BPS Monopoles Research Articles

Color superconductivity, BPS Zk strings and monopole confinement in N = 2 and N = 4 super Yang-Mills theories

We review some recent developments on BPS string solutions and monopole confinement in the Higgs (or color) superconducting phase of <IMG SRC="/img/revistas/bjp/v34n4a/caln_mai.gif" > or = 2 and <IMG SRC="/img/revistas/bjp/v34n4a/caln_mai.gif" > or = 4 super Yang-Mills theories. In particular, the monopole magnetic fluxes are shown to be always integer linear combinations of string fluxes. Moreover, a bound for the threshold length of the string breaking is obtained. When the gauge group SU(N) is broken to Z N, the BPS string tension satisfies the Casimir scaling law. Furthermore, in the SU(3) case the string solutions are such that they allow the formation of a confining system with three monopoles.

Non-Abelian monopoles

We study topological as well as dynamical properties of BPS non-Abelian magnetic monopoles of Goddard–Nuyts–Olive–Weinberg type in G = SU ( N ) , USp ( 2 N ) and SO ( N ) gauge theories, spontaneously broken to non-Abelian subgroups H. We find that monopoles transform under the group dual to H in a tensor representation of rank determined by the corresponding element in π 1 ( H ) . When the system is embedded in a N = 2 supersymmetric theory with an appropriate set of flavors with appropriate bare masses, the BPS monopoles constructed semiclassically persist in the full quantum theory. This result supports the identification of “dual quarks” found at r-vacua of N = 2 theories with the non-Abelian magnetic monopoles. We present several consistency checks of our monopole spectra.

Quantum weights of dyons and of instantons with nontrivial holonomy

We calculate exactly functional determinants for quantum oscillations about periodic instantons with non-trivial value of the Polyakov line at spatial infinity. Hence, we find the weight or the probability with which calorons with non-trivial holonomy occur in the Yang--Mills partition function. The weight depends on the value of the holonomy, the temperature, Lambda_QCD, and the separation between the BPS monopoles (or dyons) which constitute the periodic instanton. At large separation between constituent dyons, the quantum measure factorizes into a product of individual dyon measures, times a definite interaction energy. We present an argument that at temperatures below a critical one related to Lambda_QCD, trivial holonomy is unstable, and that calorons ``ionize'' into separate dyons.

Wrapped D-branes as BPS monopoles: The moduli space perspective

We study the four dimensional effective action of a system of D6-branes wrapped on the $K3$ manifold times a torus, allowing the volume of the internal manifolds to remain dynamical. An unwrapped brane is at best a Dirac monopole of the dual RR sector field to which it couples. After wrapping, a brane is expected to behave as a BPS monopole, where the Higgs vacuum expectation value is set by the size of the $K3.$ We determine the moduli space of an arbitrary number of these wrapped branes by introducing a time dependent perturbation of the static solution and expanding the supergravity equations of motion to determine the dynamics of this perturbation, in the low velocity limit. The result is the hyper-K\"ahler generalization of the Euclidean Taub-NUT metric presented by Gibbons and Manton. We note that our results also pertain to the behavior of bound states of Kaluza-Klein monopoles and wrapped NS5-branes in the ${T}^{4}$ compactified heterotic string.

A new anomalous contribution to the central charge of the N=2 monopole

We calculate the one-loop corrections to the mass and central charge of the BPS monopole in N=2 super-Yang–Mills theory in 3+1 dimensions using a supersymmetry-preserving version of dimensional regularization adapted to solitons. In the renormalization scheme where previous studies have indicated vanishing quantum corrections, we find nontrivial corrections that we identify as the (3+1)-dimensional analogue of the anomaly in the conformal central charge of the N=1 supersymmetric kink in 1+1 dimensions. As in the latter case, the associated contribution to the ordinary central charge has exactly the required magnitude to preserve BPS saturation at the one-loop level. It also restores consistency of calculations involving sums over zero-point energies with the low-energy effective action of Seiberg and Witten.

Noncommutative Monopoles and Riemann-Hilbert Problems

The Bogomolny equations for Yang-Mills-Higgs monopoles follow from a system of linear equations which may be solved through a parametric Riemann-Hilbert problem. We extend this approach to noncommutative R^3 and use it to (re)construct noncommutative Dirac, Wu-Yang, and BPS monopole configurations in a unified manner. In all cases we write down the underlying matrix-valued functions for multi-monopoles and solve the corresponding Riemann-Hilbert problems for charge one.

Calorons and BPS monopoles with non-trivial holonomy in the confinement phase of SU(2) gluodynamics

With the help of the cooling method applied to SU(2) lattice gauge theory at non-zero T ≤ T c we present numerical evidence for the existence of superpositions of Kraan-van Baal caloron (or BPS monopole pair) solutions with non-trivial holonomy, which might constitute an important contribution to the semi-classical approximation of the partition function.

Atiyah-Drinfeld-Hitchin-Manin-Nahm boundary conditions from removing monopoles

Boundary conditions play an important role in the ADHMN construction of BPS monopole solutions. In this paper we show how different types of boundary conditions can be related to each other by removing monopoles to spatial infinity. In particular, we use this method to show how the jumping data naturally emerge. The results can be interpreted in the D-brane picture and provide a better understanding of the derivation of the ADHMN construction from D-branes. We comment briefly on the cases with non-Abelian unbroken symmetry and massless monopoles.

Aq-analogue of the ADHMN construction and the axisymmetric multi-instantons

In the preceding paper, the present authors presented a q-analogue of the ADHMN construction and obtained a family of anti-selfdual configurations with a parameter q for classical SU(2) Yang-Mills theory in four-dimensional Euclidean space. The family of solutions can be seen as a q-analogue of the single BPS monopole preserving (anti-)selfduality. Further discussion is given of the relation to the axisymmetric ansatz on the anti-selfdual equation given by Witten in the late 1970s. It is found that the q-exponential functions familiar in q-analysis appear as analytic functions categorizing the anti-selfdual configurations yielded by the axisymmetric ansatz.

Duality rotations and BPS monopoles with space and time noncommutativity

We show that noncommutative electromagnetism and Dirac–Born–Infeld (DBI) theory with scalar fields are SL(2, R) self-dual when noncommutativity is light-like and we are in the slowly varying field approximation. This follows from SL(2, R) self-duality of the commutative DBI Lagrangian and of its zero slope limit that we study in detail. We study a symmetry of noncommutative static configurations that maps space-noncommutativity into space–time (and light-like) noncommutativity. SL(2, R) duality is thus extended to space-noncommutativity. Via Seiberg–Witten map we study the nontrivial action of this symmetry on commutative DBI theory. In particular space–time noncommutative BPS magnetic monopoles corresponds to commutative BPS type magnetic monopoles with both electric and magnetic B-field background. Energy, charge and tension of these configurations are computed and found in agreement with that of a D1-string D3-brane system. We discuss the dual string–brane configuration.

Superpartner solutions of a BPS monopole in noncommutative space

We construct U(2) BPS monopole superpartner solutions in N=2 non-commutative super Yang-Mills theory. Calculation to the second order in the noncommutative parameter $\theta$ shows that there is no electric quadrupole moment that is expected from the magnetic dipole structure of noncommtative U(2) monopole. This might give an example of the nature of how supersymmetry works not changing between the commutative and noncommutative theories.

Selfdual solution of classical yang—mills fields through a q-analog of ADHM construction

The ADHMN construction of selfdual Yang—Mills gauge fields is reformulated by introducing a functional space defined on a discrete point set. Using this formalism, a q-analogous object of the BPS monopole is obtained.

Enhançons, fuzzy spheres, and multimonopoles

We study the `enhancon', a spherical hypersurface apparently made of D-branes, which arises in string theory studies of large N SU(N) pure gauge theories with eight supercharges. When the gauge theory is 2+1 dimensional, the enhancon is an S^2. A relation to charge N BPS multi-monopoles is exploited to uncover many of its detailed properties. It is simply a spherical slice through an Atiyah-Hitchin-like submanifold of the charge $N$ BPS monopole moduli space. In the form of Nahm data, it is built from the N dimensional irreducible representation of SU(2). In this sense the enhancon is a non-commutative sphere, reminiscent of the spherical `dielectric' branes of Myers.

General dynamics of two distinct Bogomol’nyi-Prasad-Sommerfield monopoles

Classical and quantum dynamics of two distinct BPS monopoles in the case of non-aligned Higgs fields are studied on the basis of the recently determined low energy effective theory. Despite the presence of a specific potential together with a kinetic term provided by the metric of a Taub-NUT manifold, an O(4) or O(3,1) symmetry of the system allows for a group theoretical derivation of the bound-state spectrum and the scattering cross section.

Counting Yang-Mills dyons with index theorems

We count the supersymmetric bound states of many distinct BPS monopoles in N=4 Yang-Mills theories and in pure N=2 Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint Higgs fields are turned on. The number of purely magnetic bound states is again found to be consistent with the electromagnetic duality of the N=4 SU(n) theory, as expected. We also count dyons of generic electric charges, which correspond to 1/4 BPS dyons in N=4 theories and 1/2 BPS dyons in N=2 theories. Surprisingly, the degeneracy of dyons is typically much larger than would be accounted for by a single supermultiplet of appropriate angular momentum, implying many supermutiplets of the same charge and the same mass.

Monopole black hole skyrmions

Charged black hole solutions with pion hair are discussed.These can be used to study monopole black hole catalysisof proton decay. There also exist multi-black hole skyrmionsolutions with BPS monopole behaviour.

OSCULATION AND SINGULARITY OF CHARGE 2 (COMPLEXIFIED) BPS MONOPOLES

Hurtubise showed that the singular locus of the complexification of a BPS monopole intersects the degeneracy locus of the complex Higgs field in a null curve [Formula: see text] has a dual [Formula: see text]: by adapting a criterion of Hitchin it is shown that for a generic charge 2 monopole, [Formula: see text], the monopole's spectral curve.

Multi-instantons in Bbb R4 and Minimal Surfaces in Bbb R2,1

It is known that self-duality equations for multi-instantons on a line in four dimensions are equivalent to minimal surface equations in three dimensional Minkowski space. We extend this equivalence beyond the equations of motion and show that topological number, instanton moduli space and anti-self-dual solutions have representations in terms of minimal surfaces. The issue of topological charge is quite subtle because the surfaces that appear are non-compact. This minimal surface/instanton correspondence allows us to define a metric on the configuration space of the gauge fields. We obtain the minimal surface representation of an instanton with arbitrary charge. The trivial vacuum and the BPST instanton as minimal surfaces are worked out in detail. BPS monopoles and the geodesics are also discussed.

Singularity criteria for (complexified) BPS monopoles

In [ 7 ], Hitchin showed that the data (∇, Φ), comprising an SU(2) Yang–Mills–Higgs monopole in the Prasad–Sommerfeld limit on ℝ 3 , encodes faithfully into an auxiliary rank 2 holomorphic vector bundle E˜ over T , the total space of the holomorphic tangent bundle of ℙ 1 . In this construction ℝ 3 is viewed as a subset of H 0 (ℙ 1 , [Oscr ]( T )) ≅ [Copf ] 3 . Generically, the restriction of E˜ to a line is trivial. (The image of a global section ℙ z ⊂ T , for z ∈ [Copf ] 3 , is referred to here as a line on T .) Hence c 1 (E˜) = 0 and, for all z ∈ [Copf ] 3 , there exists m ∈ {0} ∪ ℕ such that E˜[mid ] ℙ z ≅ [Oscr ]( m ) [oplus ] [Oscr ](− m ). If m [ges ] 1 then ℙ z is a jumping line of E˜ of height m . The jumping lines are parameterized by an analytic set J ⊂ [Copf ] 3 , which is stratified by height. When the monopole has charge k , the height is bounded above by k . In this case we write J = J 1 ∪ … ∪ J k , where J i parameterizes jumping lines of height i . A priori, some J i may be empty. The analytic continuation of the monopole to [Copf ] 3 has singularities over J . To see this recall how the monopole data are recovered from E˜: very briefly, E˜ induces a sheaf [Escr ] = π 2* e*E˜ over [Copf ] 3 which is locally free away from J 2 ∪ … ∪ J k , (π 2 and e are defined in Section 2). A holomorphic connection and Higgs field are defined in [Escr ] over [Copf ] 3 null planes that cut out a given direction (see [ 1 , 7 , 9 ]). On restriction to ℝ 3 , [Escr ] gives a rank 2, SU(2) bundle and the holomorphic connection and Higgs field give the monopole data. It is easy to see that the flat connections are singular at points of J : for example, an analogous situation is described in [ 10 ].

Complete supersymmetric quantum mechanics of magnetic monopoles inN=4super Yang-Mills theory

We find the most general low energy dynamics of 1/2 BPS monopoles in the N=4 supersymmetric Yang-Mills theories (SYM) when all six adjoint Higgs expectation values are turned on. When only one Higgs is turned on, the Lagrangian is purely kinetic. When all six are turned on, however, this moduli space dynamics is augmented by five independent potential terms, each in the form of half the squared norm of a Killing vector field on the moduli space. A generic stationary configuration of the monopoles can be interpreted as stable non BPS dyons, previously found as non-planar string webs connecting D3-branes. The supersymmetric extension is also found explicitly, and gives the complete quantum mechanics of monopoles in N=4 SYM theory. We explore its supersymmetry algebra.