1 Introduction

Recently, within a newly proposed spacetime approach to dualities [1], an interesting new result emerged for the dual theory of an Abelian gauge field in two dimensions[2]. The novelty of that two dimensional model stems from the fact that there is no known dual version of it within the conventional $p$-duality approach [3]. The root of the problem is that a gauge field in two dimensions corresponds to the limiting value $p=d-1$, and the usual dualization procedure is not applicable because the corresponding Bianchi identities are undefined⁵. As a matter of fact, according to the conventional method, the dual theory of a $p$-form involves a $(d-p-2)$ field, and this correspondence leads to the constraint $p\le d-2$. Nevertheless, one wonders if there is a way to extend the $p$-duality approach to the case $p=d-1$, even though the dual field theory of the starting matter field is generally undefined. To be sure, in the absence of interactions, the dual theory of a gauge field strength of maximum rank $d$ corresponds to a background field which is constant over the spacetime manifold. In this narrow sense, one may speak of ``constant-tensor duality'' for the free theory in the limiting case. However, in a flat spacetime, such a constant can be gauged away on account of translational invariance, so that the resulting free theory is essentially empty⁶. It seems clear, therefore, that the extension of the $p$-duality approach must go beyond the case of a free theory in the limiting case. Why would these limiting cases be of any interest? Apart from the recognized importance of dualities in connection with the theory of extended objects[6], it turns out that such limiting theories have been shown to be of some phenomenological relevance in relation to the problem of confinement[7], [8] and glueball formation [9]. The question arises, then, as to what happens in the case of an interacting theory, for instance in the simplest case in which a coupling to an external current is present.