Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond

Abstract

We extend the Symmetry Topological Field Theory (SymTFT) framework beyond internal symmetries by including geometric data that encode spacetime symmetries. Concretely, we enrich the SymTFT of an internal symmetry by spacetime symmetries and study the resulting symmetry-enriched topological (SET) order, which captures the interplay between the spacetime and internal symmetries. We illustrate the framework by focusing on symmetries in $1+1$D. To this end, we first analyze how gapped boundaries of $2+1$D SETs affect the enriching symmetry, and apply this within the SymTFT framework to gauging and detecting anomalies of the $1+1$D symmetry, as well as to classifying $1+1$D symmetry-enriched phases. We then consider quantum spin chains and explicitly construct the SymTFTs for three prototypical spacetime symmetries: lattice translations, spatial reflections, and time reversal. For lattice translations, the interplay with internal symmetries is encoded in the SymTFT by translations permuting anyons, which causes the continuum description of the SymTFT to be a foliated field theory. Using this, we elucidate the relation between Lieb-Schultz-Mattis (LSM) anomalies and modulated symmetries and classify modulated symmetry-protected topological (SPT) phases. For reflection and time-reversal symmetries, the interplay can additionally be encoded by symmetry fractionalization data in the SymTFT, and we identify mixed anomalies and study gauging for such examples.

60 pages plus appendices. v2: added references + minor changes. v3: published version

References in corpus (109)

• Generalized Global Symmetries
• Generalized Symmetries in Condensed Matter
• Non-Invertible Duality Defects in 3+1 Dimensions
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Higher Gauging and Non-invertible Condensation Defects
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• Colloquium: Quantum and Classical Discrete Time Crystals
• Dynamically Generated Logical Qubits
• Universal Non-Invertible Symmetries
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
• Topological Defects on the Lattice: Dualities and Degeneracies
• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• The dipolar Bose-Hubbard model
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Lecture Notes on Generalized Symmetries and Applications
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Spontaneous breaking of multipole symmetries
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Anomalies and Symmetry Fractionalization
• Anomaly Inflow for Subsystem Symmetries
• An Introduction to Higher-Form Symmetries
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Topological symmetry in quantum field theory
• Anomalies and gauging of U(1) symmetries
• Introduction to Generalized Global Symmetries in QFT and Particle Physics
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
• Comments on Foliated Gauge Theories and Dualities in 3+1d
• Rank-2 Toric Code in Two Dimensions
• Ground state degeneracy on torus in a family of ${\mathbb{Z}}_N$ toric code
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• On the Symmetry TFT of Yang-Mills-Chern-Simons theory
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Anomalies of $(1+1)D$ categorical symmetries
• Effective Fractonic Behavior in a Two-Dimensional Exactly Solvable Spin Liquid
• Foliated BF theories and Multipole symmetries
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Anomalies of non-invertible self-duality symmetries: fractionalization and gauging
• ICTP Lectures on (Non-)Invertible Generalized Symmetries
• Foliated-Exotic Duality in Fractonic BF Theories
• Quantized axial charge of staggered fermions and the chiral anomaly
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Aspects of ${\mathbb{Z}}_N$ rank-2 gauge theory in $(2+1)$ dimensions: construction schemes, holonomies, and sublattice one-form symmetries
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Line Defect Quantum Numbers & Anomalies
• Gapped Lineon and Fracton Models on Graphs
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Emergent generalized symmetry and maximal symmetry-topological-order
• Multipole and fracton topological order via gauging foliated SPT phases
• (SPT-)LSM theorems from projective non-invertible symmetries
• Symmetric higher rank topological phases on generic graphs
• Exotic quantum liquids in Bose-Hubbard models with spatially-modulated symmetries
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Domain Wall Decorations, Anomalies and Spectral Sequences in Bosonic Topological Phases
• Classification and construction of interacting fractonic higher-order topological phases
• Anisotropic higher rank ${\mathbb{Z}}_N$ topological phases on graphs
• Symmetry TFTs for Non-Invertible Defects
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Boundary SymTFT
• Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
• Lectures on Generalized Symmetries
• Non-anomalous non-invertible symmetries in 1+1D from gapped boundaries of SymTFTs
• Symmetry TFTs and Anomalies of Non-Invertible Symmetries
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• A SymTFT for Continuous Symmetries
• Symmetry TFT for Subsystem Symmetry
• Representation Theory of Solitons
• A Gapless Phase with Haagerup Symmetry
• Topological holography for fermions
• SymSETs and self-dualities under gauging non-invertible symmetries
• Anomalies of Coset Non-Invertible Symmetries
• Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States
• Dynamics in Systems with Modulated Symmetries
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part II
• Cornering Relative Symmetry Theories
• Topological quantum chains protected by dipolar and other modulated symmetries
• Topological defects of 2+1D systems from line excitations in 3+1D bulk
• Classification of Dipolar Symmetry-Protected Topological Phases: Matrix Product States, Stabilizer Hamiltonians and Finite Tensor Gauge Theories
• Fermionic quantum criticality through the lens of topological holography
• Soft symmetries of topological orders
• (-1)-Form Symmetries and Anomaly Shifting from SymTFT
• Gauging in Parameter Space: A Top-Down Perspective
• SymTFT Fans: The Symmetry Theory of 4d N=4 Super Yang-Mills on spaces with boundaries
• Defect Networks for Topological Phases Protected By Modulated Symmetries
• Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits
• Spontaneously Broken $(-1)$-Form U(1) Symmetries
• Phases with non-invertible symmetries in 1+1D $\unicode{x2013}$ symmetry protected topological orders as duality automorphisms
• Braidings on topological operators, anomaly of higher-form symmetries and the SymTFT
• SymTFT out of equilibrium: from time crystals to braided drives and Floquet codes
• Topological Dipole Insulator
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Boundary symmetries of (2+1)D topological orders
• (-1)-form symmetries from M-theory and SymTFTs
• Exotic Field Theories for (Hybrid) Fracton Phases from Imposing Constraints in Foliated Field Theory
• Dipolar BF theory and dipolar braiding statistics
• New Field Theories with Foliation Structure and Subdimensional Particles from Godbillon-Vey Invariant
• Decomposition and (Non-Invertible) (-1)-Form Symmetries from the Symmetry Topological Field Theory

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