Xiaosen Han

Chern–Simons vortices in the Gudnason model

Abstract We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the N = 2 supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern–Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an inequality-constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.

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.

Existence Theorems for Vortices in the Aharony-Bergman-Jaferis-Maldacena Model

A series of sharp existence and uniqueness theorems are established for the multiple vortex solutions in the supersymmetric Chern–Simons–Higgs theory formalism of Aharony, Bergman, Jaferis, and Maldacena, for which the Higgs bosons and Dirac fermions lie in the bifundamental representation of the general gauge symmetry group

Magnetic impurity inspired Abelian Higgs vortices

{U(N)\times U(N)}

Non-topological Vortex Configurations in the ABJM Model

U(N)×U(N). The governing equations are of the BPS type and derived by Kim, Kim, Kwon, and Nakajima in the mass-deformed framework labeled by a continuous parameter.

Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model

An existence theorem is established for the solutions to the non-Abelian relativistic Chern-Simons-Higgs vortex equations over a doubly periodic domain when the gauge group G assumes the most general and important prototype form, G = SU(N). Mathematics subject classification (2010). 35C08, 35J50, 35Q70, 81E13

Bubbling solutions for a skew-symmetric Chern–Simons system in a torus

A bstractInspired by magnetic impurity considerations some broad classes of Abelian Higgs and Chern-Simons-Higgs BPS vortex equations are derived and analyzed.

Existence of non-Abelian vortices with product gauge groups

We study a recently developed product Abelian gauge field theory by Tong and Wong hosting magnetic impurities. We first obtain a necessary and sufficient condition for the existence of a unique solution realizing such impurities in the form of multiple vortices. We next reformulate the theory into an extended model that allows the coexistence of vortices and anti-vortices. The two Abelian gauge fields in the model induce two species of magnetic vortex-lines resulting from Ns vortices and Ps anti-vortices (s = 1, 2) realized as the zeros and poles of two complex-valued Higgs fields, respectively. An existence theorem is established for the governing equations over a compact Riemann surface S which states that a solution with prescribed N1, N2 vortices and P1, P2 anti-vortices of two designated species exists if and only if the inequalities |N1 + N2 − (P1 + P2)| < |S| π , |N1 + 2N2 − (P1 + 2P2)| < |S| π , hold simultaneously, which give bounds for the ‘differences’ of the vortex and anti-vortex numbers in terms of the total surface area of S. The minimum energy of these solutions is shown to assume the explicit value E = 4π(N1 + N2 + P1 + P2), given in terms of several topological invariants, measuring the total tension of the vortex-lines.