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.

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title: Nonlinear analysis on manifolds, Monge-Ampère equations
abstract: 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.  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.  13. Improvement of the Best Constants.- 14. The Case of the Sphere.- 15. The Exceptional Case of the Sobolev Imbedding Theorem.- 16.  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.   The Banach-Steinhaus Theorem.- 2.4.  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.  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.  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 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.

title: Vortices in holomorphic line bundles over closed Kähler manifolds
abstract: 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.

title: Geometry and Energy of Non-abelian Vortices
abstract: 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.

title: Yang–Mills–Higgs theory on a compact Riemann surface
abstract: 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.

title: Solitons in Field Theory and Nonlinear Analysis
abstract: Preface * Primer of Field Theory *  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

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abstract: 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,  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.

title: Non-Abelian Vortices in Supersymmetric Gauge Field Theory via Direct Methods
abstract: 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.

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$$\Delta u_a = - \lambda \sum\limits_{b = 1}^r {K_{ab} e^{u_b } + \lambda } \sum\limits_{b = 1}^r {\sum\limits_{c = 1}^r {e^{u_b } K_{ab} e^{u_c } K_{bc} + 4\pi \sum\limits_{j = 1}^{N_a } {\delta _{p_{aj} } } } } $$
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