Zero modes of the Dirac operator in three dimensions

Abstract

We investigate zero modes of the Dirac operator coupled to an Abelian gauge field in three dimensions. We find that the existence of a certain class of zero modes is related to a specific topological property precisely when the requirement of finite Chern--Simons action is imposed.Comment: 13 pages, 6 figures, uses the macro psbox.tex, replaced by a revised version to be published in Phys. Rev. D. The section on the Seiberg-Witten equations, which contained a sign error, has been removed. This removal leads to further issues which will appear in a future publicatio

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

“…After the work by Loss and Yau [13] was published, there have been many contributions on the study of zero modes of Weyl-Dirac operators. See Adam, Muratori and Nash [1], [2], [3], Balinsky and Evans [4], [5], [6], Bugliaro, Fefferman and Graf [7], Elton [8] and, Erdös and Solovej [9], [10], [11].…”

Particle creation via relaxing hypermagnetic knots

“…of (16). Mathematically, the curvature F = 1 2 F ij dx i dx j , F ij = ǫ ijk B k , is the pullback under the Hopf map, F = χ * Ω, of the standard area two-form Ω on the target S 2 .…”

“…three-dimensional) Abelian Dirac operator, at all. The first example of such a zero mode was given only in 1986 in [15], and some further results including the feature of zero mode degeneracy were obtained recently in [16]- [21].…”

Asymptotics for Erdős–Solovej Zero Modes in Strong Fields

“…The operator D R 3 ,B acts on two-component spinor fields, which, on R 3 , can be viewed simply as C 2 valued functions. Standard arguments (see [19,Theorem 4.3] for example) show that D R 3 ,B is Zero modes have been studied in a number of contexts in mathematical physics including the stability of matter [13,17] and chiral gauge theories [1,2]. Most early works concentrated on the construction of explicit examples, including the original example [17], examples with arbitrary multiplicity [2], compact support [6] and a certain rotational type of symmetry ( [10]; further details below).…”

“…This line bundle determines Ψ up to isomorphism (note that H 2 (S 2 ; Z) ∼ = Z which has no two-torsion). On S 2 , there are infinitely many mutually non-isomorphic spin c spinor bundles which we denote as Ψ (k) for k ∈ Z, labelled so that the first Chern number of the associated line bundle satisfies c 1 …”