Resolution of Chern–Simons–Higgs Vortex Equations

Abstract

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. IntrodcutionThe main result of this paper is a proof of a general existence theorem for the doublyperiodic solutions of the relativistic non-Abelian Chern-Simons-Higgs vortex equations whose Cartan matrix K is that of an arbitrary simple Lie algebra. The motivation of our work originates from theoretical physics and the development of new methods of mathematical analysis to tackle systems of nonlinear equations of physical interests. Below, we begin with a description of some of the field-theory aspects of our study.Field-theoretical origins. In 1957, Abrikosov [1] predicted in the context of the Ginzburg-Landau theory [18,30,56,65] that vortices of a planar lattice structure may appear in superconductors of the second type. Due to the complexity of the problem, a complete mathematical understanding of such Abrikosov vortices has not been obtained yet, although in an over simplified situation, known as the Bogomol'nyi-PrasadSommerfield (BPS) limit [9,54], an existence and uniqueness theorem is proved under a necessary and sufficient condition [68] where the Abrikosov lattice structure is realized through gauge-periodic boundary conditions conceptualized and formulated by 't Hooft

Magnetic impurity inspired Abelian Higgs vortices

“…Assuming the zero set of φ is as given in (6.1) and taking u = ln |φ| 2 as before, we reduce (4.9)-(4.10) into ∆u = 4 κ 2 e u e u − ζ − σ(x) + 4π Integrating (6.12), we arrive at the constraint 13) similar to that in the Abelian Higgs model case, which leads us to the necessary condition…”

“…12) 13) whose solutions carry the energy or vortex tension 14) which is seen to be indifferent to the magnetic source term. It may be examined directly that (2.12)-(2.13) imply (2.8)-(2.9).…”

“…Using a recently developed approach in [13] we obtain the following existence results, assuming ac + bd < 0 in addition, for technical reasons.…”

Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations

“…To formulate the system (2.14) in a variational form, as in [25], we rewrite (2.14) equivalently as follows:…”

“…In [25] the above equations (2.22) are shown to be uniquely solvable when one takes ε j = 1, ∀j = 1, . .…”