Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries

TL;DR

This work extends topological holography to 2+1-D gapped and gapless phases with generalized (fusion 2-category) symmetries via the SymTFT framework. It develops sandwiches with minimal gapped boundaries to classify gapped phases and introduces the club-sandwich paradigm for gapless phases, linking gapless topological skeletons to non-Lagrangian condensable algebras and their cohomological data. A central result is the duality between minimal condensable algebras in $ ext{Z}_1[2 extbf{Vec}_G]$ and 2+1-D $G$-gSPTs, with emergent anomalies captured by the $d_4$ obstruction in an LHS spectral sequence. The paper provides concrete examples for groups like $ ext{Z}_2$, $S_3$, and $S_4$, derives exact sequences that classify minimal algebras, and constructs intrinsically non-invertible SPTs (igSPTs) with both gapped and gapless sectors, broadening the landscape of gapped/gapless phases under generalized symmetries and enabling computable cohomological classifications. Overall, it furnishes a computable bridge between higher category theory and condensed-matter phases, yielding new intrinsically non-invertible SPTs and a robust toolkit for exploring gapless topological phenomena.

Abstract

We study topological holography for 2+1-D gapped and gapless phases with generalized symmetries using tools from higher linear algebra and higher condensation theory. We focus on bosonic fusion 2-category symmetries, where the Symmetry Topological Field Theory (SymTFT) are 3+1D Dijkgraaf-Witten theories. (1). Gapped phases are obtained from the sandwich construction with gapped symmetry and physical boundaries. A gapped boundary of the 3+1D SymTFT is called minimal if it has no intrinsic 2+1-D topological order. We derive the general structure of a sandwich construction with minimal gapped symmetry and physical boundaries, including the underlying topological order and the symmetry action. We also study some concrete examples with 2-group or non-invertible symmetries. (2). For gapless phases, we show that the SymTFT provides a complete description of the \textit{topological skeleton} of a gapless phase. The topological skeleton of a gapless phase is the higher categorical structure of its topological defects. We rigorously establish this relation for 2+1-D gapless phases with finite group symmetries. For a gapless phase with a finite group symmetry, its topological skeleton(also known as gapless SPT(gSPT)) can be characterized by the decorated domain wall construction. We give a precise formulation of this using spectral sequence. We show that certain class of condensable algebras in the SymTFT $\mathcal{Z}_1[2\mathbf{Vec}_G]$, which we call minimal condensable algebras, has exactly the same structure. We further give a cohomological classification of minimal condensable algebras, which enables us to compute the classification of 2+1-D $G$-gSPTs via ordinary group cohomology. Finally we use SymTFT to construct 2+1-D gSPT with generalized symmetries, including an intrinsically gSPT(igSPT) with exact non-invertible fusion 2-category symmetry and anomalous 2-group IR symmetry.

Figures (7)

• Figure 1: The sandwich construction, mathematically given by a relative tensor product.
• Figure 2: The club-sandwich construction captures the topological skeleton of a gapless phase with symmetry. The left physical boundary encodes all the local information of the gapless phase such as OPE of local fields. The domain wall $\mathfrak{M}_{}$ encodes the topological skeleton of the gapless phase. By fusing the domain wall $\mathfrak{M}_{}$ with the symmetry boundary $\mathfrak{B}_{\mathbf{sym}}$, we obtain an ordinary sandwich that describes the IR symmetry of the gapless phase. This process defines a functor $\EuScript{C}\to \EuScript{C}^{\mathbf{IR}}$.
• Figure 3: If the physical and symmetry boundaries happen to be Morita equivalent, then we can bend them to make a a gapped domain wall $\EuScript{M}$. Therefore the dimension reduction of the sandwich is the bulk of the domain wall $\EuScript{M}$. By "bulk=center of boundary", we see that the dimension reduction of the sandwich must be $\mathcal{Z}_0(\EuScript{M})=\text{End}(\EuScript{M})$. The symmetry $\EuScript{C}$ then acts on $\text{End}(\EuScript{M})$ according to the module 2-category structure on $\EuScript{M}$.
• Figure 4: Domain walls in the toric code topological order. A solid line represents a smooth boundary and a dashed line represents a rough boundary. There is also an invertible domain wall corresponding to a non-trivial braided auto-equivalence. As module category over $\mathbf{Rep}(\mathbb{Z}_2)$, the rough boundary is $\mathbf{Vec}$ and the smooth boundary is $\mathbf{Rep}(\mathbb{Z}_2)$.
• Figure 5: The club-sandwich construction separates the topological skeleton from the local quantum symmetry of a gapless phase. The left boundary encodes all the local data of the gapless phase such as OPE of local fields. The domain wall $\mathfrak{M}$ encodes the topological skeleton of the gapless phase, including the topological sectors of states and operators. By fusing the domain wall $\mathfrak{M}$ with the symmetry boundary $\mathfrak{B}_{\mathbf{sym}}$, we obtain an ordinary sandwich that describes the IR symmetry of the gapless phase. In particular, the new symmetry boundary describes the IR symmetry of the gapless phase, denoted as $\mathfrak{B}_{\mathbf{sym}}^\mathbf{IR}$.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Example 2.1
• Example 2.2
• Example 2.3
• Definition 3.1
• Example 3.1
• Corollary 4.1
• Example 4.1
• Remark