A Sharp Existence Theorem for Vortices in the Theory of Branes

Abstract

We investigate the BPS equations arising from the theory of multiply intersecting D-branes. By using the direct minimization method, we establish a sharp existence and uniqueness theorem for multiple vortex solutions of the BPS equations over a doubly periodic domain and over the full plane, respectively. In particular, we obtain an explicit necessary and sufficient condition for the existence of a unique solution for the doubly periodic domain case. (2000). 35A05, 58E50. Mathematics Subject Classification

Existence and uniqueness of solutions to Bogomol'nyi-Prased-Sommerfeld equations on graphs

“…Taubes established the multiple vortex static solutions for the Abelian Higgs model for the first time in [31,32,41]. After that, a large number of work related to vortex equations has been accomplished; see, for example, [4,15,33,35,39] and the references therein. Recently, Lin and Yang pursued a systematic research [22,23] of the multiple vortex equations obtained in [7,8,9,12,27,28,29,30].…”

“…In [40], Yang and Lieb presented a series of sharp existence and uniqueness theorems for the solutions of some non-Abelian vortex equations derived in [5,6], which provide an essential mechanism for linear confinement. Recently, Han [15] established the existence of multipe vortex solutions for the BPS equations derived in [26] from the theory of multi-intersection of D-branes. Chen and Yang [6] proved two sharp existence theorems for the non-Abelian BPS vortex equations arising in the supersymmetric U(1) × SU(N) gauge theory.…”

“…Chen and Yang [6] proved two sharp existence theorems for the non-Abelian BPS vortex equations arising in the supersymmetric U(1) × SU(N) gauge theory. In this paper, we investigate the existence and uniqueness of the solutions to the two BPS equations in [6,15] on a connected finite graph.…”