Walls and vortices in supersymmetric non-abelian gauge theories

We review recent results on the BPS multi-wall solutions in supersymmetric U(N_C) gauge theories in five dimensions with N_F(>N_C) hypermultiplets in the fundamental representation. Total moduli space of the BPS non-Abelian walls is found to be the complex Grassmann manifold SU(N_F)/[SU(N_C)xSU(N_F-N_C)xU(1)]. Exact solutions are obtained with full generic moduli for infinite gauge coupling. A 1/4 BPS equation is also solved, giving vortices together with the non-Abelian walls and monopoles in the Higgs phase attached to the vortices. The full moduli space of the 1/4 BPS solutions is found to be holomorphic maps from a complex plane to the wall moduli space.Comment: 10pages, 2figures, Contribution to the proceedings of ``NathFest'' at PASCOS conference, Northeastern University, Boston, Ma, August 200

Solitons in Supersymmetric Gauge Theories: Moduli Matrix Approach

We review our recent works on solitons in U(Nc) gauge theories with Nf (>Nc) Higgs fields in the fundamental representation, which possess eight supercharges. The moduli matrix is proposed as a crucial tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Since vacua are in the Higgs phase, we find domain walls (kinks) and vortices as the only elementary solitons. Stable monopoles and instantons can exist as composite solitons with vortices attached. Webs of walls are also found as another composite soliton. The moduli space of all these elementary as well as composite solitons are found in terms of the moduli matrix. The total moduli space of walls is given by the complex Grassmann manifold SU(Nf)/[SU(Nc)x SU(Nf-Nc) x U(1)] and is decomposed into various topological sectors corresponding to boundary conditions specified by particular vacua. We found charges characterizing composite solitons contribute negatively (either positively or negatively) in Abelian (non-Abelian) gauge theories. Effective Lagrangians are constructed on walls and vortices in a compact form. The power of the moduli matrix is illustrated by an interaction rule of monopoles, vortices, and walls, which is difficult to obtain in other methods. More thorough description of the moduli matrix approach can be found in our review article (hep-th/0602170).Comment: 14 pages, 9 figures, proceedings of CAQC

Moduli Space of Non-Abelian Vortices

We completely determine the moduli space M_{N,k} of k-vortices in U(N) gauge theory with N Higgs fields in the fundamental representation. Its open subset for separated vortices is found as the symmetric product (C x CP^{N-1})^k / S_k. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. Relation to Kahler quotient construction is discussed.Comment: 11 pages, no figures, references added, v3: typos corrected, references added, the final version in PR

Non-Abelian Vortices on Cylinder -- Duality between vortices and walls

We investigate vortices on a cylinder in supersymmetric non-Abelian gauge theory with hypermultiplets in the fundamental representation. We identify moduli space of periodic vortices and find that a pair of wall-like objects appears as the vortex moduli is varied. Usual domain walls also can be obtained from the single vortex on the cylinder by introducing a twisted boundary condition. We can understand these phenomena as a T-duality among D-brane configurations in type II superstring theories. Using this T-duality picture, we find a one-to-one correspondence between the moduli space of non-Abelian vortices and that of kinky D-brane configurations for domain walls.Comment: 33 pages, 17 figures, v2: references added, typos corrected, the final version published in PR

Non-Abelian Walls in Supersymmetric Gauge Theories

The Bogomol'nyi-Prasad-Sommerfield (BPS) multi-wall solutions are constructed in supersymmetric U(N_C) gauge theories in five dimensions with N_F(>N_C) hypermultiplets in the fundamental representation. Exact solutions are obtained with full generic moduli for infinite gauge coupling and with partial moduli for finite gauge coupling. The generic wall solutions require nontrivial configurations for either gauge fields or off-diagonal components of adjoint scalars depending on the gauge. Effective theories of moduli fields are constructed as world-volume gauge theories. Nambu-Goldstone and quasi-Nambu-Goldstone scalars are distinguished and worked out. Total moduli space of the BPS non-Abelian walls including all topological sectors is found to be the complex Grassmann manifold SU(N_F) / [SU(N_C) x SU(N_F-N_C) x U(1)] endowed with a deformed metric.Comment: 62 pages, 17 figures, the final version in PR

Instantons in the Higgs Phase

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 U(Nc) gauge theory with Nf(>=Nc) fundamental hypermultiplets. We solve the hypermultiplet BPS equation and show that all 1/4 BPS solutions are generated by an Nc x Nf 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.Comment: 32 pages, 6 figures, typos corrected, comments and references adde