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

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.