SymTFT, Protected Gaplessness, and Spontaneous Breaking of Non-invertible Symmetries

TL;DR

This work develops a coarse, first-principles classification of non-invertible KW-like and KW-inspired zero-form symmetries in $3+1$-dimensional QFTs using the SymTFT framework, clarifying when such symmetries enforce gaplessness or remain unbroken in the IR. It ties 3+1D non-invertible symmetries to 4+1D BF-type topological orders, develops a systematic method to classify invariant Lagrangian subgroups under finite symmetry groups $ ext G$, and analyzes how these data constrain symmetry-preserving RG flows. The authors introduce a robust computational program—grounded in p-adic analysis, cyclotomic factorization, and Galois theory—to determine the existence (or obstruction) of $ ext G$-invariant boundary conditions, providing explicit criteria and examples, including deformations of class $ ext S$ theories and $ ext N=1^*$ vacua. Overall, the paper establishes a tractable coarse-grained framework for predicting IR phases (gapless or symmetry-broken) in the presence of non-invertible symmetries and points toward richer global structures and potential non-Lagrangian realizations at Argyres-Douglas-type fixed points.

Abstract

In recent years we have learned that several four-dimensional field theories can manifest non-invertible zero-form symmetries generalizing the Kramers-Wannier duality defect of the 2d critical Ising model. Several recent works by various groups have observed a deep interplay among such non-invertible symmetries in 3+1 dimensions, their anomalies, and the properties of the ground state(s). The purpose of this work is to present a first coarse classification of all possible classes of non-invertible symmetries of this type that can either enforce gaplessness or be spontaneously broken in the infrared exploiting the topological symmetry theory formalism. Our methods also generalize to non-invertible KW-like duality symmetries graded by non-abelian finite subgroups. As a first applications of our results we present examples in the context of supersymmetric models. Along the way we notice the potential for further global structures that could be realized by non-SUSY versions of Argyres-Douglas type fixed points.

Figures (8)

• Figure 1: Topological symmetry theory. The $d$-dimensional field theory $\mathcal{T}_{\mathcal{B}}$ is realized via an isomorphism with a $(d+1)$-dimensional topological field theory $\mathcal{F}$ on a $d$-strip. On the left side of the $d$-strip, we have a $d$-dimensional topological boundary $\mathcal{B}$ condition of $\mathcal{F}$, on the right side of the $d$-strip we have a $d$-dimensional $\mathcal{F}$-relative theory $\widehat{\mathcal{T}}$. Since the whole $d$-strip construction is topological, we can stack the two boundaries contracting the interval, thus obtaining the desired isomorphism leading to $\mathcal{T}_{\mathcal{B}}$.
• Figure 2: The boundary changing interfaces. If the theory $\mathcal{F}$ has a 0-form symmetry group $\mathbb G$, we can represent the latter using topological codimension 1 defects $\mathbb D_g$ for $g\in \mathbb G$. These can transform the boundaries non-trivially and if these can end along a codimension 2 defect $\sigma_g$, this results into a collection of interfaces among the $d$-dimensional QFTs $\mathcal{T}_{\mathbb D_g\mathcal{B}}$ for $g\in \mathbb G$.
• Figure 3: The $\mathbb G$-duality symmetry. When $\mathbb D_g \widehat{\mathcal{T}} \simeq \widehat{\mathcal{T}}$, this induces a non-trivial duality isomorphism $\mathsf{S}_{g^{-1}}: \mathcal{T}_{\mathbb D_g\mathcal{B}} \simeq \mathcal{T}_{\mathcal{B}}$. The composition of $\sigma_g$ with $\mathsf{S}_{g^{-1}}$ gives rise to a non-invertible $\mathbb G^{(0)}$-duality symmetry $\mathcal{N}_g = \sigma_g \circ \mathsf{S}_{g^{{-1}}}$ with $g \in \mathbb G^{(0)}$.
• Figure 4: SymTFT for anomalous 1-form symmetries.
• Figure 5: SymTFT and RG flow: the case $\mathcal{T}_{IR}$ is gapped.

Theorems & Definitions (1)

• Theorem 1