Zero-modes of non-Abelian solitons in three-dimensional gauge theories

Abstract

We study non-Abelian solitons of the Bogomol'nyi type in N = 2 (d = 2 + 1) supersymmetric Chern-Simons (CS) and Yang-Mills (YM) theory with a generic gauge group. In CS theory, we find topological, non-topological and semi-local (non-)topological vortices of non-Abelian kinds in unbroken, broken and partially broken vacua. We calculate the number of zero-modes using an index theorem and then we apply the moduli matrix formalism to realize the moduli parameters. For the topological solitons we exhaust all the moduli while we study several examples of the non-topological and semi-local solitons.We find that the zero-modes of the topological solitons are governed by the moduli matrix H 0 only and those of the non-topological solitons are governed by both H 0 and the gauge invariant field Ω. We prove local uniqueness of the master equation in the YM case and finally, compare all results between the in the early '90s and we shall refer to all these as solitons of the Abelian kind. 1The second era of the CS solitons recently started when a new type of the toplogical CS soliton of the non-Abelian (NA) kind 2 was found in the U(N) CS theory coupled with N Higgs fields in the fundamental representation [21]. The NA CS vortex found so far is quite similar to the NA YM vortex [17,18,19,20]. It lives in the color-flavor locked, broken vacuum (i.e. the Higgs vacuum) and has internal orientational zero-modes, CP N −1 . These internal orientational modes are related to the color and flavor degrees of freedom, which make the solitons being truly of the NA kind. Soon after its discovery, the NA semi-local vortices were found in the U(N) CS theory with N f > N fundamental Higgs fields [22,23]. The dynamics of the NA vortices was studied in Ref.[24] and the NA vortices with a globally conserved charge was found in Ref. [25].The topological solitons of both local, semi-local and fractional kind in the NA CS theory with an arbitrary gauge group of the form U(1) times a simple group have also been studied very recently in Refs. [26,27]. Thus, the study on the CS solitons is being revived now.The NA CS solitons found so far are all topological solitons in the broken vacuum (the Higgs vacuum). This is because their discovery [21] was inspired by the NA YM vortex which is indeed a topological soliton living in the broken (Higgs) vacuum. However, we know that there are also non-topological solitons in the Abelian CS models [7]. So it is natural to ask ourselves if nontopological or other solitons of the NA kind exist or not, in NA CS theories. With this question in mind, we will be concerned in this paper with vortices of all possible non-Abelian kinds in the NA CS gauge theories with the gauge group G being general; G = U(1) × G ′ with G ′ is an arbitrary simple group. As if it was not broad enough a spectrum, we will make every statement throughout the paper be independent of the gauge group. Beyond our naive expectation, we will find semi-local NA vortices of both topological and non-topological types in the models where the number of ...

The moduli space of non-Abelian vortices in Yang-Mills-Chern-Simons-Higgs theory

“…The index that we will calculate here is an extension of the index that was calculated in ref. [38], which in turn is an extension of the Atiyah-Singer index to R 2 , i.e. of Callias type, specialized to non-Abelian systems whose fluctuations are described by D. The index is formally given by…”

“…The symbolic manipulation of the terms given in equations (3.39) and (3.40) follow analogously to that in refs. [14,38], because we have that…”

“…Although we have not proved from the PDE point-of-view that the field Ω and Υ are uniquely determined by the above PDEs for given H 0 (z), we know that the number of moduli parameters available in the moduli matrix are (see refs. [29,30,36,38,41,42])…”

“…The parallel development in non-Abelian Chern-Simons-Higgs theory of generalizing the gauge group from U(N ) to U(1) × G was made in refs. [36][37][38].…”

“…[29,30,35] where it was applied to the non-Abelian vortices in Yang-Mills-Higgs theory and in refs. [36][37][38] in non-Abelian Chern-Simons-Higgs theory, both with gauge groups U(1) × G . All these well-studied vortices are included in our BPS theory as several limits.…”