Strings in supersymmetric U(1)N gauge theories with N hypermultiplets are studied in the generic setting of an arbitrary Fayet–Iliopoulos triplet of parameters for each gauge group and an invertible charge matrix. Although the string tension is generically of a square-root form, it turns out that all existing Bogomol’nyi–Prasad–Sommerfield solutions have a tension that is linear in the magnetic fluxes, which, in turn, are linearly related to the winding numbers. The main result is a series of theorems establishing three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in SU(2)R space. We further prove for all cases that a seemingly vanishing Bogomol’nyi bound cannot have solutions. Finally, we write down the most general vortex equations in both master form and Taubes-like form. Remarkably, the final vortex equations essentially look Abelian in the sense that there is no trace of the SU(2)R symmetry in the equations after the constraint equations have been solved.
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BPS is an abbreviation for Bogomol’nyi–Prasad–Sommerfield who were the first to discover a reduction from second-order Euler–Lagrange equations for a magnetic monopole to first-order partial differential equations (PDEs), now known as BPS equations. BPS equations and solutions are now used in many areas of theoretical physics, from gauge theory to string theory.
Actually, there is a vacuum modulus (a complex phase) for each gauge group in the case of M = N, which corresponds to the base point of the broken U(1) symmetry. This parameter has no physical relevance.
Strictly speaking, we should take the limiting value of U for m = (0, 0, ±1), i.e., .
Like for U, should be taken as the limiting value for m3 = ±1, i.e., , which for m3 → 1 yields . However, the limit m3 → −1 is ambiguous, and this introduces a spurious phase α, i.e., , which survives the limit of m3 → −1. Noticing that m3 → −1 corresponds simply to swapping ψA with , we can fix the ambiguous phase α, so the limit reads .
We do not allow n = 0, Q = 0 and, therefore, neither k = 0 since that is the (trivial) vacuum solution.
