119740800
Magnetic impurity inspired Abelian Higgs vortices
A bstractInspired by magnetic impurity considerations some broad classes of Abelian Higgs and Chern-Simons-Higgs BPS vortex equations are derived and analyzed.
119836660
Vortex Line Models for Dual Strings
We call attention to the possibility of constructing field theories for the dual string. As an example, we show that a Higgs type of Lagrangian allows for vortex-line solutions, in analogy with the vortex lines in a type II superconductor. These vortex lines can approximately be identified with the Nambu string. In the strong coupling limit we speculate that the vortex lines make up all low energy phenomena. It turns out that this strong coupling limit is “super quantum mechanical” in the sense that the typical action of the theory is very small in comparison with Plank's constant.
120549115
Vortices and quark confinement in non-abelian gauge theories
Abstract It is shown that finite-length vortices in an SU(n) Nielsen-Olesen model require explicit introduction of monopoles, which are confined in multiples of n by the Meissner effect. The model therefore possesses a natural explanation of quark confinement.
123270044
Strings, Monopoles and Gauge Fields
The Nielsen-Olesen interpretation of dual strings as Abrikosov flux lines is extended to the case of open-ended strings by adapting Dirac's description of magnetic monopoles to a London-type theory. The mathematical formalism turns out to be similar to that of Kalb and Ramond. Translated to hadron physics, it implies that the quarks will act as carriers of magnetic charge, permanently bound in pairs by the string bonds. However, massive axial-vector gluons can be created by hadrons.
120803053
Magnetic monopoles in unified gauge theories
It is shown that in all those gauge theories in which the electromagnetic group U(1) is taken to be a subgroup of a larger group with a compact covering group, like SU(2) or SU(3), genuine magnetic monopoles can be created as regular solutions of the field equations. Their mass is calculable and of order 137 MW, where MW is a typical vector boson mass.
122168573
On the Phase Transition Towards Permanent Quark Confinement
In quantized gauge field theories one can introduce sets of operators that modify the gauge-topological structure of the fields but whose physical effect is essentially local. In 2 + 1 dimensional non-Abelian gauge theories these operators form scalar fields and it is argued that when the local gauge symmetry is not spontaneously broken then these topological fields develop a vacuum expectation value and their mutual symmetry breaks spontaneously. It is shown that quarks are then permanently confined. In 3 + 1 dimensional non-Abelian gauge theories one finds that the topological operators and the gauge field operators form a closed algebra from which it is deduced that this system can be in one of the four different phases: (i) spontaneous breakdown via an explicit or composite Higgs field, (ii) no Higgs field but permanent confinement of gauge quantum numbers, (iii) Higgs effect and still confinement, presumably only if there is an unbroken subgroup, and (iv) an intermediate phase (critical point?) with massless particles. Finally, the algebra can be realized in a simple model where phases (i) and (ii) can be obtained from each other by dual transformation.
118606636
Constructing Non-Abelian Vortices with Arbitrary Gauge Groups
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 use of holomorphic invariants extends the scope of the powerful moduli matrix formalism in which the vortex moduli spaces have been determined.
119294007
Multiple layer structure of non-Abelian vortex
Abstract Bogomolny–Prasad–Sommerfield (BPS) vortices in U ( N ) gauge theories have two layers corresponding to non-Abelian and Abelian fluxes, whose widths depend nontrivially on the ratio of U ( 1 ) and SU ( N ) gauge couplings. We find numerically and analytically that the widths differ significantly from the Compton lengths of lightest massive particles with the appropriate quantum number.
45617597
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 (CxCP(N-1))k/(see text)k. Orbifold singularities of this space correspond to coincident vortices and are resolved resulting in a smooth moduli manifold. The relation to Kähler quotient construction is discussed.
119111240
Non-Abelian vortex dynamics: effective world-sheet action
The low-energy vortex effective actionis 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) × U(1) or USp(2N) × 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 ofthe color-flavor SO(2N)C+F or USp(2N)C+F symmetry, i.e. of SO(2N) or of SO(2N + 1). Applied to the benchmark U(N) model our procedure reproduces the known $ \mathbb{C}{P^{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.
119465286
Non-Abelian Confinement via Abelian Flux Tubes in Softly Broken N=2 SUSY QCD
We study confinement in softly broken N=2 SUSY QCD with gauge group SU(N_c) and N_f hypermultiplets of fundamental matter (quarks) when the Coulomb branch is lifted by small mass of adjoint matter. Concentrating mostly on the theory with SU(3) gauge group we discuss the N=1 vacua which arise in the weak coupling at large values of quark masses and study flux tubes and monopole confinement in these vacua. In particular we find the BPS strings in SU(3) gauge theory formed by two interacting U(1) gauge fields and two scalar fields generalizing ordinary Abrikosov-Nielsen-Olesen vortices. Then we focus on the SU(3) gauge theories with N_f=4 and N_f=5 flavors with equal masses. In these theories there are N=1 vacua with restored SU(2) gauge subgroup in quantum theory since SU(2) subsectors are not asymptotically free. We show that although the confinement in these theories is due to Abelian flux tubes the multiplicity of meson spectrum is the same as expected in a theory with non-Abelian confinement.
118956344
Non-Abelian string junctions as confined monopoles
Various dynamical regimes associated with confined monopoles in the Higgs phase of $\mathcal{N}=2$ two-flavor QCD are studied. The microscopic model we deal with has the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge group, with a Fayet\char21{}Iliopoulos term of the U(1) factor, and large and (nearly) degenerate mass terms of the matter hypermultiplets. We present a complete quasiclassical treatment of the BPS sector of this model, including the full set of the first-order equations, derivations of all relevant zero modes, and derivation of an effective low-energy theory for the corresponding collective coordinates. The macroscopic description is provided by a ${\mathrm{CP}}^{1}$ model with or without twisted mass. The confined monopoles\char22{}string junctions of the microscopic theory\char22{}are mapped onto BPS kinks of the ${\mathrm{CP}}^{1}$ model. The string junction is $\frac{1}{4}$ BPS. Masses and other characteristics of the confined monopoles are matched with those of the ${\mathrm{CP}}^{1}$-model kinks. The matching demonstrates the occurrence of an anomaly in the monopole central charge in 4D Yang\char21{}Mills theory. We study what becomes of the confined monopole in the bona fide non-Abelian limit of degenerate mass terms where a global SU(2) symmetry is restored. The solution of the macroscopic model is known, e.g., from the mirror description of the ${\mathrm{CP}}^{1}$ model. The monopoles, aka ${\mathrm{CP}}^{1}$-model kinks, are stabilized by nonperturbative dynamics of the ${\mathrm{CP}}^{1}$ model. We explain an earlier rather puzzling observation of a correspondence between the BPS kink spectrum in the ${\mathrm{CP}}^{1}$ model and the Seiberg\char21{}Witten solution.
118032129
TOPICAL REVIEW: Solitons in the Higgs phase: the moduli matrix approach
We review our recent work on solitons in the Higgs phase. We use U(NC) gauge theory with NF Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices, which are the only elementary solitons in the Higgs phase, are found in terms of the moduli matrix. Stable monopoles and instantons can exist in the Higgs phase if they are attached by vortices to form composite solitons. The moduli spaces of these composite solitons are also worked out in terms of the moduli matrix. Webs of walls can also be formed with characteristic difference between Abelian and non-Abelian gauge theories. Instanton–vortex systems, monopole–vortex–wall systems, and webs of walls in Abelian gauge theories are found to admit negative energy objects with the instanton charge (called intersectons), the monopole charge (called boojums) and the Hitchin charge, respectively. We characterize the total moduli space of these elementary as well as composite solitons. In particular the total moduli space of walls is given by the complex Grassmann manifold SU(NF)/[SU(NC) × SU(NF − NC) × U(1)] and is decomposed into various topological sectors corresponding to boundary condition specified by particular vacua. The moduli space of k vortices is also completely determined and is reformulated as the half ADHM construction. Effective Lagrangians are constructed on walls and vortices in a compact form. We also present several new results on interactions of various solitons, such as monopoles, vortices and walls. Review parts contain our works on domain walls (Isozumi Y et al 2004 Phys. Rev. Lett. 93 161601 (Preprint hep-th/0404198), Isozumi Y et al 2004 Phys. Rev. D 70 125014 (Preprint hep-th/0405194), Eto M et al 2005 Phys. Rev. D 71 125006 (Preprint hep-th/0412024), Eto M et al 2005 Phys. Rev. D 71 105009 (Preprint hep-th/0503033), Sakai N and Yang Y 2005 Comm. Math. Phys. (in press) (Preprint hep-th/0505136)), vortices (Eto M et al 2005 Phys. Rev. Lett. 96 161601 (Preprint hep-th/0511088), Eto M et al 2006 Phys. Rev. D 73 085008 (Preprint hep-th/0601181)), domain wall webs (Eto M et al 2005 Phys. Rev. D 72 085004 (Preprint hep-th/0506135), Eto M et al 2006 Phys. Lett. B 632 384 (Preprint hep-th/0508241), Eto M et al 2005 AIP Conf. Proc. 805 354 (Preprint hep-th/0509127)), monopole–vortex–wall systems (Isozumi Y et al 2005 Phys. Rev. D 71 065018 (Preprint hep-th/0405129), Sakai N and Tong D 2005 J. High Energy Phys. JHEP03(2005)019 (Preprint hep-th/0501207)), instanton–vortex systems (Eto M et al 2005 Phys. Rev. D 72 025011 (Preprint hep-th/0412048)), effective Lagrangian on walls and vortices (Eto M et al 2006 Phys. Rev. D (in press) (Preprint hep-th/0602289)), classification of BPS equations (Eto M et al 2005 Preprint hep-th/0506257) and Skyrmions (Eto M et al 2005 Phys. Rev. Lett. 95 252003 (Preprint hep-th/0508130)).
117948932
An Introduction to the Confinement Problem
Introduction.- Global Symmetry, Local Symmetry, and the Lattice.- What is Confinement?- Order Parameters for Confinement.- Properties of the Confining Force.- Confinement From Center Vortices I.- Monopoles, Calorons, and Dual Superconductivity.- Coulomb Confinement.- Ghosts, Gluons, and Dyson-Schwinger Equations.- Large-N, Planar Diagrams, and the Gluon-Chain Model.- The Vacuum Wavefunctional.- Anti-deSitter Space and Confinement.- Concluding Remarks.- Index.
15231250
Advent of Non-Abelian Vortices and Monopoles : Further Thoughts about Duality and Confinement(30 Years of Mathematical Methods in High Energy Physics-In Honor of Professor Tohru Eguchi's 60th Birthday-)
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.
53580399
Confinement in Yang–Mills: Elements of a Big Picture
This is a combined and slightly expanded version of talks delivered at 14th International QCD Conference “QCD 08,” 7-12th July 2008, Montpellier, France, the International Conference “Quark Confinement and the Hadron Spectrum,” Mainz, Germany, September 1-6, 2008 (Confinement 08), and the International Conference “Approaches to Quantum Chromodynamics,” Oberwolz, Austria, September 7-13, 2008.
119330850
Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories
In the last decade it became clear that methods and techniques based on supersymmetry provide deep insights in quantum chromodynamics and other supersymmetric and non-supersymmetric gauge theories at strong coupling. In this review we summarize major advances in the critical (Bogomol'nyi-Prasad-Sommerfeld-saturated, BPS for short) solitons in supersymmetric theories and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After a brief introduction in the theory of critical solitons (including a historical introduction) we focus on three topics: (i) non-Abelian strings in N=2 and confined monopoles; (ii) reducing the level of supersymmetry; and (iii) domain walls as D brane prototypes.
7271479
Quantum vortex strings: A review
Abstract The quantum worldsheet dynamics of vortex strings contains information about the 4d non-Abelian gauge theory in which the string lives. Here I tell this story. The string worldsheet theory is typically some variant of the CP N - 1 sigma-model, describing the orientation of the string in a U ( N ) gauge group. Qualitative parallels between 2d sigma-models and 4d non-Abelian gauge theories have been known since the 1970s. The vortex string provides a quantitative link between the two. In 4d theories with N = 2 supersymmetry, the exact BPS spectrum of the worldsheet coincides with the bulk spectrum in 4d. Moreover, by tuning parameters, the CP N - 1 sigma-model can be coaxed to flow to an interacting conformal fixed point which is related to the 4d Argyres–Douglas fixed point. For theories with N = 1 supersymmetry, the worldsheet theory suffers dynamical supersymmetry breaking and, more interestingly, supersymmetry restoration, in a way which captures the physics of Seiberg’s quantum deformed moduli space.
123452983
Exact Classical Solution for the 't Hooft Monopole and the Julia-Zee Dyon
We present an exact solution to the nonlinear field equations which describe a classical excitation possessing magnetic and electric charge. This solution has finite energy and exhibits explicitly those properties which have previously been found by numerical analysis.
119267782
Supersymmetric defect models and mirror symmetry
A bstractWe study supersymmetric field theories in three space-time dimensions doped by various configurations of electric charges or magnetic fluxes. These are supersymmetric avatars of impurity models. In the presence of additional sources such configurations are shown to preserve half of the supersymmetries. Mirror symmetry relates the two sets of configurations. We discuss the implications for impurity models in 3d $ \mathcal{N} $ = 4 QED with a single charged hypermultiplet (and its mirror, the theory of a free hypermultiplet) as well as 3d $ \mathcal{N} $ = 2 QED with one flavor and its dual, a supersymmetric Wilson-Fisher fixed point. Mirror symmetry allows us to find backreacted solutions for arbitrary arrays of defects in the IR limit of $ \mathcal{N} $ = 4 QED. Our analysis, complemented with appropriate string theory brane constructions, sheds light on various aspects of mirror symmetry, the map between particles and vortices and the emergence of ground state entropy in QED at finite density.
46452677
Vortices and impurities
A bstractWe describe the BPS dynamics of vortices in the presence of impurities. We argue that a moduli space of solitons survives the addition of both electric and magnetic impurities. However, dynamics on the moduli space is altered. In the case of electric impurities, the metric remains unchanged but the dynamics is accompanied by a connection term, acting as an effective magnetic field over the moduli space. We give an expression for this connection and compute the vortex-impurity bound states in simple cases. In contrast, magnetic impurities distort the metric on the moduli space. We show that magnetic impurities can be viewed as vortices associated to a second, frozen, gauge group. We provide a D-brane description of the dynamics of vortices in product gauge groups and show how one can take the limit such that a subset of the vortices freeze.
27905458
Multivortex solutions of the Abelian Chern-Simons-Higgs theory.
We have examined vortex solutions in (2+1)D Chern-Simons-Higgs theory which has no usual Maxwell term. It is shown that the Bogomol'nyi-type equations can be derived for a simple sixth-order Higgs potential and corresponding general {ital n}-vortex solutions should contain 2{ital n} free parameters. Various characteristics of Chern-Simons vortices are discussed briefly.
40349828
Self-dual Chern-Simons vortices.
We study vortex solutions in an Abelian Chern-Simons theory with spontaneous symmetry breaking. We show that for a specific choice of the Higgs potential the vortex satisfies a set of Bogomol'nyi-type, or self-duality,'' equations.
124700572
Poles with Both Magnetic and Electric Charges in Nonabelian Gauge Theory
We show that a non-Abelian gauge theory with Higgs fields exhibits classical solutions which are both electrically and magnetically charged. This represents a specific realization of the dyons discussed some years ago by Schwinger. At the classical level the electric charge of the dyon does not appear to be quantized. We present some remarks in this connection.
14379214
Proof of the Julia–Zee Theorem
It is a well accepted principle that finite-energy static solutions in the classical relativistic gauge field theory over the (2 + 1)-dimensional Minkowski spacetime must be electrically neutral. We call such a statement the Julia–Zee theorem. In this paper, we present a mathematical proof of this fundamental structural property.
119162094
Resolution of Chern–Simons–Higgs Vortex Equations
It is well known that the presence of multiple constraints of non-Abelian relativisitic Chern–Simons–Higgs vortex equations makes it difficult to develop an existence theory when the underlying Cartan matrix K of the equations is that of a general simple Lie algebra and the strongest result in the literature so far is when the Cartan subalgebra is of dimension 2. In this paper we overcome this difficulty by implicitly resolving the multiple constraints using a degree-theorem argument, utilizing a key positivity property of the inverse of the Cartan matrix deduced in an earlier work of Lusztig and Tits, which enables a process that converts the equality constraints to inequality constraints in the variational formalism. Thus this work establishes a general existence theorem that settles a long-standing open problem in the field regarding the general solvability of the equations.
121110068
A property of electric and magnetic flux in non-Abelian gauge theories
Abstract Pure non-Abelian gauge models with gauge group SU( N ) are considered in a box with periodic boundary conditions at various temperatures β −1 . Electric and magnetic flux are defined in a gauge-invariant way. The free energy of the system satisfies an exact duality equation, following from Euclidean invariance. The equation relates properties of the electric and the magnetic fields. Conclusions that can be drawn for instance are that for N ⩽ 3 one cannot have both electric and magnetic confinement, and that the infrared structure of the Georgi-Glashow model is self-dual.
122327586
Abrikosov's vortices in the critical coupling
A necessary and sufficient condition is obtained for the existence of multivortex solutions of the Bogomol’nyi system arising in the abelian Higgs theory defined on a rectangular domain and subject to a ’t Hooft type periodic boundary condition. In particular, the number of vortices of a solution is confined by the size of the domain. Such solutions realize the magnetic periodic cell structure in a superconductor predicted by Abrikosov. If the periodic boundary condition is removed, the Bogomol’nyi equations on a bounded domain possess solutions with an arbitrary number of vortices, and these solutions may be used to approximate the unique finite energy solution over the full plane. Moreover, it is shown that, for any given vortex distribution in the plane, the Bogomol’nyi system has a continuous family of gauge-distinct solutions with infinite energy.
117896961
Nonlinear analysis on manifolds, Monge-Ampère equations
1 Riemannian Geometry.- 1. Introduction to Differential Geometry.- 1.1 Tangent Space.- 1.2 Connection.- 1.3 Curvature.- 2. Riemannian Manifold.- 2.1 Metric Space.- 2.2 Riemannian Connection.- 2.3 Sectional Curvature. Ricci Tensor. Scalar Curvature.- 2.4 Parallel Displacement. Geodesic.- 3. Exponential Mapping.- 4. The Hopf-Rinow Theorem.- 5. Second Variation of the Length Integral.- 5.1 Existence of Tubular Neighborhoods.- 5.2 Second Variation of the Length Integral.- 5.3 Myers' Theorem.- 6. Jacobi Field.- 7. The Index Inequality.- 8. Estimates on the Components of the Metric Tensor.- 9. Integration over Riemannian Manifolds.- 10. Manifold with Boundary.- 10.1. Stokes' Formula.- 11. Harmonic Forms.- 11.1. Oriented Volume Element.- 11.2. Laplacian.- 11.3. Hodge Decomposition Theorem.- 11.4. Spectrum.- 2 Sobolev Spaces.- 1. First Definitions.- 2. Density Problems.- 3. Sobolev Imbedding Theorem.- 4. Sobolev's Proof.- 5. Proof by Gagliardo and Nirenberg.- 6. New Proof.- 7. Sobolev Imbedding Theorem for Riemannian Manifolds.- 8. Optimal Inequalities.- 9. Sobolev's Theorem for Compact Riemannian Manifolds with Boundary.- 10. The Kondrakov Theorem.- 11. Kondrakov's Theorem for Riemannian Manifolds.- 12. Examples.- 13. Improvement of the Best Constants.- 14. The Case of the Sphere.- 15. The Exceptional Case of the Sobolev Imbedding Theorem.- 16. Moser's Results.- 17. The Case of the Riemannian Manifolds.- 18. Problems of Traces.- 3 Background Material.- 1. Differential Calculus.- 1.1. The Mean Value Theorem.- 1.2. Inverse Function Theorem.- 1.3. Cauchy's Theorem.- 2. Four Basic Theorems of Functional Analysis.- 2.1. Hahn-Banach Theorem.- 2.2. Open Mapping Theorem.- 2.3. The Banach-Steinhaus Theorem.- 2.4. Ascoli's Theorem.- 3. Weak Convergence. Compact Operators.- 3.1. Banach's Theorem.- 3.2. The Leray-Schauder Theorem.- 3.3. The Fredholm Theorem.- 4. The Lebesgue Integral.- 4.1. Dominated Convergence Theorem.- 4.2. Fatou's Theorem.- 4.3. The Second Lebesgue Theorem.- 4.4. Rademacher's Theorem.- 4.5. Fubini's Theorem.- 5. The LpSpaces.- 5.1. Regularization.- 5.2. Radon's Theorem.- 6. Elliptic Differential Operators.- 6.1. Weak Solution.- 6.2. Regularity Theorems.- 6.3. The Schauder Interior Estimates.- 7. Inequalities.- 7.1. Holder's Inequality.- 7.2. Clarkson's Inequalities.- 7.3. Convolution Product.- 7.4. The Calderon-Zygmund Inequality.- 7.5. Korn-Lichtenstein Theorem.- 7.6. Interpolation Inequalities.- 8. Maximum Principle.- 8.1. Hopf's Maximum Principle.- 8.2. Uniqueness Theorem.- 8.3. Maximum Principle for Nonlinear Elliptic Operator of Order Two.- 8.4. Generalized Maximum Principle.- 9. Best Constants.- 9.1. Application to Sobolev Spaces.- 4 Green's Function for Riemannian Manifolds.- 1. Linear Elliptic Equations.- 1.1. First Nonzero Eigenvalue ? of ?.- 1.2. Existence Theorem for the Equation ?? = f.- 2. Green's Function of the Laplacian.- 2.1. Parametrix.- 2.2. Green's Formula.- 2.3. Green's Function for Compact Manifolds.- 2.4. Green's Function for Compact Manifolds with Boundary.- 5 The Methods.- 1. Yamabe's Equation.- 1.1. Yamabe's Method.- 2. Berger's Problem.- 2.1. The Positive Case.- 3. Nirenberg's Problem.- 3.1. A Nonlinear Theorem of Fredholm.- 3.2. Open Questions.- 6 The Scalar Curvature.- 1. The Yamabe Problem.- 1.1. Yamabe's Functional.- 1.2. Yamabe's Theorem.- 2. The Positive Case.- 2.1. Geometrical Applications.- 2.2. Open Questions.- 3. Other Problems.- 3.1. Topological Meaning of the Scalar Curvature.- 3.2. Kazdan and Warner's Problem.- 7 Complex Monge-Ampere Equation on Compact Kahler Manifolds.- 1. Kahler Manifolds.- 1.1 First Chern Class.- 1.2. Change of Kahler Metrics. Admissible Functions.- 2. Calabi's Conjecture.- 3. Einstein-Kahler Metrics.- 4. Complex Monge-Ampere Equation.- 4.1. About Regularity.- 4.2. About Uniqueness.- 5. Theorem of Existence (the Negative Case).- 6. Existence of Kahler-Einstein Metric.- 7. Theorem of Existence (the Null Case).- 8. Proof of Calabi's Conjecture.- 9. The Positive Case.- 10. A Priori Estimate for ??.- 11. A Priori Estimate for the Third Derivatives of Mixed Type.- 12. The Method of Lower and Upper Solutions.- 8 Monge-Ampere Equations.- 1. Monge-Ampere Equations on Bounded Domains of ?n.- 1.1. The Fundamental Hypothesis.- 1.2. Extra Hypothesis.- 1.3. Theorem of Existence.- 2. The Estimates.- 2.1. The First Estimates.- 2.2. C2-Estimate.- 2.3. C3-Estimate.- 3. The Radon Measure ?(?).- 4. The Functional ? (?).- 4.1. Properties of ? (?).- 5. Variational Problem.- 6. The Complex Monge-Ampere Equation.- 6.1. Bedford's and Taylor's Results.- 6.2. The Measure M(?).- 6.3. The Functional J(?).- 6.4. Some Properties of J(?).- 7. The Case of Radially Symmetric Functions.- 7.1. Variational Problem.- 7.2. An Open Problem.- 8. A New Method.- Notation.
59456762
Vortices in holomorphic line bundles over closed Kähler manifolds
We apply a modified Yang-Mills-Higgs functional to unitary bundles over closed Kähler manifolds and study the equations which govern the global minima. The solutions represent vortices in holomorphic bundles and are direct analogs of the vortices overR2. We obtain a complete description of the moduli space of these new vortices where the bundle is of rank one. The description is in terms of a class of divisors in the base manifold. There is also a dependence on a real valued parameter which can be attributed to the compactness of the base manifold.
119323771
Geometry and Energy of Non-abelian Vortices
We study pure Yang--Mills theory on $\Sigma\times S^2$, where $\Sigma$ is a compact Riemann surface, and invariance is assumed under rotations of $S^2$. It is well known that the self-duality equations in this set-up reduce to vortex equations on $\Sigma$. If the Yang--Mills gauge group is $\SU{2}$, the Bogomolny vortex equations of the abelian Higgs model are obtained. For larger gauge groups one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang--Mills theory with gauge group $\SU{N}/\ZZ_N$ and a special reduction which yields only one non-abelian Higgs field. One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator $N$, we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of non-trivial lower and upper bounds for the energy of the reduced theory on $\Sigma$. These bounds are proportional to the area of $\Sigma$. We give special solutions of the theory on $\Sigma$ by embedding solutions of the abelian Higgs model into the non-abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang--Mills theory.
120507041
Yang–Mills–Higgs theory on a compact Riemann surface
Jaffe and Taubes [Vortices and Monopoles (Birkhauser, Boston, 1980)] have shown the existence and uniqueness of n‐vortex solutions on the complex plane. In this paper, their results are generalized to an arbitrary U(1) bundle over a compact Riemann surface with a Hermitian metric. Berger’s ‘‘nonlinear analysis’’ [Nonlinearity and Functional Analysis (Academic, New York, 1977)] has provided an effective method to prove the existence part of the main theorem of this paper.
117945254
Solitons in Field Theory and Nonlinear Analysis
Preface * Primer of Field Theory * Sigma Models * Multiple Instantons and Characteristic Classes * Generalized Abelian Higgs Equations * Chern-Simons Systems: Abelian Case * Chern-Simons Systems: Non-Abelian Case * Electroweak Vortices * Dyons * Ordinary Differential Equations * Strings in Cosmology * Vortices and Antivortices * Born-Infeld Solutions * References * Bibliography * Index
120842110
Vortex condensation in the Chern-Simons Higgs model: An existence theorem
It is shown that there is a critical value of the Chern-Simons coupling parameter so that, below the value, there exists self-dual doubly periodic vortex solutions, and, above the value, the vortices are absent. Solutions of such a nature indicate the existence of dyon condensates carrying quantized electric and magnetic charges.
122902919
Multiple condensate solutions for the Chern–Simons–Higgs theory
We study the existence of condensate solutions for the Chern‐Simons‐Higgs model with the choice of a potential field where both the symmetric and asymmetric vacua occur as ground states [see Hong, Kim, and Pac, Phys. Rev. Lett. 64, 2230 (1990) and Jackiw and Weinberg, ibid. 64, 2234 (1990)]. We show that if the Chern‐Simons coupling parameter k is above a critical value, no such solutions can exist, while for k≳0 below this critical value there exist at least two condensate solutions carrying the same quantized energy, as well as electric and magnetic charge. This multiplicity result accounts for the two vacua states present in the model. In fact, as k→0+ it is shown that the two solutions found ‘‘bifurcate’’ from the asymmetric and symmetric vacuum states respectively.
119144392
Non-Abelian Vortices in Supersymmetric Gauge Field Theory via Direct Methods
Vortices in supersymmetric gauge field theory are important constructs in a basic conceptual phenomenon commonly referred to as the dual Meissner effect which is responsible for color confinement. Based on a direct minimization approach, we present a series of sharp existence and uniqueness theorems for the solutions of some non-Abelian vortex equations governing color-charged multiply distributed flux tubes, which provide an essential mechanism for linear confinement. Over a doubly periodic domain, existence results are obtained under explicitly stated necessary and sufficient conditions that relate the size of the domain, the vortex numbers, and the underlying physical coupling parameters of the models. Over the full plane, existence results are valid for arbitrary vortex numbers and coupling parameters. In all cases, solutions are unique.
121989581
The Relativistic non-abelian Chern-Simons Equations
AbstractWe study ther xr system of nonlinear elliptic equations
$$\Delta u_a = - \lambda \sum\limits_{b = 1}^r {K_{ab} e^{u_b } + \lambda } \sum\limits_{b = 1}^r {\sum\limits_{c = 1}^r {e^{u_b } K_{ab} e^{u_c } K_{bc} + 4\pi \sum\limits_{j = 1}^{N_a } {\delta _{p_{aj} } } } } $$
,a=1,2,...,r,x∈R2, where λ τ 0 is a constant parameter,K = (Kab) is the Cartan matrix of a semi-simple Lie algebra, and βp is the Dirac measure concentrated atpR2. This system of equations arises in the relativistic non-Abelian Chern-Simons theory and may be viewed as a nonintegrable deformation of the integrable Toda system. We establish the existence of a class of solutions known as topological multivortices. The crucial step in our method is the use of the decomposition theorem of Cholesky for positive definite matrices so that a variational principle can be formulated.