(-1)-Form Symmetries and Anomaly Shifting from SymTFT

TL;DR

This paper extends the SymTFT framework to $(-1)$-form symmetries in the absence of bulk topological point operators, showing that codimension-one defects arising from higher gauging can generate the $(-1)$-form symmetry of the absolute theory. It clarifies how such defects can shift ‘t Hooft anomalies across different universes and unifies the decomposition and parameter-space viewpoints within SymTFT. The work analyzes invertible and non-invertible $(-1)$-form symmetries, provides explicit constructions in low dimensions (including gauge defects in the free boson and non-invertible condensations in 3d $\mathbb{Z}_2$ gauge theory), and uses dimensional reduction and orbifold groupoids to illustrate anomaly shifting and universes connected by discrete torsion. These results pave the way for applying SymTFT techniques to higher-dimensional theories and call for a more rigorous treatment of defect normalizations and fusion rules.”

Abstract

We investigate (-1)-form symmetries using the framework of symmetry topological field theories. Previous studies of (-1)-form symmetries have primarily focused on SymTFTs with topological point operators. Here we examine SymTFTs devoid of point operators, constructed to realize zero-form symmetries of some physical theory. In this context we identify codimension-one defects within the bulk of SymTFT constructed via higher gauging which can be interpreted as the generators of the (-1)-form symmetry of the absolute theory. In addition, we present examples where (-1)-form symmetries exhibit the novel ability to shift the 't Hooft anomalies of the theory.

Figures (4)

• Figure 1: Imagine the horizontal line as the parameter space. Each vertical line is QFT. For example, these are Pure Yang Mills theories at different $\theta$-value. If we start with QFT A, then by the action of operator $\mathcal{U}_{AB}$ we can move from A to B.
• Figure 2: We have an insertion of $\mathcal{D}$ within the ${\mathbb{Z}}_2$ gauge theory. If we pierce a $e$ line through the defect, it gets modified into a $m$ line.
• Figure 3: The action of $\mathcal{D}$, modifies a $W^1_1$ line operator into a $U^1_{\frac{1}{2}}$ line.
• Figure 4: We have condensed the $e$ line to construct $S_e$, which means the $e$ line can now terminate on the defect.