Vortices carrying truly non-Abelian flux moduli, which do not dynamically reduce to Abelian vortices, are found in the context of softly-broken ${\cal N}=2$ supersymmetric chromodynamics (SQCD). By tuning the bare quark masses appropriately we identify the vacuum in which the underlying SU(N) gauge group is partially broken to $SU(n) \times SU(r) \times U(1)/{\mathbbm Z}_{K}$, where $K$ is the least common multiple of $(n, r)$, and with $N_{f}^{su(n)}=n$ and $N_{f}^{su(r)}=r$ flavors of light quark multiplets. At much lower energies the gauge group is broken completely by the squark VEVs, and vortices develop which carry non-Abelian flux moduli $CP^{n-1}\times CP^{r-1}$. For $n>r$ we argue that the SU(n) fluctuations become strongly coupled and Abelianize, while leaving weakly fluctuating $SU(r)$ flux moduli. This allows us to recognize the semi-classical origin of the light non-Abelian monopoles found earlier in the fully quantum-mechanical treatment of 4D SQCD.
We show that a given set of first class constraints becomes abelian if one maps each constraint to the surface of other constraints. There is no assumption that first class constraints satisfy a closed algebra. The explicit form of the projection map is obtained at least for irreducible first class constraints. Using this map we give a method to obtain gauge fixing conditions such that the set of abelian first class constraints and gauge fixing conditions satisfy the symplectic algebra.
In applications of AdS/CFT to condensed matter systems in 2+1 dimensions, the ABJM model is often used, however the condensed matter models are usually abelian and contain charged fields. We show that a naive reduction of the ABJM model to N=1 does not have the desired features, but we can find an abelian reduction that has most features, and we can also add fundamental fields to the ABJM model to obtain other models with similar properties.
We apply the BFFT formalism to a prototypical second-class system, aiming to convert its constraints from second- to first-class. The proposed system admits a consistent initial set of second-class constraints and an open potential function providing room for applications to mechanical models as well as field theory such as the non-linear sigma model. The constraints can be arbitralily non-linear, broadly generalizing previously known cases. We obtain a sufficient condition for which a simple closed expression for the Abelian converted constraints and modified involutive Hamiltonian can be ach…
We count Higgs "phase" BPS states of general non-Abelian quiver, possibly with loops, by mapping the problem to its Abelian, or toric, counterpart and imposing Weyl invariance later. Precise Higgs index computation is particularly important for quivers with superpotentials; the Coulomb "phase" index is recently shown to miss important BPS states, dubbed intrinsic Higgs states or quiver invariants. We demonstrate how the refined Higgs index is naturally decomposed to a sum over partitions of the charge. We conjecture, and show in simple cases, that this decomposition expresses the Higgs index a…
We generalize the non-abelianization of Gaiotto-Moore-Neitzke from the case of $SL(n)$ and $GL(n)$ to arbitrary reductive algebraic groups. This gives a map between a moduli space of certain $N$-shifted weakly $W$-equivariant $T$-local systems on an open subset of a cameral cover $\tilde{X}\rightarrow X$ to the moduli space of $G$-local systems on a punctured Riemann surface $X$. For classical groups, we give interpretations of these moduli spaces using spectral covers. Non-abelianization uses a set of lines on the Riemann surface $X$ called a spectral network, defined using a point in the Hit…
It is shown that SU(N) gauge theory coupled to adjoint Higgs can be explicitly re-written in terms of SU(N) gauge invariant dynamical variables with local physical interactions. The resultant theory has a novel compact abelian $U(1)^{(N - 1)}$ gauge invariance. The above abelian gauge invariance is related to the adjoint Higgs field and not to the gauge group SU(N). In this abelianized version the magnetic monopoles carrying the magnetic charges of $(N-1)$ types have a natural origin and therefore appear explicitly in the partition function as Dirac monopoles along with their strings. The gaug…
We explore the idea that in some class of strongly-coupled chiral $SU(N)$ gauge theories the infrared dynamics might be characterized by a bifermion condensate in the adjoint representation of the color gauge group. As an illustration, in this work we revisit an $SU(N)$ chiral gauge theory with Weyl fermions in a symmetric ($ψ$) and anti-antisymmetric ($χ$) tensor representations, together with eight fermions in the anti-fundamental representations ($η$), which we called $ψχη$ model in the previous investigations. We study the infrared dynamics of this system more carefully, by assuming dynami…
We present an embedding of the 3-dimensional relativistic Landau-Ginzburg model for condensed matter systems in an $\mathcal{N}=6$, $U(N)\times U(N)$ Chern-Simons-matter theory (the ABJM model) by consistently truncating the latter to an abelian effective field theory encoding the collective dynamics of ${\cal O}(N)$ of the ${\cal O}(N^2)$ modes. In fact, depending on the VEV on one of the ABJM scalars, a mass deformation parameter $μ$ and the Chern-Simons level number $k$, our abelianization prescription allows us to interpolate between the abelian Higgs model with its usual multi-vortex solu…
Recent development on non-Abeliann vortices and monopoles is reviewed with an emphasis on their relevance on confinement and duality. A very recent construction of non-Abelian vortices which do not dynamically Abelianize is crucial in this context.
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. In addition to the usual type I/II Abelian superconductors, we find other two new regimes: type I*/II*.
Various aspects including the construction and the symmetries of Abelian Chern-Simons vortices are reviewed. Extended version of the Lectures delivered at NIKHEF (Amsterdam), July 2006. Typos corrected, some refernces added.
After briefly reviewing the problems associated with non-Abelian monopoles, we turn our attention to the development in our understanding of non-Abelian {\it vortices} in the last several years. In the U(N) model with $N_{f}=N$ flavors in which they were first found, the fluctuations of the orientational modes along the vortex length and in time become strongly coupled at long distances. They effectively reduce to Abelian ANO vortices. We discuss then a very recent work on non-Abelian vortices with $CP^{n-1}\times CP^{r-1}$ orientational moduli, which, unlike the ones so far extensively studie…
We discuss some of the latest results concerning the non-Abelian vortices. The first concerns the construction of non-Abelian BPS vortices based on general gauge groups of the form G= G' x U(1). In particular detailed results about the vortex moduli space have been obtained for G'=SO(N) or USp(2N). The second result is about the "fractional vortices", i.e., vortices of the minimum winding but having substructures in the tension (or flux) density in the transverse plane. Thirdly, we discuss briefly the monopole-vortex complex.
We find, by an appropriate extension of the standard holographic superconductor setup, static bulk solutions which describe holographic duals to non-Abelian vortices. In the core of these vortices a scalar field condenses, breaking a non-Abelian global symmetry which leads to additional zero modes called orientational moduli. These moduli appear in the bulk as Goldstone bosons associated to the condensation of a neutral scalar field.
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-Ab…
We compute the Euler characteristics of the moduli spaces of abelian vortices on curves with nodal and cuspidal singularities. This generalizes our previous work where only nodes were taken into account. The result we obtain is again consistent with the expected reconciliation between the vortex picture of D2-D0 branes and the proposal by Gopakumar and Vafa.
We propose a novel and simple method to compute the partition function of statistical mechanics of local and semi-local BPS vortices in the Abelian-Higgs model and its non-Abelian extension on a torus. We use a D-brane realization of the vortices and T-duality relation to domain walls. We there use a special limit where domain walls reduce to gas of hard (soft) one-dimensional rods for Abelian (non-Abelian) cases. In the simpler cases of the Abelian-Higgs model on a torus, our results agree with exact results which are geometrically derived by an explicit integration over the moduli space of v…
Fermionic zero modes around non-abelian vortices are shown that they constitute two $N=2$, $d=1$ supersymmetric quantum mechanics algebras. These two algebras can be combined under certain circumstances to form a central charge extended $N=4$ supersymmetric quantum algebra. We thoroughly discuss the implications of the existence of supersymmetric quantum mechanics algebras, in the quantum Hilbert space of the fermionic zero modes.
We study periodic arrays of non-Abelian vortices in an $SU(N) \times U(1)$ gauge theory with $N_f$ flavors of fundamental matter multiplets. We carefully discuss the corresponding twisted boundary conditions on the torus and propose an ansatz to solve the first order Bogomolnyi equations which we find by looking to a bound of the energy. We solve the equations numerically and construct explicit vortex solutions.
The magnetic order of the triangular lattice with antiferromagnetic interactions is described by an SO(3) field and allows for the presence of Z2 magnetic vortices as defects. In this work we show how these Z2 vortices can be fitted into a local SU(2) gauge theory. We propose simple Ansatzes for vortex configurations and calculate their energies using well-known results of the Abelian gauge model. We comment on how Dzyaloshinskii-Moriya interactions could be derived from a non-Abelian gauge theory and speculate on their effect on non trivial configurations.
We calculate the Kähler potential for the Samols metric on the moduli space of Abelian Higgs vortices on $\mathbbm{R}^{2}$, in two different ways. The first uses a scaling argument. The second is related to the Polyakov conjecture in Liouville field theory. The Kähler potential on the moduli space of vortices on $\mathbbm{H}^{2}$ is also derived, and we are led to a geometrical reinterpretation of these vortices. Finally, we attempt to find the Kähler potential for vortices on $\mathbbm{R}^{2}$ in a third way by relating the vortices to SU(2) Yang-Mills instantons on $\mathbbm{R}^{2}\times S^{…
The vortex solutions of various classical planar field theories with (Abelian) Chern-Simons term are reviewed. Relativistic vortices, put forward by Paul and Khare, arise when the Abelian Higgs model is augmented with the Chern-Simons term. Adding a suitable sixth-order potential and turning off the Maxwell term provides us with pure Chern-Simons theory with both topological and non-topological self-dual vortices, as found by Hong-Kim-Pac, and by Jackiw-Lee-Weinberg. The non-relativistic limit of the latter leads to non-topological Jackiw-Pi vortices with a pure fourth-order potential. Explici…
We study non-Abelian Chern-Simon BPS-saturated vortices enjoying N=2 supersymmetry in d=2+1 dimensions, with generic gauge groups of the form U(1) x G', with G' being a simple group, allowing for orientational modes in the solutions. We will keep the group as general as possible and utilizing the powerful moduli matrix formalism to provide the moduli spaces of vortices and derive the corresponding master equations. Furthermore, we study numerically the vortices applying a radial Ansatz to solve the obtained master equations and we find especially a splitting of the magnetic fields, when the co…
We construct exact wavefunctions of two vortices on a plane, a single vortex on the cylinder and a vortex on the torus. In each case, the physics is shown to be equivalent to a particle moving in a covering space, something simple to solve in those examples. We describe how our solutions fit into the general theory of quantum mechanics of $N$ particles on a two-dimensional space and attribute our success to the fact that the fundamental groups are Abelian in those simple cases that we are considering.
Non-Abelian BPS vortex solutions are constructed in N=2 theories with gauge groups SO(N)\times U(1). The model has N_f flavors of chiral multiplets in the vector representation of SO(N), and we consider a color-flavor locked vacuum in which the gauge symmetry is completely broken, leaving a global SO(N)_{C+F} diagonal symmetry unbroken. Individual vortices break this symmetry, acquiring continuous non-Abelian orientational moduli. By embedding this model in high-energy theories with a hierarchical symmetry breaking pattern such as SO(N+2) --> SO(N)\times U(1) --> 1, the correspondence between …
We show that Abelian Higgs Models with dielectric function defined on the noncommutative plane enjoy self-dual vorticial solutions. By choosing a particular form of the dielectric function, we provide a family of solutions whose Higgs and magnetic fields interpolate between the profiles of the noncommutative Nielsen-Olesen and Chern-Simons vortices. This is done both for the usual $U(1)$ model and for the $SU(2)\times U(1)$ semilocal model with a doublet of complex scalar fields. The variety of known noncommutative self-dual vortices which display a regular behaviour when the noncommutativity …
We construct the general vortex solution in a fully-Higgsed, color-flavor locked vacuum of a non-Abelian gauge theory, where the gauge group is taken to be the product of an arbitrary simple group and U(1), with a Fayet-Iliopoulos term. The strict correspondence between vortices and lumps in the associated Non-Linear Sigma-Model which arise in the limit of strong coupling is pointed out. The construction of the vortex moduli space is derived here as a consequence of this correspondence.
We construct the general vortex solution in the fully-Higgsed, color-flavor locked vacuum of a non-Abelian gauge theory, where the gauge group is taken to be the product of an arbitrary simple group and U(1), with a Fayet-Iliopoulos term. The vortex moduli space is determined.
A particular dimensional reduction of SU(2N) Yang--Mills theory on $Σ\times S^2$, with $Σ$ a Riemann surface, yields an $S(U(N) \times U(N))$ gauge theory on $Σ$, with a matrix Higgs field. The SU(2N) self-dual Yang--Mills equations reduce to Bogomolny equations for vortices on $Σ$. These equations are formally integrable if $Σ$ is the hyperbolic plane, and we present a subclass of solutions.
We study three-dimensional Yang-Mills-Higgs theories with and without a Chern-Simons interaction. We find that these theories admit a rich spectrum of vortex solitons carrying both a topological charge and a global flavour charge. We further derive a low-energy description of the vortex dynamics from a gauged linear sigma model on the vortex worldline.
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 prob…
Non-Abelian BPS vortices in SO(N) x U(1) and USp(2N) x U(1) gauge theories are constructed in maximally color-flavor locked vacua. We study in detail their moduli and transformation properties under the exact symmetry of the system. Our results generalize non-trivially those found earlier in supersymmetric U(N) gauge theories. The structure of the moduli spaces turns out in fact to be considerably richer here than what was found in the U(N) theories. We find that vortices are generally of the semi-local type, with power-like tails of profile functions.
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 …
We study the interactions of non-abelian vortices in two spatial dimensions. These interactions have novel features, because the Aharonov-Bohm effect enables a pair of vortices to exchange quantum numbers. The cross section for vortex-vortex scattering is typically a multi-valued function of the scattering angle. There can be an exchange contribution to the vortex-vortex scattering amplitude that adds coherently with the direct amplitude, even if the two vortices have distinct quantum numbers. Thus two vortices can be ``indistinguishable'' even though they are not the same.
In this Letter we present new, genuinely non-Abelian vortex solutions in SU(2) Yang-Mills--Higgs theory with only one {\it isovector} scalar field. These non-Abelian solutions branch off their Abelian counterparts (Abrikosov-Nielsen-Olesen vortices) for precise values of the Higgs potential coupling constant $λ$. For all values of $λ$, their energies lie below those of the Abelian energy profiles, the latter being logarithmically divergent as $λ\to\infty$. The non-Abelian branches plateau in the limit $λ\to\infty$ and their number increases with $λ$, this number becoming infinite. For each vor…
We study domain walls and vortices in chiral symmetry breaking in a QCD-like theory with N flavors in the chiral limit. If the axial anomaly is absent, there exist stable Abelian axial vortices winding around the spontaneously broken U(1)_A symmetry and non-Abelian axial vortices winding around both the U(1)_A and non-Abelian SU(N) chiral symmetries. In the presence of the axial anomaly term, metastable domain walls are present and Abelian axial vortices must be attached by N domain walls, forming domain wall junctions. We show that a domain wall junction decays into N non-Abelian vortices att…
Many properties of the moduli space of abelian vortices on a compact Riemann surface are known. For non-abelian vortices the moduli space is less well understood. Here we consider non-abelian vortices on the Riemann sphere CP^1, and we study their moduli spaces near the Bradlow limit. We give an explicit description of the moduli space as a Kahler quotient of a finite-dimensional linear space. The dimensions of some of these moduli spaces are derived. Strikingly, there exist non-abelian vortex configurations on CP^1, with non-trivial vortex number, for which the moduli space is a point. This i…
In this brief report we consider a non-local Abelian Higgs model in the presence of a neutralizing uniform background charge. We show that such a system possesses vortices which key feature is a strong radial electric field. We estimate the basic properties of such an object and characteristic length scales in this model.
We find chiral non-Abelian vortices having windings only in one of the diquark condensations of left-handed and right-handed quarks in the color-flavor locked phase of dense QCD. They are the minimum vortices carrying half color magnetic fluxes of those of non-Abelian semi-superfluid vortices (color magnetic flux tubes) and 1/6 quantized superfluid circulations of Abelian superfluid vortices. These vortices carry ${\mathbb C}P^2$ orientational moduli in the internal space corresponding to their fluxes. The ${\mathbb C}P^2$ moduli of two chiral non-Abelian vortices with chiralities opposite to …
The moduli space metric and its Kahler potential for well-separated non-Abelian vortices are obtained in U(N) gauge theories with N Higgs fields in the fundamental representation.
Inspired by magnetic impurity considerations some broad classes of Abelian Higgs and Chern--Simons--Higgs BPS vortex equations are derived and analyzed.
We continue the study of $U(1)$ vortices with cholesteric vacuum structure. A new class of solutions is found which represent global vortices of the internal spin field. These spin vortices are characterized by a non-vanishing angular dependence at spatial infinity, or winding. We show that despite the topological $\mathbb{Z}_2$ behavior of $SO(3)$ windings, the topological charge of the spin vortices is of the $\mathbb{Z}$ type in the cholesteric. We find these solutions numerically and discuss the properties derived from their low energy effective field theory in $1+1$ dimensions.
Non-Abelian vortices in six spacetime dimensions are obtained for a supersymmetric U(N) gauge theory with N hypermultiplets in the fundamental representation. Massless (moduli) fields are identified and classified into Nambu-Goldstone and quasi-Nambu-Goldstone fields. Effective gauge theories for the moduli fields are constructed on the four-dimensional world volume of vortices. A systematic method to obtain the most general form of the effective Lagrangian consistent with symmetry is proposed. The moduli space for the multi-vortices is found to be a vector bundle over the complex Grassmann ma…
We extend our previous analysis on the exchange statistics of vortices having a single Dirac fermion trapped in each core, to the case where vortices trap two Dirac fermions with U(2) symmetry. Such a system of vortices with non-Abelian Dirac fermions appears in color superconductors at extremely high densities, and in supersymmetric QCD. We show that the exchange of two vortices having doublet Dirac fermions in each core is expressed by non-Abelian representations of a braid group, which is explicitly verified in the matrix representation of the exchange operators when the number of vortices …
In this note we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-abelian vortex equations over C . Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I .
We consider a generalization of abelian Chern-Simons-Higgs model by introducing a nonstandard kinetic term. In particular we show that the Bogomolnyi equations of the abelian Higgs theory may be obtained, being its solutions Nielsen-Olesen vortices with electric charge. In addition we study the self-duality equations for a generalized non-relativistic Maxwell-Chern-Simons model.
Topological superconductors classified as type D admit zero-energy Majorana fermions inside vortex cores, and consequently the exchange statistics of vortices becomes non-Abelian, giving a promising example of non-Abelian anyons. On the other hand, types C and DIII admit zero-energy Dirac fermions inside vortex cores. It has been long believed that an essential condition for the realization of non-Abelian statistics is non-locality of Dirac fermions made of two Majorana fermions trapped inside two well-separated vortices as in the case of type D. Contrary to this conventional wisdom, however, …
A detailed study of vortices is presented in Ginzburg-Landau (or Abelian Higgs) models with two complex scalars (order parameters) assuming a general U(1)$\times$U(1) symmetric potential. Particular emphasis is given to the case, when only one of the scalars obtains a vacuum expectation value (VEV). It is found that for a significantly large domain in parameter space vortices with a scalar field condensate in their core (condensate core, CC) coexist with Abrikosov-Nielsen-Olesen (ANO) vortices. Importantly CC vortices are stable and have lower energy than the ANO ones. Magnetic bags or giant v…
The scattering is studied using moduli space metric for well-separated vortices of non-Abelian vortices in (2+1)-dimensional U(N) gauge theories with N Higgs fields in the fundamental representation. Unlike vortices in the Abelian-Higgs model, dynamics of non-Abelian vortices has a lot of new features; The kinetic energy in real space can be transfered to that of internal orientational moduli and vice versa, the energy and charge transfer between two vortices, the scattering angle of collisions with a fixed impact parameter depends on the internal orientations, and some resonances appear due t…
We argue that in the infrared regime of continuum Yang-Mills theory, the possibility of a mass gap in the charged sector is closely associated with the center vortex sector. The analysis of the possible consequences of the ensembles of defects is done by showing that the description of center vortices and monopoles is naturally unified by means of a careful treatment of Cho decomposition. If on the one hand confinement is usually associated with monopole condensation in a compact abelian model, in this scenario, the previous decoupling of the off-diagonal degrees of freedom, for the abelian mo…
Non-Abelian vortices are topologically stable objects in the color-flavor locked (CFL) phase of dense QCD. We derive a dual Lagrangian starting with the Ginzburg-Landau effective Lagrangian for the CFL phase, and obtain topological interactions of non-Abelian vortices with quasiparticles such as $U(1)_B$ Nambu-Goldstone bosons (phonons) and massive gluons. We find that the phonons couple to the translational zero modes of the vortices while the gluons couple to their orientational zero modes in the internal space.
We investigate the collision dynamics of two non-Abelian vortices and find that, unlike Abelian vortices, they neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. Our predictions are verified using the model of the cyclic phase of a spin-2 spinor Bose-Einstein condensate.
Faddeev and Niemi (FN) have introduced an abelian gauge theory which simulates dynamical abelianization in Yang-Mills theory (YM). It contains both YM instantons and Wu-Yang monopoles and appears to be able to describe the confining phase. Motivated by the meson degeneracy problem in dynamical abelianization models, in this note we present a generalization of the FN theory. We first generalize the Cho connection to dynamical symmetry breaking pattern SU(N+1) -> U(N), and subsequently try to complete the Faddeev-Niemi decomposition by keeping the missing degrees of freedom. While it is not poss…
We study topologically stable non-Abelian global vortices in the U(N) linear sigma model. The profile functions of the solutions are numerically obtained. We investigate the behaviour of vortices in two limits in which masses of traceless or trace parts of massive bosons are much larger than the others. In the limit that the traceless parts are much heavier, we find a somewhat bizarre vortex solution carrying a non-integer U(1) winding number 1/\sqrt{N} which is irrational in general.
We successfully exhaust the complete set of exact solutions of non-Abelian vortices in a quiver gauge theory, that is, the S[U(N) x U(N)] gauge theory with a bi-fudamental scalar field on a hyperbolic plane with a certain curvature, from SO(3)-invariant SU(2N) Yang-Mills instanton solutions. This work provides, for the first time, exact non-Abelian vortex solutions. We establish the ADHM construction for non-Abelian vortices and identify all the moduli parameters and the complete moduli space.
