Topological Manipulations On $\mathbb{R}$ Symmetries Of Abelian Gauge Theory

Performing topological manipulations is a fruitful way to understand global aspects of Quantum Field Theory (QFT). Such modifications are controlled by the notion of Topological QFT (TQFT) coupling across different codimensions. Motivated by the recent developments involving non-compact TQFTs as the Symmetry Topological Field Theory (SymTFT) for continuous symmetries, we explore topological manipulations on global $\mathbb{R}$ symmetries in the simple contexts of abelian gauge theories. We do so by turning on $\mathbb{R}$ background fields in these models, which are then inserted into a BF theory with a conjugate field valued either in $\mathbb{R}$ or $U(1)$, depending on the manipulation. When the conjugate field is $\mathbb{R}$-valued, it gives a flat gauging prescription of the $\mathbb{R}$ global symmetry. For certain cases, this gauging can be used to establish path-integral dualities, with which one can construct topological duality defects and condensation defects. The defects being defined by employing self-duality under $\mathbb{R}$ gauging, they are anticipated to obey a Tambara-Yamagami like fusion algebra. Due to the challenges arising from the non-compactness of $\mathbb{R}$, we do not rigorously establish this expectation. For the TQFT coupling with $U(1)$ valued conjugate field, we demonstrate a $2\pi \mathbb{Z} \subset \mathbb{R}$ subgroup gauging, which we connect to the results obtained from SymTFT recently. Executing this gauging in the 2d real scalar and the 4d $\mathbb{R}$ Maxwell, we respectively obtain the 2d compact scalar and the 4d $U(1)$ Maxwell theory. Finally, we give an exposition on the $\mathbb{R}$ gauging ideas applied to $(-1)-$form and $(d-1)-$form $\mathbb{R}$ symmetries. In particular, using the SymTFT picture, we identify the possible manipulations and realize them explicitly in the context of $p$-form gauge theory in $d=p+1$ dimensions.