Consistency Conditions for $p$-Form Fields Localization on Codimension two Braneworlds

TL;DR

This paper investigates the localization of $p$-form fields in codimension two braneworlds, revealing that only scalar fields and their duals can be consistently localized when considering Einstein's equations and Hodge duality.

Contribution

It introduces a new consistency condition linking field localization to duality symmetry, showing that many previously localized fields are incompatible with these constraints.

Findings

1-form and its dual 3-form can be localized in certain models

Localization of fields must satisfy both Einstein's equations and Hodge duality

Most previously localized $p$-forms in codimension two are ruled out by these conditions

Abstract

Recently, in (Eur.Phys.J.C 80 (2020) 5, 432), the present authors obtained general stringent conditions on the localization of fields in braneworlds by imposing that its zero-mode must satisfy Einstein's equations (EE). Here, we continue this study by considering free $p$-form. These fields present an on-shell equivalency relation between a $p$-form and a $(D-p-2)$-form, provided by Hodge duality (HD). This symmetry will impose a new consistency condition, namely, confinement of a $p$-form must imply the localization of its dual. We apply the above conditions to $6$D braneworlds. With this, we find that in global string-like defects, for example, the $1$-form has a normalizable zero-mode. By using the HD, we show that its bulk dual $3$-form also has a normalizable zero-mode, making the confinement consistent with HD. However, these solutions cannot be made consistent with EE, therefore, these fields must be ruled out. In fact, by imposing both conditions, only the scalar and its dual field can be consistently localized. In this way, all the literature so far in which the free $1$-form is localized in codimension two models should be reviewed. These results also point to the fact that the symmetries of the fields can be used to verify the consistency of their localization and even prohibit it.