abelian and non-Abelian axial vortices, axial wall-vortex composites, and Skyrmions.

We make a detailed study of the moduli space of winding number two (k = 2) axially symmetric vortices (or equivalently, of co-axial composite of two fundamental vortices), occurring in U(2) gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto-Tong and Auzzi-Shifman-Yung. We find that it is a weighted projective space W CP 2 (2,1,1) ≃ CP 2 /Z 2 . This manifold contains an A 1 -type (Z 2 ) orbifold singularity even though the full moduli space including the relative position moduli is smooth. The SU(2) transformation properties of such vortices are studied. Our results are then generalized to U(N) gauge theory with N flavors, where the internal moduli space of k = 2 axially symmetric vortices is found to be a weighted Grassmannian manifold. It contains singularities along a submanifold.