Topology of Higgs fields

Magnetic monopole dynamics, supersymmetry, and duality

“…Hence, terms in D and D † that fall exponentially with distance can be ignored. The potentially dangerous 15 On a compact space, where D † D and DD † would have discrete spectra with identical nonzero eigenvalues, I(M 2 ) would be independent of M 2 . The fact that it is not is a consequence of the continuum spectra.…”

“…15 Each normalizable zero mode of D † D contributes 1 to the right-hand side of Eq. (4.2.9), while each normalizable zero mode of DD † contributes −1.…”

Regularity of static axially symmetric solutions inSU(2)Yang-Mills-dilaton theory

“…with H to leading order in the expansion Eq. (22). As for winding number n = 2 the gauge transformed functions for n = 3 and n = 4 fulfill the requirement, Eqs.…”

Existence of Dyons in the Coupled Georgi–Glashow–Skyrme Model

“…The Higgs field Φ maps the infinity of R 3 which is topological the 2-sphere S 2 into the SU (2) or SO(3) vacuum manifold, modulo the residual symmetry U (1) or SO (3) and, hence, is classified by the homotopy group π 2 (SU (2)/ U (1)) = Z. In other words, the solution corresponds to an integer class called the monopole charge, Q M , which is actually the Brouwer degree or "winding number" of the normalized mapΦ = Φ |Φ| (5.8) from a 2-sphere near infinity of R 3 into S 2 and has been identified with the magnetic charge [5,31], resulting in the conclusion Q M = 1. This fact can also be verified directly when we insert (2.10) into (5.8) and use the well-known as desired.…”

“…Skyrme baryon charges of the solutions obtained. We will see that, while the magnetic charge is uniquely determined by the monopole charge which is a homotopy invariant [5,31], the electric charge, Q e , remains undetermined even with the added topological invariant-the Skyrme baryon charge. In fact, this study shows that, in the context of the formulation of Brihaye et al [12], Q e may be prescribed in an open interval.…”