Ghost free dual vector theories in 2+1 dimensions

Abstract

We explore here the issue of duality versus spectrum equivalence in abelian vector theories in 2 + 1 dimensions. Specifically we examine a generalized self-dual (GSD) model where a Maxwell term is added to the self-dual model. A gauge embedding procedure applied to the GSD model leads to a Maxwell-Chern-Simons (MCS) theory with higher derivatives. We show that the latter contains a ghost mode contrary to the original GSD model. On the other hand, the same embedding procedure can be applied to N f fermions minimally coupled to the self-dual model. The dual theory corresponds to N f fermions with an extra Thirring term coupled to the gauge field via a Pauli-like term. By integrating over the fermions at N f → ∞ in both matter coupled theories we obtain effective quadratic theories for the corresponding vector fields. On one hand, we have a nonlocal type of the GSD model. On the other hand, we have a nonlocal form of the MCS theory. It turns out that both theories have the same spectrum and are ghost free. By figuring out why we do not have ghosts in this case we are able to suggest a new master action which takes us from the local GSD to a nonlocal MCS model with the same spectrum of the original GSD model and ghost free. Furthermore, there is a dual map between both theories at classical level which survives quantum correlation functions up to contact terms. The remarks made here may be relevant for other applications of the master action approach.

Higher derivative massive spin-3 models inD=2+1

“…where we have used the self-adjoint property of the operator G μνλ and the commutativity between the operators S μνλ and G μνλ in the sense that when integrated S μνλ ðψÞG μνλ ðϕÞ ¼ S μνλ ðϕÞG μνλ ðψÞ. In (20) one can notice that it is possible to rewrite the last two terms in terms of the dual field F μνλ given by (17), giving us a unique term G μνλ ðFÞ which is the equivalent of the first term in (19). The rest of the equivalence can be achieved by observing that using (13) we have…”

“…showing us that the equations of motion derived from the fourth-order model can be written in the same form of the Singh-Hagen equations of motion (19). For a complete proof of equivalence, we also need to compare the equations of motion of the scalar field W in both formulations.…”

Quantum equivalence between the self-dual and the Maxwell–Chern–Simons models nonlinearly coupled toU(1) scalar fields

“…Now in order to verify whether our quadratic truncation furnishes sensible theories we check the spectrum of both quadratic theories (37) and (22). It is a general result, see [17], that due to the fact that (37) and (22) are connected via a Chern-Simons mixing term, see (18), the propagators coming from both theories will have the same pole structure except for a non-physical, gauge dependent, massless pole k 2 = 0 associated with the ChernSimons term which will appear in the propagator of the gauge field as one can explicitly check from (22). Consequently, we only need to check the spectrum of (37).…”

Generalized duality between local vector theories inD= 2+1

“…In order to suggest a new master action which would produce a local gauge theory dual to (1) we recall previous attempts. Both suggestions of [9] and [10] can be cast in the form of a gauge invariant second order master equation:…”

“…A direct generalization of the master action approach leads, quite surprisingly, to a gauge theory [9] which now includes a ghost mode in the spectrum, so the existence of a master action which interpolates between two theories does not guarantee spectrum equivalence a priori. As explained in [10] if we insist in the spectrum equivalence a new master action can be suggested which leads however, to a non-local vector theory. It is seems that we have glanced the old problem of formulating massive theories in a gauge invariant way.…”