Topological Holography for 2+1-D Gapped and Gapless Phases with Generalized Symmetries (2503.13685v1)

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.

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

This paper presents a comprehensive paper of topological holography for two-dimensional systems, focusing on both gapped and gapless phases where generalized symmetries are considered. By employing sophisticated mathematical frameworks such as higher linear algebra and higher condensation theory, this work successfully unravels the complex interplay between topology, symmetry, and phases of matter in condensed matter physics.

Gapped Phases

In the context of gapped phases, the paper utilizes the sandwich construction, a pivotal mechanism where a topological order in a higher dimension is compactified to produce phases with symmetry in lower dimensions. The primary aim is to describe gapped phases without intrinsic topological order on boundaries via minimal gapped boundaries. The methodology revolves around the relative tensor product, offering a concrete mathematical expression for gapped phases, particularly when the boundaries are Morita equivalent.

The paper introduces several examples to demonstrate its claims, including gapped phases with generalized symmetries emerging from Dijkgraaf-Witten theories such as $Z_1[2Vec_{Z_2}]$, $Z_1[2Vec_{S_3}]$, and the more complex $Z_1[2Vec_{S_4}]$. These examples illustrate how non-invertible symmetries can lead to distinct gapped phases, including those with multiple vacua hosting different gauge theories.

Gapless Phases and Topological Skeleton

For gapless phases, which are inherently challenging due to the lack of established theoretical frameworks, the paper explores the concept of the topological skeleton—a higher categorical structure of topological defects—which is described using condensable algebras in the SymTFT. The novel club-sandwich construction helps isolate the topological features of gapless phases, allowing for a systematic description of such systems.

Crucially, the paper establishes rigorous duality between certain condensable algebras in $Z_1[2Vec_G]$ and gapless symmetry-protected topological phases (gSPTs) in 2+1D. This is accomplished by using the Lyndon-Hochschild-Serre spectral sequence to articulate the intricate relationships between various components involved in 2+1D gSPTs, resulting in a cohomological classification.

Implications and Future Work

The implications of understanding these systems stretch both practical and theoretical horizons. On a practical level, the work provides insights into constructing new materials with bespoke topological features or symmetries, potentially useful in quantum computing and exotic state formation. Theoretically, the robust frameworks developed are anticipated to guide future explorations in higher-dimensional topological phases, inviting developments in categorical symmetries and condensed matter physics.

While the paper sets a foundation for understanding the relationship between generalized symmetries and topological phases in two-dimensional spaces, future research may extend these ideas to three-dimensional systems. Moreover, the exploration of fermionic systems and their categorization, as hinted at, would broaden the scope of topological holography, capturing a wider array of physical phenomena.

In conclusion, this paper marks a significant step in the systematic unraveling of complex topological structures within 2+1D systems, paving the way for broader adoption of generalized symmetries in analyzing phase transitions and topological orders. It invites further research to deepen the connections discovered and to explore untouched regions within this fascinating domain.