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

TL;DR

The work develops a local, non-compact SymTFT framework to topologically manipulate continuous $\mathbb{R}$ and related symmetries in Abelian gauge theories, focusing on a 2d real scalar and 4d Maxwell theories. By coupling background fields to non-compact BF- and related TQFTs, it identifies flat gaugings that preserve the spectrum and constructs half-space topological defects, revealing invertible actions and Heisenberg-type mixings with shift symmetries. A novel deformation extends gaugings to $(d-1)$-form $\mathbb{R}$ symmetries, capturing all Lagrangian algebras of the corresponding SymTFT and enabling subgroup gaugings $\mathbb{Z}_k^{(p)}\times\mathbb{Z}_q^{(d-p-2)}$ in a controlled manner. The analysis connects to the SymTFT perspective, discusses implications for bosonic string theory and target-space topology, and outlines generalizations to $p$-form theories in arbitrary dimensions. Overall, the results offer a cohesive, theory-agnostic toolkit for topological manipulations of non-compact symmetries with potential applications to strings, dualities, and higher-form global structures.

Abstract

Performing topological manipulations is a fruitful way to understand global aspects of Quantum Field Theory (QFT). Such modifications are typically 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 realize topological manipulations on global $\mathbb{R}$ symmetries via TQFT coupling in the simple context of non-compact abelian gauge theory. Namely, by inserting the background fields for $\mathbb{R}$ symmetries into non-compact TQFTs on spacetime, we study the topological gaugings. Furthermore, we explore topological defects in non-compact theories by employing the said manipulations on the half-space, which are analogous to duality defects in compact gauge theories. We examine the action of these defects on the local and extended operators, and discuss their algebra. As opposed to the duality defects, our defects act invertibly on the spectrum. To further understand their role, we discuss the way they mix with other topological defects and the resulting global symmetries. We also comment on possible applications of these ideas to the bosonic String Theory. After studying defects, we provide a detailed inspection of manipulations on $\mathbb{R}^{(-1)}$ and $\mathbb{R}^{(d-1)}$ symmetries in abelian gauge theory. Notably, we develop a novel deformation tailored for flat gauging subgroups of $\mathbb{R}^{(d-1)}$ symmetries, and provide an extension of it for $\mathbb{R}^{(p)}$ symmetries. We show that this new simple deformation captures all the topological boundary conditions of the corresponding non-compact SymTFTs as the deformation parameter is varied, which the ordinary non-compact BF coupling cannot do.

Figures (7)

• Figure 1: We divide the spacetime into two parts, joined along an interface boundary $\mathfrak B$. The left and right hand sides of $\mathfrak B$ are denoted $M_-$ and $M_+$ respectively. We gauge the $\mathbb{R}$ shift symmetry on the $M_+$ part, which produces the same theory. Then, Dirichlet boundary conditions on $C$ near $\mathfrak B$ defines a topological defect since the gauging operation is topological.
• Figure 2: Due to invariance under gauging $\phi(\mathcal{X} _\mathbb{R} ) \simeq \mathcal{X} _\mathbb{R}$, it is possible to define a topological defect $\mathfrak D$ realized by a Chern-Simons like coupling between the fields of $M_-$ and $M_+$.
• Figure 3: From the defect $\mathfrak D_r$ we stretch out a straight line $L$ ending on the support of the $V_s$ operator. We can then attach the invisible line $\exp( is \int_{L} dX )$ to $V_s$. This does not make $V_s$ a non-genuine local operator because the line is trivial.
• Figure 4: As the topological defect $\mathfrak D_r$ is swept across the $V_s(y)$ insertion, the line $L$ is deformed to $L'$, and the $\mathfrak D_r$ passing over $V_s$ produces a new operator $V'_s(y)$ which we argued to be $V_{rs}(y)$. To this operator, a trivial line $\exp( i rs \int_{L'} dY )$ is attached. As the operator inserted on $L'$ is transparent, $V'_s(y)$ is also a genuine local operator.
• Figure 5: The defect $\mathfrak D_r$ and its dual inserted along an infinitesimal interval $I_\varepsilon$.