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

TL;DR

The paper develops spacetime symmetry-enriched SymTFTs by treating spacetime symmetries as part of the enrichment of an internal symmetry’s SET, yielding a bulk that encodes geometric data (e.g., foliation) and background structures. It provides a general framework (quiche and sandwich) to diagnose anomalies and classify phases in 1+1D and 2+1D settings, including lattice translations, reflections, and time-reversal, with explicit lattice-model realizations via stabilizer codes. The work unifies gauging, anomaly inflow, and modulated SPT classifications, linking LSM anomalies to modulated symmetries and showing how dual or non-invertible symmetries emerge upon gauging. It demonstrates that translation, reflection, and TRS enrichments can be captured by foliated field theories and stability analyses of Lagrangian algebras, enabling systematic phase classification and anomaly detection. The results provide a foundation for extending SymTFT to Floquet dynamics and higher dimensions, with potential applications to crystalline, subsystem, and out-of-equilibrium systems.

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.

Figures (4)

• Figure 1: The ${\cal S}$-symmetric theory $\mathfrak{T}^ {\cal S}$ on ${(d+1)}$D spacetime ${X_{d+1}}$ is related to the ${((d+1)+1)}$D SymTFT $\mathfrak{Z}( {\cal S} )$ of ${\cal S}$ by the interval compactification of the slab theory ${(\mathfrak{B}^{\text{sym}}_ {\cal S} , \mathfrak{Z}( {\cal S} ), \mathfrak{B}^{\text{phys}}_{\mathfrak{T}^ {\cal S} })}$ on spacetime ${X_{d+1}\times [0,1]}$. The $\mathfrak{B}^{\text{sym}}_ {\cal S}$ boundary realizes the ${\cal S}$ symmetry defects and the boundary $\mathfrak{B}^{\text{phys}}_{\mathfrak{T}^ {\cal S} }$ encodes the dynamics of $\mathfrak{T}^ {\cal S}$. The interval compactification is an exact relation due to the topological properties of $\mathfrak{Z}( {\cal S} )$. More specifically, the interval compactification induces an isomorphism from the "sandwich" ${(\mathfrak{B}^{\text{sym}}_ {\cal S} , \mathfrak{Z}( {\cal S} ), \mathfrak{B}^{\text{phys}}_{\mathfrak{T}^ {\cal S} })}$ to $\mathfrak{T}^ {\cal S}$.
• Figure 2: Consider a gapped ${1+1}$D boundary (shown in gray) of a ${2+1}$D $Q$-enriched topological order. When $Q$ symmetry defects (shown in purple) end on the boundary, they have two distinct types of interplay. (Left) When the $Q$-action $\rho$ on anyons is non-trivial, it induces a $Q$ action on the boundary Lagrangian algebra ${\cal L}$, denoted by ${\rho_q( {\cal L} )}$. (Right) When the symmetry fractionalization class $[\eta]$ is non-trivial, a trivalent junction of $Q$ symmetry defects can end on the boundary, sourcing a $\eta(q_1,q_2)$ topological defect line on the boundary.
• Figure 3: The classification of quantum phases characterized by a symmetry ${\cal S}$ using the SymTFT $\mathfrak{Z}( {\cal S} )$ is based on interfaces $\mathcal{I}_\mathcal{A}$ with condensable algebras $\mathcal{A}$ of the SymTFT. See Appendix \ref{['SymTFTReview']} for an introduction.
• Figure 4: Shows how, in the quantum code description of the SymTFT, the cells of the spatial lattice for a ${2+1}$D SymTFT are organized into the SymTFT sandwich ${(\mathfrak{B}^{\text{sym}}_ {\cal S} , \mathfrak{Z}( {\cal S} ), \mathfrak{B}^{\text{phys}}_{\mathfrak{T}^ {\cal S} })}$. The stabilizers defining the SymTFT ${\mathfrak{Z}( {\cal S} )}$ act only on purple edges in the bulk, and the boundary stabilizers defining the symmetry boundary act on at least one top boundary edges belonging to ${\mathfrak{B}^{\text{sym}}_ {\cal S} }$. Stabilizers or terms in the Hamiltonian acting on the bottom boundary edges ${\mathfrak{B}^{\text{phys}}_{\mathfrak{T}^ {\cal S} }}$ encode the dynamics of the ${1+1}$D theory $\mathfrak{T}^ {\cal S}$. After the interval compactification, only the black edges remain, and they become the edges and sites, respectively, of the ${1+1}$D spatial lattice for $\mathfrak{T}^ {\cal S}$.