We propose to use the second Hopf map for the reduction (via SU(2) group action) of the eight-dimensional N=8 supersymmetric mechanics to five-dimensional supersymmetric systems specified by the presence of an SU(2) Yang monopole. For our purpose we develop the relevant Lagrangian reduction procedure. The reduced system is characterized by its invariance under the N=5 or N=4 supersymmetry generators (with or without an additional conserved BRST charge operator) which commute with the su(2) generators.
Motivated by Gibbons and Townsend's recent work, we construct Yang-Type monopoles in maximally symmetric space-time. We then analyze the dependence of horizon structure of the space-times around the 5-dimensional monopoles on the relative strength of gravitations to Yang-Mills interactions. We also analyze the stability of the monopoles against tensor type perturbations on metrics.
A supersymmetric collective coordinate expansion of the monopole solution of $N=4$ Yang-Mills theory is performed resulting in an $N=4$ supersymmetric quantum mechanics on the moduli space of monopole solutions.
The Dirac-Yang monopoles are singular Yang--Mills field configurations in all Euclidean dimensions. The regular counterpart of the Dirac monopole in D=3 is the t Hooft-Polyakov monopole, the former being simply a gauge transform of the asymptotic fields of the latter. Here, regular counterparts of Dirac-Yang monopoles in all dimensions, are described. In the first part of this talk the hierarchy of Dirac--Yang (DY) monopoles will be defined, in the second part the motivation to study these in a topoical context will be briefly presented, and in the last part, two classes of regular counterpart…
A possible mechanism accounting for monopole configurations in continuum Yang-Mills theories is discussed. The presence of the gauge fixing term is taken into account.
Axially symmetric monopole anti-monopole dipole solutions to the second order equations of a simple SU(2) Yang-Mills-Higgs model featuring a quartic Skyrme-like term are constructed numerically. The effect of varying the Skyrme coupling constant on these solutions is studied in some detail.
Based on the result of string/string duality, we construct the six dimensional Yang monopole in terms of Type IIA wrapped D-branes. In particular, we show that all the information of such a magnetic solution can be encoded in the K3 surface compactification in the presence of D2 and D4-branes wrapping its non trivial cycles. We give a geometrical and physical interpretations for the {+1,-1} Yang monopole charges. Lifting to eleven dimensions, we relate this Type IIA configuration with the heterotic M-theory one, given in hep-th/0607193. The nature of the black Yang monopole is also discussed.
In this short note we suggest that the singular behavior of large gauge transformations preserving the vacuum at null infinity in Yang-Mills theory implies monopoles into the bulk, as well as that the inclusion of a theta term induces a decoupling between holomorphic and anti-holomorphic currents associated to those large gauge transformations
We show that a Jackiw-Nohl-Rebbi solution, as the most general two-instanton, generates a circular loop of magnetic monopole in four-dimensional Euclidean SU(2) Yang-Mills theory.
We compute the quark-monopole potential for ${\cal N}=4$ super Yang-Mills in the large $N$ limit. We find an attractive potential that falls off as 1/L and is manifestly invariant under $g\to 1/g$. The strength of the potential is less than the quark-antiquark and monopole-antimonopole potentials.
2+1-dimensional Yang-Mills theory is reinterpreted in terms of metrics on 3-manifolds. The dual gluons are related to diffeomorphisms of the 3-manifold. Monopoles are identified with points where the Ricci tensor has triply degenerate eigenvalues. The dual gluons have the desired interaction with these monopoles. This would give a mass for the dual gluons resulting in confinement.
A path integral description of an effective action of monopoles in Abelian projections of Yang-Mills theories is discussed and used to establish a projection independence of the effective action. A dynamic regime in which the effective dynamics may contain massive solitonic excitations is described.
he Wu-Yang monopole for pure SU(2) Yang-Mills theory is revisited. New classical solutions with finite energy are found for a generalized Wu-Yang configuration. Our method relies on known asymptotic series solutions and explores the conformal invariance of the theory to set the scale for the monopole configurations. Furthermore, by breaking the scale invariance of the pure Yang-Mills theory by a term which simulates the coupling to quark fields, four different monopole configurations with an energy of 2.9 GeV and spatial extent of 1.2 fm, 1.4 fm, 2.4 fm and 2.6 fm are obtained. These configura…
As shown by Taubes, in the Bogomol'nyi-Prasad-Sommerfield limit the SU(2) Yang-Mills-Higgs model possesses smooth finite energy solutions, which do not satisfy the first order Bogomol'nyi equations. We construct numerically such a non-Bogomol'nyi solution, corresponding to a monopole-antimonopole pair, and extend the construction to finite Higgs potential.
We consider an SU(4) Yang-Mills-Higgs theory spontaneously broken to $SO(4)$ by a scalar field in the $n\times n$ representation and the unbroken algebra invariant under Cartan automorphism. We obtain that $\mathbb Z_2$ monopoles in this theory belong to different classes of $su(2)$ embeddings associated to continuous families of up to four parameters. This result generalizes the discrete families of $\mathbb Z_2$ monopoles previously known as well as it shows new classes of embeddings.
A brane picture in Type IIA superstring for the Yang Monopole is reconsidered. It makes use of D2 and D4-branes wrapped on cycles in the K3 surface. When the model was first presented some problems concerning the charges of the monopoles arised. In this paper, they are shown to be cured by the model itself. Surprisingly, the incompatibility between the multi-charge configuration and the spherical symmetry of the Yang Monopole is seen in the brane description as the emergence of the enhancon shell and the fuzzy geometry. This consistency is deep and surprising, and is the point that triggered t…
Motivated by the correlated monopoles and vortices observed in lattice QCD, using Cho decomposition method and two successive gauge transformations which lead to the observation of configurations that include both monopoles and center vortices simultaneously, we obtain a Lagrangian density that explicitly indicates these two magnetic defects and the interaction between them.
We express the Pontryagin index in Polyakov gauge completely in terms of magnetically charged gauge fixing defects, namely magnetic monopoles, lines, and domain walls. Open lines and domain walls are topologically equivalent to monopoles, which are the genuine defects. The emergence of non-genuine magnetically charged closed domain walls can be avoided by choosing the temporal gauge field smoothly. The Pontryagin index is then exclusively determined by the magnetic monopoles.
In this paper, we show the existence of magnetic monopoles in the pure $SU(2)$ Yang--Mills theory even in absence of scalar fields when the gauge-invariant mass term is introduced. This result follows from the recent proposal for obtaining the gauge field configurations in the Yang-Mills theory from the solutions of the field equations in the "complementary" gauge-scalar model with the radial length of the scalar field being fixed. The gauge-invariant mass term is obtained through a change of variables and a gauge-independent description of the Brout-Englert-Higgs mechanism which neither relie…
We consider the dynamics of the classical $SU(2)$ Wu-Yang monopole problem and show a set of new directions for its analysis starting from a variational setting. This allows us to give a new interpretation of the monopole charge as a string in a centrifugal potential. The field equations are solved using both standard power series, as well as a new Padé approach. Furthermore, we discuss non-existence, and give a proof that rule out even singular non-trivial smooth finite energy solutions. Due to its non-vanishing chromomagnetic fields, the monopole can be interpreted as a collection of magneti…
We show that dyon and magnetic monopole can be constructed in the gauge-independent way for the $SU(2)$ Yang--Mills theory even in the absence of the scalar field. This result is derived from the recent proposal for obtaining non-trivial topological configurations responsible for quark confinement in the Yang-Mills theory based on the Confinement-Higgs complementary relationship between the pure Yang-Mills theory and the gauge-scalar model with an adjoint scalar field of the fixed length. We discuss how such configurations have the implications for quark confinement.
We study the structure of the monopole configuration in U(2) non-commutative super Yang-Mills theory. Our analysis consists of two steps: solving the BPS equation and then the eigenvalue equation in the non-commutative space. Calculation to the first non-trivial order in the non-commutativity parameter theta shows that the monopole exhibits a certain non-locality. This structure is precisely the one expected from the recent predictions by the brane-configuration technique.
We derive the low energy dynamics of monopoles and dyons in N=2 supersymmetric Yang-Mills theories with hypermultiplets in arbitrary representations by utilizing a collective coordinate expansion. We consider the most general case that Higgs fields both in the vector multiplet and in the hypermultiplets have nonzero vacuum expectation values. The resulting theory is a supersymmetric quantum mechanics which has been obtained by a nontrivial dimensional reduction of two-dimensional (4,0) supersymmetric sigma models with potentials.
We review work on construction of Monopoles in higher dimensions. These are solutions to a particular class of models descending from Yang--Mills systems on even dimensional bulk, with Spheres as codimensions. The topological lower bounds on the Yang-Mills action translate to Bogomol'nyi lower bounds on the residual Yang-Mills-Higgs systems. Mostly, consideration is restricted to 8 dimensional bulk systems, but extension to the arbitrary case follows systematically. After presenting the monopoles, the corresponding dyons are also constructed. Finally, new Chern-Simons densities expressed in te…
A field theoretic description of monopole condensation in strongly coupled gauge theories is given by actions involving antisymmetric tensors B_{μν} of rank 2. We rederive the corresponding action for 4d compact QED, summing explicitly over all possible monopole configurations. Its gauge symmetries and Ward identities are discussed. Then we consider the Wilsonian RGs for Yang-Mills theories in the presence of collective fields (again tensors B_{μν}) for the field strengths F_{μν} associated to the U(1) subgroups. We show that a ``vector-like'' Ward identity for the Wilsonian action involving B…
We twist the monopole equations of Seiberg and Witten and show how these equations are realized in topological Yang-Mills theory. A Floer derivative and a Morse functional are found and are used to construct a unitary transformation between the usual Floer cohomologies and those of the monopole equations. Furthermore, these equations are seen to reside in the vanishing self-dual curvature condition of an $OSp(1|2)$-bundle. Alternatively, they may be seen arising directly from a vanishing self-dual curvature condition on an $SU(2)$-bundle in which the fermions are realized as spanning the tange…
We investigate static axially symmetric monopole and black hole solutions with magnetic charge n > 1 in Einstein-Yang-Mills-Higgs theory. For vanishing and small Higgs selfcoupling, multimonopole solutions are gravitationally bound. Their mass per unit charge is lower than the mass of the n=1 monopole. For large Higgs selfcoupling only a repulsive phase exists. The static axially symmetric hairy black hole solutions possess a deformed horizon with constant surface gravity. We consider their properties in the isolated horizon framework, interpreting them as bound states of monopoles and black h…
We study the low-energy effective Hamiltonian of N=2 super Yang-Mills theories. We find that the BPS equations are unchanged outside a quantum core where higher dimension contributions are expected to be important. We find two quantum generalizations of the BPS soliton. The leading higher-derivative correction to the effective action is shown not to contribute to the BPS mass formula.
A version of non-Abelian monopole equations is explored through dimensional reductions, with often the addition of algebraic conditions. On zero curvature spaces, spinor related extensions of integrable systems have been generated, and certain reduced one-dimensional systems have been discussed with respect to integrability, as well as solutions found.
We discuss the construction of non-Abelian black holes and globally regular monopole solutions to N=2, d=4 EYM theories. Special emphasis is put on how the attractor mechanism works for the non-Abelian black holes.
We propose Lagrangian and Hamiltonian formulations of a N=4 supersymmetric three-dimensional isospin-carrying particle moving in the non-Abelian field of a Wu-Yang monopole and in some specific scalar potential. This additional potential is completely fixed by N=4 supersymmetry and in the simplest case of flat metrics it coincides with that which provides the existence of the Runge-Lenz vector for the bosonic subsector. The isospin degrees of freedom are described on the Lagrangian level by bosonic auxiliary variables forming N=4 supermultiplet with additional, also auxiliary fermions. Being q…
We investigate the relation between instantons and monopoles in the Laplacian Abelian Gauge using analytical methods in the continuum. Our starting point is the fact that the 't Hooft instanton with its high symmetry leads to a pointlike defect with Hopf invariant one. In order to generalise this result we partly break the symmetry by a local perturbation. We find that for generic configurations near the 't Hooft instanton the defects become loops. The analytical results show explicitly that these defects are magnetic monopoles with unit charge. In addition, the monopoles are twisted to accoun…
It is well known that the Dirac monopole solution with the U(1) gauge group embedded into the group SU(2) is equivalent to the SU(2) Wu-Yang point monopole solution having no Dirac string singularity. We consider a multi-center configuration of m Dirac monopoles and n anti-monopoles and its embedding into SU(2) gauge theory. Using geometric methods, we construct an explicit solution of the SU(2) Yang-Mills equations which generalizes the Wu-Yang solution to the case of m monopoles and n anti-monopoles located at arbitrary points in R^3.
We have studied a modified Yang-Mills-Higgs system coupled to Einstein gravity. The modification of the Einstein-Hilbert action involves a direct coupling of the Higgs field to the scalar curvature. In this modified system we are able to write a Bogomol'nyi type condition in curved space and demonstrate that the positive static energy functional is bounded from below. We then investigate non-Abelian sperically symmetric static solutions in a similar fashion to the `t Hooft-Polyakov monopole. After reviewing previously studied monopole solutions of this type, we extend the formalism to included…
In this paper we study wrapped brane configurations that give rise to three dimensional pure Yang-Mills theory with eight supercharges. The corresponding supergravity solution is singular and it was conjectured that the singularity is removed by an enhancon mechanism. Instead, we incorporate non-perturbative gauge fields into supergravity and find a smooth solution for these configurations. Along the way we derive a non-abelian supergravity Lagrangian for type IIA on K3 and explicit formulae for the toroidal reduction of the heterotic string with non-abelian gauge fields. We proceed to analyse…
Faddeev and Niemi have proposed a reformulation of SU(2) Yang-Mills theory in terms of new variables, appropriate for describing the theory in its infrared limit based on the intuitive picture of colour confinement due to monopole condensation. I generalize their proposal (with some differences) to SU(N) Yang-Mills theory. The natural variables are $N-1$ mutually commuting traceless $N\times N$ Hermitian matrices, an element of the maximal torus defined by these commuting matrices, $N-1$ Abelian gauge fields for the maximal torus gauge group, and an invariant symmetric two-index tensor on the …
Monopoles in topologically massive gauge theories in 2+1 dimensions with a Chern-Simon mass term have been studied by Pisarski some years ago. He investigated the SU(2) Yang-Mills-Higgs model with an additional Chern-Simon mass term in the action. Pisarski argued that there is a monopole solution that is regular everywhere, but found that it does not possess finite action. There were no exact or numerical solutions being presented by Pisarski. Hence it is our purpose to further investigate this solution in more detail. We obtained numerical regular solutions that smoothly interpolates between …
We construct the component action and supersymmetry transformations of the $\mathcal{N} = 3$ Chern-Simons-Yang-Mills theory that flows in the infra red to the $\mathcal{N} = 3$ super conformal quiver Chern-Simons gauge theory obtained by Jafferis and Tomasiello in \cite{Jafferis:2008qz}. We then obtain classical $\frac{1}{3}$-BPS (and anti-BPS) solutions all along the RG flow. The solutions display a rich structure, for example, there is a non-trivial moduli space. Various properties of supersymmetric monopole operators in the infra red super conformal field theory, such as global charges and …
An efficient way of resolving Gauss' law in Yang-Mills theory is presented by starting from the projected gauge invariant partition function and integrating out one spatial field variable. In this way one obtains immediately the description in terms of unconstrained gauge invariant variables which was previously obtained by explicitly resolving Gauss' law in a modified axial gauge. In this gauge, which is a variant of 't Hooft's Abelian gauges, magnetic monopoles occur. It is shown how the Pontryagin index of the gauge field is related to the magnetic charges. It turns out that the magnetic mo…
In the presence of a background supergravity flux, N M2-branes will expand via the Myers effect into M5-branes wrapped on a fuzzy three-sphere. In previous work the fluctuations of the M2-branes were shown to be described by the five-dimensional Yang-Mills gauge theory associated to D4-branes. We show that the ABJM prescription for eleven-dimensional momentum in terms of magnetic flux lifts to an instanton flux of the effective five-dimensional Yang-Mills theory on the sphere, giving an M-theory interpretation for these instantons.
We give a gauge-covariant decomposition of the Yang-Mills field with an exceptional gauge group $G(2)$, which extends the field decomposition invented by Cho, Duan-Ge, and Faddeev-Niemi for the $SU(N)$ Yang-Mills field. As an application of the decomposition, we derive a new expression of the non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation of $G(2)$. The resulting new form is used to define gauge-invariant magnetic monopoles in the $G(2)$ Yang-Mills theory. Moreover, we obtain the quantization condition to be satisfied by the resulting magnetic charge. Th…
Dual Feynman rules for Dirac monopoles in Yang-Mills fields are obtained by the Wu-Yang (1976) criterion in which dynamics result as a consequence of the constraint defining the monopole as a topological obstruction in the field. The usual path-integral approach is adopted, but using loop space variables of the type introduced by Polyakov (1980). An anti-symmetric tensor potential $L_{μν}[ξ|s]$ appears as the Lagrange multiplier for the Wu-Yang constraint which has to be gauge-fixed because of the ``magnetic'' $\widetilde{U}$-symmetry of the theory. Two sets of ghosts are thus introduced, whic…
We show that gravity cures the infra-red divergence of the Yang monopole when a proper definition of conserved quantities in curved backgrounds is used, i.e. the gravitating Yang monopole in cosmological Einstein theory has a finite mass in generic even dimensions (including time). In addition, we find exact Yang-monopole type solutions in the cosmological Einstein-Gauss-Bonnet-Yang-Mills theory and briefly discuss their properties.
Using the second Hopf map, we perform the reduction of the eight-dimensional (pseudo)spherical (Higgs)oscillator to a five-dimensional system interacting with a Yang monopole. Then, using a standard trick, we obtain, from the latter system, the pseudospherical and spherical generalizations of the Yang-Coulomb system (the five dimensional analog of MICZ-Kepler system). We present the whole set of its constants of motions, including the hidden symmetry generators given by the analog of Runge-Lenz vector. In the same way, starting from the eight-dimensional anisotropic inharmonic Higgs oscillator…
By making use of the decomposition theory of gauge potential, the inner structure of SU(2) and SO(4) gauge theory is discussed in detail. We find the SO(4) monopole can be given via projecting the SO(4) gauge field onto an antisymmetric tensor. This projection fix the coset $% SU(2)/U(1)\bigotimes SU(2)/U(1)$ of SO(4) gauge group. The generalized Hopf map is given via a Dirac spinor. Further we prove that this monopole can be consider as a new topological invariant. Which is composed of two monopole structures. Local topological structure of the SO(4) monopole is discussed in detail, which is …
The (2k+2)-dimensional Einstein-Yang-Mills equations for gauge group SO(2k) (or SU(2) for k=2 and SU(3) for k=3) are shown to admit a family of spherically-symmetric magnetic monopole solutions, for both zero and non-zero cosmological constant Lambda, characterized by a mass m and a magnetic-type charge. The k=1 case is the Reissner-Nordstrom black hole. The k=2 case yields a family of self-gravitating Yang monopoles. The asymptotic spacetime is Minkowski for Lambda=0 and anti-de Sitter for Lambda<0, but the total energy is infinite for k>1. In all cases, there is an event horizon when m>m_c, …
We investigate the validity of the Dirac Quantization Condition (DQC) for magnetic monopoles in noncommutative space-time. We use an approach which is based on an extension of the method introduced by Wu and Yang. To study the effects of noncommutativity of space-time, we consider the gauge transformations of $U_\star(1)$ gauge fields and use the corresponding deformed Maxwell's equations. Using a perturbation expansion in the noncommutativity parameter $θ$, we show that the DQC remains unmodified up to the first order in the expansion parameter. The result is obtained for a class of noncommut…
A self-contained study of monopole configurations of pure Yang-Mills theories and a discussion of their charges is carried out in the language of principal bundles. A n-dimensional monopole over the sphere S^n is a particular type of principal connection on a principal bundle over a symmetric space K/H which is K-invariant, where K=SO(n+1) and H=SO(n). It is shown that principal bundles over symmetric spaces admit a unique K-invariant principal connection called canonical, which also satisfy Yang-Mills equations. The geometrical framework enables us to describe their associated field strengths…
We study the quark confinement problem in 2+1 dimensional pure Yang-Mills theory using euclidean instanton methods. The instantons are regularized and dressed Wu-Yang monopoles. The dressing of a monopole is due to the mean field of the rest of the monopoles. We argue that such configurations are stable to small perturbations unlike the case of singular, undressed monopoles. Using exact non-perturbative results for the 3-dim. Coulomb gas, where Debye screening holds for arbitrarily low temperatures, we show in a self-consistent way that a mass gap is dynamically generated in the gauge theory. …
We construct static axially symmetric solutions of SU(2) Einstein-Yang-Mills-Higgs theory in the topologically trivial sector, representing gravitating monopole--antimonopole pairs, linked to the Bartnik-McKinnon solutions.
It is well known that there are no static non-Abelian monopole solutions in pure Yang-Mills theory on Minkowski space R^{3,1}. We show that such solutions exist in SU(N) gauge theory on the spaces R^2\times S^2 and R^1\times S^1\times S^2 with Minkowski signature (-+++). In the temporal gauge they are solutions of pure Yang-Mills theory on T^1\times S^2, where T^1 is R^1 or S^1. Namely, imposing SO(3)-invariance and some reality conditions, we consistently reduce the Yang-Mills model on the above spaces to a non-Abelian analog of the ϕ^4 kink model whose static solutions give SU(N) monopole (-…
We give a definition of gauge-invariant magnetic monopoles in Yang-Mills theory without using the Abelian projection due to 't Hooft. They automatically appear from the Wilson loop operator. This is shown by rewriting the Wilson loop operator using a non-Abelian Stokes theorem. The magnetic monopole defined in this way is a topological object of co-dimension 3, i.e., a loop in four-dimensions. We show that such magnetic loops indeed exist in four-dimensional Yang-Mills theory. In fact, we give an analytical solution representing circular magnetic monopole loops joining a pair of merons in the …
We determine the low energy dynamics of monopoles in pure N=2 Yang-Mills theories for points in the vacuum moduli space where the two Higgs fields are not aligned. The dynamics is governed by a supersymmetric quantum mechanics with potential terms and four real supercharges. The corresponding superalgebra contains a central charge but nevertheless supersymmetric states preserve all four supercharges. The central charge depends on the sign of the electric charges and consequently so does the BPS spectrum. We focus on the SU(3) case where certain BPS states are realised as zero-modes of a Dirac …
The well-known Bogomol'nyi-Prasad-Sommerfield (BPS) monopole is considered in the limit of the infinite mass of the Higgs field as a basis of the Yang-Mills field vacuum with the finite energy density. In this limit, the Higgs field disappears, but it leaves its traces as the BPS monopole transforms into the Wu-Yang monopole obtained in the pure Yang-Mills theory by a spontaneous scale symmetry breaking in the class of functions with the zero value of a topological charge. The topological degeneration of the BPS monopole manifests itself as the Gribov copies of the covariant Coulomb gauge in t…
The coulomb branch of $N=4$ supersymmetric Yang-Mills gauge theories in $d=2+1$ is studied. A direct connection between gauge theories and monopole moduli spaces is presented. It is proposed that the hyper-Kähler metric of supersymmetric $N=4$ $SU(N)$ Yang-Mills theory is given by the charge $N$ centered moduli space of BPS monopoles in $SU(2)$. The theory is compared to $N=2$ supersymmetric Yang-Mills theory in four dimensions through compactification on a circle of the latter. It is found that rational maps are appropriate to this comparison. A BPS mass formula is also written for particles …
Monopole solutions in SU(2) Yang-Mills theory which includes spinor fields described by the nonlinear Dirac equation are obtained. It is demonstrated that the energy spectrum of such a system possesses a global minimum whose appearance is brought about solely by the nonlinear spinor fields. It is shown that the monopole solution obtained differs in principle from the 't~Hooft-Polyakov monopole in that it is topologically trivial.
Attractive and repulsive Yang-Mills--Higgs magnetic monopoles on $\mathbb{R}^3$
Universidad Complutense de Madrid（马德里康普顿斯大学）；Universidade de Aveiro（阿威罗大学）；Center for Research and Development in Mathematics and Applications（数学与应用研究中心）；Dublin Institute for Advanced Studies（都柏林高等研究院）；National University of Ireland Maynooth（爱尔兰国立大学梅努斯分校）
