What Maxwell Theory in D<>4 teaches us about scale and conformal invariance

TL;DR

This work demonstrates that unitary scale invariance does not universally imply conformal invariance by analyzing Maxwell theory in dimensions other than four. It shows that in $d=3$ conformality can be restored through a unitary extension by introducing a free scalar, while in $d\ge5$ one can achieve a non-unitary conformal extension via a primary vector $Y_{\mu}$ with $F_{\mu\nu}$ as its descendant; the authors further reveal BRST-like structures and potential $OSp(d,2|2)$-related symmetry, though exact closure faces obstacles. These results refine the understanding of when scale invariance implies conformality and illuminate how extended operator algebras and gauge choices influence conformal structure. The findings have implications for the general conjecture relating scale and conformal invariance and motivate further exploration of extended symmetry algebras in nontrivial QFTs.

Abstract

The free Maxwell theory in D<>4 dimensions provides a physical example of a unitary, scale invariant theory which is NOT conformally invariant. The easiest way to see this is that the field strength operator F_mn is neither a primary nor a descendant. We show how conformal multiplets can be completed, and conformality restored, by adding new local operators to the theory. In D>=5, this can only be done by sacrificing unitarity of the extended Hilbert space. We analyze the full symmetry structure of the extended theory, which turns out to be related to the OSp(D,2|2) superalgebra.