Self-Dual Chern-Simons Theories

Abstract

In these lectures I review classical aspects of the self-dual Chern-Simons systems which describe charged scalar fields in $2+1$ dimensions coupled to a gauge field whose dynamics is provided by a pure Chern-Simons Lagrangian. These self-dual models have one realization with nonrelativistic dynamics for the scalar fields, and another with relativistic dynamics for the scalars. In each model, the energy density may be minimized by a Bogomol'nyi bound which is saturated by solutions to a set of first-order self-duality equations. In the nonrelativistic case the self-dual potential is quartic, the system possesses a dynamical conformal symmetry, and the self-dual solutions are equivalent to the static zero energy solutions of the equations of motion. The nonrelativistic self-duality equations are integrable and all finite charge solutions may be found. In the relativistic case the self-dual potential is sixth order and the self-dual Lagrangian may be embedded in a model with an extended supersymmetry. The self-dual potential has a rich structure of degenerate classical minima, and the vacuum masses generated by the Chern-Simons Higgs mechanism reflect the self-dual nature of the potential.

42 pages LaTeX

References in corpus (10)

• Aharonov-Bohm Scattering, Contact Interactions and Scale Invariance
• Extended Superconformal Galilean Symmetry in Chern-Simons Matter Systems
• Self-Dual Chern-Simons Higgs Systems with an N=3 Extended Supersymmetry
• Self-DUal SU(3) Chern-Simons Higgs Systems
• Conformal symmetry of the coupled Chern-Simons and gauged nonlinear Schrödinger equations
• Non-Abelian Chern-Simons Particles and their Quantization
• Vacuum Mass Spectra for SU(N) Self-Dual Chern-Simons-Higgs Systems
• Perturbative Renormalizations of Anyon Quantum Mechanics
• The Chern-Simons Coefficient in the Higgs Phase
• Symmetry Breaking in the Schrödinger Representation for Chern-Simons Theories

Cited by in corpus (133)

• Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources
• Aspects of Chern-Simons Theory
• Generalized self-dual Chern-Simons vortices
• An Investigation of the $K_F$-type Lorentz-Symmetry Breaking Gauge Models in $N=1$-Supersymmetric Scenario
• Non-Abelian Vortices in Supersymmetric Gauge Field Theory via Direct Methods
• Maxwell-Chern-Simons Scalar Electrodynamics at Two Loop
• Classification of blowup limits for SU(3) singular Toda systems
• Standing waves for a gauged nonlinear Schrödinger equation with a vortex point
• Exotic galilean symmetry and non-commutative mechanics
• Chern--Simons Vortices in the Gudnason Model
• Topological self-dual configurations in a Lorentz-violating gauged O(3) sigma model
• Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in ${\mathbb{R}}^2$
• An improved geometric inequality via vanishing moments, with applications to singular Liouville equations
• Aspects of a planar nonbirefringent and CPT-even electrodynamics stemming from the Standard Model Extension
• Self-duality and soldering in odd dimensions
• Resolution of Chern--Simons--Higgs Vortex Equations
• Existence and multiplicity result for the singular Toda system
• Existence Theorems for Vortices in the Aharony--Bergman--Jaferis--Maldacena Model
• Two Species of Vortices in a massive Gauged Non-linear Sigma Model
• Uncharged compactlike and fractional Lorentz-violating BPS vortices in the CPT-even sector of the standard model extension
• Pauli's Term and Fractional Spin
• Supersymmetric D-brane Bound States with B-field and Higher Dimensional Instantons on Noncommutative Geometry
• Topological vortices in generalized Born-Infeld-Higgs electrodynamics
• A Moser-Trudinger inequality for the singular Toda system
• Galilean noncommutative gauge theory: symmetries & vortices
• A topological join construction and the Toda system on compact surfaces of arbitrary genus
• Degree counting and shadow system for $SU(3)$ Toda system: one bubbling
• Consistency analysis of a nonbirefringent Lorentz-violating planar model
• The asymptotic quasi-stationary states of the two-dimensional magnetically confined plasma and of the planetary atmosphere
• Supersymmetric Chern-Simons Vortex Systems and Extended Supersymmetric Quantum Mechanics Algebras
• Topological charged BPS vortices in Lorentz-violating Maxwell-Higgs electrodynamics
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles
• Vortices and Jacobian varieties
• Noncommutative Chern-Simons Soliton
• Quantum Description of Anyons: Role of Contact Terms
• Vortex Condensates in the Relativistic Self-Dual Maxwell-Chern-Simons-higgs System
• On non-topological solutions for planar Liouville Systems of Toda-type
• Compact Chern-Simons vortices
• Global wellposedness of the equivariant Chern-Simons-Schrödinger equation
• A Typology for Quantum Hall Liquids
• Non-Yang-Mills Gauge Theories
• Chiral Solitons in a Current Coupled Schrödinger Equation With Self Interaction
• Moving vortices in noncommutative gauge theory
• The symmetries of the Manton superconductivity model
• Classification of solutions of a Toda system in R^2
• Some Comments on the Spin of the Chern - Simons Vortices
• Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in ${\mathbb{R}}^2$
• On a nonlinear elliptic system from Maxwell-Chern-Simons vortex theory
• On the solitons of the Chern-Simons-Higgs model
• All timelike supersymmetric solutions of three-dimensional half-maximal supergravity
• Casimir force due to condensed vortices in a plane
• Topology of non-topological Chern-Simons vortices
• New blow-up phenomena for SU(n+1) Toda system
• On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation: existence, uniqueness, and instability
• One-loop mass shift formula for kinks and self-dual vortices
• The Non-abelian Chern-Simons Coefficient in the Higgs Phase
• Monopoles, vortices and kinks in the framework of non-commutative geometry
• Lowest weight representations of super Schrodinger algebras in low dimensional spacetime
• Non-degeneracy of Gauss curvature equation with negative conic singularity
• Chern-Simons Solitons in Quantum Potential
• Spontaneous Symmetry Breaking in the Non-Abelian Anyon Fluid
• On Non-topological Solutions of the ${\mathbf{G}}_2$ Chern-Simons System
• Vortices in the Landau--Ginzburg Model of the Quantized Hall Effect
• Galilean Supersymmetry in 2+1 Dimensions
• Gravitating Self-dual Chern-Simons Solitons
• Non-topological Vortex Configurations in the ABJM Model
• Degree counting for Toda system with simple singularity : one point blow up
• Introduction to Grassmann Manifolds and Quantum Computation
• Chern--Simons Terms as an Example of the Relations Between Mathematics and Physics
• A continuum of solutions for the SU(3) Toda System exhibiting partial blow-up
• Bäcklund transformation for non-relativistic Chern-Simons vortices
• Renormalization In Coupled-Abelian Self-Dual Chern-Simons Models
• On Liouville systems at critical parameters, Part 1: one bubble
• The Chern-Simons Coefficient in Supersymmetric Non-abelian Chern-Simons Higgs Theories
• Boundary charges and integral identities for solitons in $(d+1)$-dimensional field theories
• On the Topological degree of the Mean field equation with two parameters
• Mass Spectra of N=2 Supersymmetric SU(n) Chern-Simons-Higgs Theories
• Vortex solutions of the Liouville equation
• New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces
• Almost Optimal Local Well-Posedness of the Chern-Simons-Dirac System in the Coulomb Gauge
• Singular mean field equations on compact Riemann surfaces
• Asymmetric blow-up for the SU(3) Toda System
• Multiplicity for a nonlinear elliptic fourth order equation in Maxwell-Chern-Simons vortex theory
• About the self-dual Chern-Simons system and Toda field theories on the noncommutative plane
• Quantum phase transitions of fractons (a fractal scaling theory for the FQHE)
• On the general Toda system with multiple singular points
• Degree counting theorems for singular Liouville systems
• Field theoretical formulation of the asymptotic relaxation states of two-dimensional ideal fluids
• Physico--Mathematical Interactions: The Chern--Simons Story
• On the late phase of relaxation of two-dimensional fluids: turbulence of unitons
• Dimensionally Reduced Chern-Simons Terms and their Solitons
• Existence of Non-Abelian Vortices in the Aharony--Bergman--Jafferis--Maldacena Theory
• Bogomol'nyi equations in gauge theories
• Local profile of fully bubbling solutions to SU(n+1) Toda Systems
• Self-dual configurations in a generalized Abelian Chern-Simons-Higgs model with explicit breaking of the Lorentz covariance
• Periodic Maxwell-Chern-Simons vortices with concentrating property
• Convergence rate, location and ${\partial }_z^2$ condition for fully bubbling solutions to SU(n+1) Toda Systems
• Solitons in a gauged Landau-Lifshitz model
• Uniqueness of topological solutions of self-dual Chern-Simons equation with collapsing vortices
• Classification of Radial Solutions to Liouville Systems with Singularities
• Polymer plats and multicomponent anyon gases
• Existence results for mean field equations with turbulence
• Asymptotic Behavior of Blowup Solutions for Elliptic Equations with Exponential Nonlinearity and Singular Data
• The Field Theory of Generalized Ferromagnet on the Hermitian Symmetric Spaces
• Fractal sets of dual topological quantum numbers
• Exact vortex solution of the Jacobs-Rebbi equation for ideal fluids
• Extended Supersymmetries and 2+1 Dimensional Supersymmetric Chern Simons Theories
• Low Dimensional Supersymmetries in SUSY Chern-Simons Systems and Geometrical Implications
• Prescribing the Gaussian curvature in a subdomain of S^2 with Neumann boundary condition
• Non-topological condensates for the self-dual Chern-Simons-Higgs model
• A new type of non-topological bubbling solutions to a competitive Chern-Simons model
• Small data global existence and decay for relativistic Chern-Simons equations
• The Existence of Multi-vortices for a Generalized Self-dual Chern-Simons Model
• Topological quantization of self-dual Chern-Simons vortices on Riemann Surfaces
• Symmetry of solutions of a mean field equation on flat tori
• Wave equations associated to Liouville-type problems: global existence in time and blow up criteria
• Vanishing estimates for fully bubbling solutions of $SU(n+1)$ Toda Systems at a singular source
• A variational Analysis of the Toda System on Compact Surfaces
• Multiple Solutions for the Non-Abelian Chern--Simons--Higgs Vortex Equations
• Almost critical regularity of non-abelian Chern-Simons-Higgs system in the Lorenz gauge
• On Threshold Solutions of equivariant Chern-Simons-Schrödinger Equation
• Physical aspects of the field-theoretical description of two-dimensional ideal fluids
• Topological Multivortex Solutions of the Self-Dual Maxwell-Chern-Simons-Higgs System
• Critical gauged Schrödinger equations in R^2 with vanishing potentials
• Chern-Simons Solitons, Chiral Model, and (affine) Toda Model on Noncommutative Space
• The moduli space of non-Abelian vortices in Yang-Mills-Chern-Simons-Higgs theory
• The blow-up analysis of an affine Toda system corresponding to superconformal minimal surfaces in ${\mathbb{S}}^4$
• Degree Counting Theorems for 2x2 non-symmetric singular Liouville Systems
• On Liouville systems at critical parameters, Part 2: Multiple bubbles
• Blow up at infinity in the SU(3) Chern-Simons model, part I
• Well-posedness in a critical space of Chern-Simons-Dirac system in the Lorenz gauge
• Estimates of bubbling solutions of $SU(3)$ Toda systems at critical parameters-Part 1
• Analytic aspects of the Toda system: I. A Moser-Trudinger inequality