Group Theory of Non-Abelian Vortices

We investigate the structure of the moduli space of multiple BPS non-Abelian vortices in U(N) gauge theory with N fundamental Higgs fields, focusing our attention on the action of the exact global (color-flavor diagonal) SU(N) symmetry on it. The moduli space of a single non-Abelian vortex, CP(N-1), is spanned by a vector in the fundamental representation of the global SU(N) symmetry. The moduli space of winding-number k vortices is instead spanned by vectors in the direct-product representation: they decompose into the sum of irreducible representations each of which is associated with a Young tableau made of k boxes, in a way somewhat similar to the standard group composition rule of SU(N) multiplets. The K\"ahler potential is exactly determined in each moduli subspace, corresponding to an irreducible SU(N) orbit of the highest-weight configuration.Comment: LaTeX 46 pages, 4 figure

Zero-modes of Non-Abelian Solitons in Three Dimensional Gauge Theories

We study non-Abelian solitons of the Bogomol'nyi type in N=2 (d=2+1) supersymmetric Chern-Simons (CS) and Yang-Mills (YM) theory with a generic gauge group. In CS theory, we find topological, non-topological and semi-local (non-)topological vortices of non-Abelian kinds in unbroken, broken and partially broken vacua. We calculate the number of zero-modes using an index theorem and then we apply the moduli matrix formalism to realize the moduli parameters. For the topological solitons we exhaust all the moduli while we study several examples of the non-topological and semi-local solitons. We find that the zero-modes of the topological solitons are governed by the moduli matrix H_0 only and those of the non-topological solitons are governed by both H_0 and the gauge invariant field \Omega. We prove local uniqueness of the master equation in the YM case and finally, compare all results between the CS and YM theories.Comment: 54 pages, 1 figur

Vortices on Orbifolds

The Abelian and non-Abelian vortices on orbifolds are investigated based on the moduli matrix approach, which is a powerful method to deal with the BPS equation. The moduli space and the vortex collision are discussed through the moduli matrix as well as the regular space. It is also shown that a quiver structure is found in the Kahler quotient, and a half of ADHM is obtained for the vortex theory on the orbifolds as the case before orbifolding.Comment: 25 pages, 4 figures; references adde

Supersymmetry Breaking on Gauged Non-Abelian Vortices

There are a large number of systems characterized by a completely broken gauge symmetry, but with an unbroken global color-flavor diagonal symmetry, i.e., systems in the so-called color-flavor locked phase. If the gauge symmetry breaking supports vortices, the latter develop non-Abelian orientational zero-modes and become non-Abelian vortices, a subject of intense study in the last several years. In this paper we consider the effects of weakly gauging the full exact global flavor symmetry in such systems, deriving an effective description of the light excitations in the presence of a vortex. Surprising consequences are shown to follow. The fluctuations of the vortex orientational modes get diffused to bulk modes through tunneling processes. When our model is embedded in a supersymmetric theory, the vortex is still 1/2 BPS saturated, but the vortex effective action breaks supersymmetry spontaneously.Comment: Latex, 24 pages, 1 figur

Vortex counting from field theory

The vortex partition function in 2d N = (2,2) U(N) gauge theory is derived from the field theoretical point of view by using the moduli matrix approach. The character for the tangent space at each moduli space fixed point is written in terms of the moduli matrix, and then the vortex partition function is obtained by applying the localization formula. We find that dealing with the fermionic zero modes is crucial to obtain the vortex partition function with the anti-fundamental and adjoint matters in addition to the fundamental chiral multiplets. The orbifold vortex partition function is also investigated from the field theoretical point of view.Comment: 21 pages, no figure

Non-Abelian vortex dynamics: Effective world-sheet action

The low-energy vortex effective action is constructed in a wide class of systems in a color-flavor locked vacuum, which generalizes the results found earlier in the context of U(N) models. It describes the weak fluctuations of the non-Abelian orientational moduli on the vortex worldsheet. For instance, for the minimum vortex in SO(2N) x U(1) or USp(2N) x U(1) gauge theories, the effective action found is a two-dimensional sigma model living on the Hermitian symmetric spaces SO(2N)/U(N) or USp(2N)/U(N), respectively. The fluctuating moduli have the structure of that of a quantum particle state in spinor representations of the GNO dual of the color-flavor SO(2N) or USp(2N) symmetry, i.e. of SO(2N) or of SO(2N+1). Applied to the benchmark U(N) model our procedure reproduces the known CP(N-1) worldsheet action; our recipe allows us to obtain also the effective vortex action for some higher-winding vortices in U(N) and SO(2N) theories.Comment: LaTeX, 25 pages, 0 figure

Solitons in Supersymmety Breaking Meta-Stable Vacua

In recently found supersymmetry-breaking meta-stable vacua of the supersymmetric QCD, we examine possible exsitence of solitons. Homotopy groups of the moduli space of the meta-stable vacua show that there is no nontrivial soliton for SU(N_c) gauge group. When U(1)_B symmetry present in the theory is gauged, we find non-BPS solitonic (vortex) strings whose existence and properties are predicted from brane configurations. We obtain explicit classical solutions which reproduce the predicitions. For SO(N_c) gauge group, we find there are solitonic strings for N = N_f-N_c+4 = 2, and Z_2 strings for the other N. The strings are meta-stable as they live in the meta-stable vacua.Comment: 30 pages, 14 figures, Comments on stability of non-BPS vortices are added, Comments on sigma model solitons are added, An appendix is adde

Type I Non-Abelian Superconductors in Supersymmetric Gauge Theories

Non-BPS non-Abelian vortices with CP^1 internal moduli space are studied in an N=2 supersymmetric U(1) x SU(2) gauge theory with softly breaking adjoint mass terms. For generic internal orientations the classical force between two vortices can be attractive or repulsive. On the other hand, the mass of the scalars in the theory is always less than that of the vector bosons; also, the force between two vortices with the same CP^1 orientation is always attractive: for these reasons we interpret our model as a non-Abelian generalization of type I superconductors. We compute the effective potential in the limit of two well separated vortices. It is a function of the distance and of the relative colour-flavour orientation of the two vortices; in this limit we find an effective description in terms of two interacting CP^1 sigma models. In the limit of two coincident vortices we find two different solutions with the same topological winding and, for generic values of the parameters, different tensions. One of the two solutions is described by a CP^1 effective sigma model, while the other is just an Abelian vortex without internal degrees of freedom. For generic values of the parameters, one of the two solutions is metastable, while there are evidences that the other one is truly stable.Comment: 35 pages, 8 figures. v2: fixed typos and added small comments, v3 removed an unecessary figur

Electronic States in Silicon Quantum Dots: Multivalley Artificial Atoms

Electronic states in silicon quantum dots are examined theoretically, taking into account a multivalley structure of the conduction band. We find that (i) exchange interaction hardly works between electrons in different valleys. In consequence electrons occupy the lowest level in different valleys in the absence of Hund's coupling when the dot size is less than 10 nm. High-spin states are easily realized by applying a small magnetic field. (ii) When the dot size is much larger, the electron-electron interaction becomes relevant in determining the electronic states. Electrons are accommodated in a valley, making the highest spin, to gain the exchange energy. (iii) In the presence of intervalley scattering, degenerate levels in different valleys are split. This could result in low-spin states. These spin states in multivalley artificial atoms can be observed by looking at the magnetic-field dependence of peak positions in the Coulomb oscillation.Comment: 18 pages, 5 figure

Static Interactions of non-Abelian Vortices

Interactions between non-BPS non-Abelian vortices are studied in non-Abelian U(1) x SU(N) extensions of the Abelian-Higgs model in four dimensions. The distinctive feature of a non-Abelian vortex is the presence of an internal CP^{N-1} space of orientational degrees of freedom. For fine-tuned values of the couplings, the vortices are BPS and there is no net force between two static parallel vortices at arbitrary distance. On the other hand, for generic values of the couplings the interactions between two vortices depend non-trivially on their relative internal orientations. We discuss the problem both with a numerical approach (valid for small deviations from the BPS limit) and in a semi-analytical way (valid at large vortex separations). The interactions can be classified with respect to their asymptotic property at large vortex separation. In a simpler fine-tuned model, we find two regimes which are quite similar to the usual type I/II Abelian superconductors. In the generic model we find other two new regimes: type I*/II*. Unlike the type I (type II) case, where the interaction is always attractive (repulsive), the type I* and II* have both attractive and repulsive interactions depending on the relative orientation. We have found a rich variety of interactions at small vortex separations. For some values of the couplings, a bound state of two static vortices at a non-zero distance exists.Comment: 36 pages, 13 figures; v2 a small comment and a reference adde