Yisong Yang

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.

The existence of non-topological solitons in the self-dual Chern-Simons theory

In the recently discovered (2+1)-dimensional relativistic Chern-Simons model, self-duality can be achieved when the Higgs potential density assumes a special form for which both the asymmetric and symmetric vacua are ground state solutions. This important feature may imply the coexistence of static topological and non-topological vortex-like solutions inR2 but the latter have been rather elusive to a rigorous construction. Our main purpose in this paper is to prove the existence of non-topological radially symmetricN-vortex solutions in the self-dual Chern-Simons model. By a shooting method, we obtain a continuous family of gauge-distinctN-vortex solutions. Moreover, we are also able to classify all possible bare (or 0-vortex) solutions.

Topological solutions in the self-dual Chern-Simons theory: existence and approximation

Abstract In this paper a globally convergent computational scheme is established to approximate a topological multivortex solution in the recently discovered self-dual Chern-Simons theory in R 2 . Our method which is constructive and numerically efficient finds the most superconducting solution in the sense that its Higgs field has the largest possible magnitude. The method consists of two steps: first one obtains by a convergent monotone iterative algorithm a suitable solution of the bounded domain equations and then one takes the large domain limit and approximates the full piane solutions. It is shown that with a special choice of the initial guess function, the approximation sequence approaches exponentially fast a solution in R 2 . The convergence rate implies that the truncation errors away from local regions are insignificant.

A nonlinear elliptic equation arising from gauge field theory and cosmology

We study radially symmetric solutions of a nonlinear elliptic partial differential equation in R2 with critical Sobolev growth, i. e. the nonlinearity is of exponential type. This problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology. Our results lead to many interesting implications for the physical problems considered. For example, for the self-dual Chern–Simons theory, we are able to conclude that the electric charge, magnetic flux, or energy of a non-topological N-vortex solution may assume any prescribed value above an explicit lower bound. For the Einstein-matter-gauge equations, we find a necessary and sufficient condition for the existence of a self-dual cosmic string solution. Such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant.

The Relativistic non-abelian Chern-Simons Equations

AbstractWe study ther xr system of nonlinear elliptic equations

A necessary and sufficient condition for the existence of multisolitons in a self-dual gauged sigma model

\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} } } } }

Electrically and magnetically charged vortices in the Chern-Simons-Higgs theory

,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.

Dually charged particle-like solutions in the Weinberg-Salam theory

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with NF charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(NC) gauge theory with NF hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(NC) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.