SymTFT actions, Condensable algebras and Categorical anomaly resolutions

TL;DR

This work advances the understanding of symmetry topological field theories (SymTFTs) for non-abelian and non-invertible symmetries by systematically enumerating condensable algebras in Drinfeld centers, with a detailed analysis of $\mathcal{Z}(Q_8)$. It provides explicit 3d SymTFT actions and Lagrangian algebras for $\text{Rep}(D_4)$ and $\text{Rep}(Q_8)$, clarifying how discrete gaugings map to different symmetry boundaries and how igSPTs emerge from non-Lagrangian condensates. The paper also develops a concrete anomaly-resolution framework via club sandwiches, identifying categorical short exact sequences that relate anomalous and non-anomalous symmetry extensions for both group-like and categorical symmetries. Overall, the results offer a concrete toolbox for analyzing anomaly resolution and igSPT phases in theories with rich symmetry categories, with potential extensions to $\text{Rep}(\mathcal{H}_8)$ and connections to RG flows and emergent symmetries.

Abstract

We investigate symmetry topological field theories (SymTFTs) of non-abelian and non-invertible symmetries and the different Lagrangian algebras associated with a given Drinfeld center. For several examples we analyze the condensable algebras of the Drinfeld center to identify the intrinsically gapless symmetry protected topological (igSPT) phases. In previous work, the relation between igSPT phases and resolving anomalies by embedding an anomalous symmetry inside a larger fusion category was demonstrated. Here we present more examples of this mechanism that involve both group-like and categorical symmetries.

Figures (6)

• Figure 1: The $D_4$ symmetry web in the space of $c=1$ theories. We are still viewing $D_4$ as an extension of ${\mathbb{Z}}^{(a)}_2 \times {\mathbb{Z}}^{(b)}_2$ by $\hat{{\mathbb{Z}}}^{(c)}_2$. The horizontal line represents the circle branch and the vertical line is the orbifold branch. The arrows represents different gaugings that connects the theories. Similar figures have also appeared in Perez-Lona:2023djoPutrov:2024uorDiatlyk:2023fwfThorngren:2021ysoBartsch:2022ytjBhardwaj:2022maz.
• Figure 2: The club sandwich with the respective Drinfeld centers of $\text{Rep}({D}_4)$ and the reduced topological order, ${{\mathbb{Z}}}^\omega_2$.
• Figure 3: The club sandwich with the respective Drinfeld centers of $({D}_4)$ and the reduced topological order, ${{\mathbb{Z}}}_4$.
• Figure 4: Once we perform the interval compactification on the blue side of the club sandwich, we obtain the modified boundary condition for $\mathfrak{Z}({\mathbb{Z}}_4)$, which corresponds to gauging a ${\mathbb{Z}}_2$ subgroup of ${\mathbb{Z}}_4$. This gives rise to a theory with mixed anomaly.
• Figure :