SymTFTs: Unifying Symmetry and Topological Phases

• SymTFTs are higher-dimensional topological field theories that encode symmetry data, including anomalies, of lower-dimensional quantum field theories.
• They utilize fusion category symmetry and fiber functors to classify symmetry-protected topological phases and determine consistent boundary conditions.
• Their framework incorporates time-reversal symmetry, duality defects, and mixed anomalies, offering insights for condensed matter and high-energy physics.

Symmetry Topological Field Theories (SymTFTs) are higher-dimensional topological quantum field theories that universally encode the symmetry data, including anomalies, of a given lower-dimensional quantum field theory (QFT). SymTFTs provide a unifying algebraic and geometric framework for the classification and construction of symmetry-protected topological (SPT) phases, including systems with invertible and non-invertible (fusion category) symmetries, time-reversal invariance, and duality phenomena. In the (1+1)d bosonic setting with fusion category symmetry, the input data for a consistent SPT phase is a refinement of fiber functor data, tightly coupled to how boundary theories match bulk symmetry actions, and is critical for the realization of such phases as gapped boundaries of SymTFTs. The structure of SymTFTs naturally connects with recent work on categorical symmetries, duality defects, mixed anomalies, and their role in quantum field theory and condensed matter.

1. Fusion Category Symmetry and Fiber Functors

The notion of symmetry in (1+1)d is generalized from groups to unitary fusion categories 𝒞. The symmetry data is encoded in the algebraic structure of defect lines, both invertible and non-invertible, with fusion rules and associators prescribed by the fusion category. Symmetry protected topological phases with fusion category symmetry are classified, in the oriented case, by isomorphism classes of fiber functors: $(Z, M, i): \mathcal{C} \rightarrow \mathrm{Vec}$ where $Z$ assigns a finite-dimensional vector space to each simple object (defect label), $M$ specifies the tensor product (fusion) maps, and $i$ gives the associativity and unit structure, consistent with the fusion category axioms. The fiber functor implements the “symmetry action” in the Hilbert space of the phase and forgets the categorical (fusion) data, reducing to an anomaly-free, trivial realization for group-like symmetries.

2. Incorporating Time-Reversal Symmetry: Algebraic Quintuple Data

When time-reversal symmetry is imposed (unoriented setting), the classification of SPT phases is refined. The complete algebraic datum becomes a quintuple: $(Z, M, i, s, \varphi)$ where $(Z, M, i)$ is as above, $s \in \mathbb{Z}_2$ is a sign capturing an intrinsic time-reversal SPT invariant (analogous to the $\mathbb{Z}_2$ index in bosonic time-reversal-protected phases), and

$\varphi_x: Z(x) \rightarrow Z(x^*)$

is a collection of isomorphisms encoding the action of orientation-reversal on the vector spaces, required to satisfy an involutive relation up to the pivotal structure $a_x$ of the fusion category: $\varphi_{x^*} \circ \varphi_x = Z(a_x)$ This data are necessary to define consistent partition functions on non-orientable manifolds and ensure that the actions of symmetry defects and orientation reversal are compatible, forming the boundary data for unoriented TFTs.

3. SymTFTs, Boundary Realization, and Gapped Phases

In the bulk-boundary correspondence paradigm, many gapped phases with symmetry are realized as boundary conditions of a (d+1)d SymTFT, which encodes the universal symmetry action and potential 't Hooft anomaly inflow. In (1+1)d, the quintuple $(Z, M, i, s, \varphi)$ specifies the topological boundary condition for the (2+1)d SymTFT, ensuring anomaly cancellation and modular invariance. The bulk SymTFT can be invertible or non-invertible, depending on the defect sector, and the allowed boundary conditions are governed by the compatibility constraints derived from the fusion category and time-reversal/orientation data. This formalism enables the systematic classification of all such boundary theories and their fusion structures, with applications to higher-categorical data and enriching the “higher Landau paradigm.”

4. Duality Symmetries, Noninvertible Defects, and Mixed Anomalies

In cases admitting duality symmetries (like Kramers–Wannier duality in the Ising model), the symmetry is noninvertible and described by Tambara–Yamagami fusion categories. The classification by quintuple data restricts which SPT phases admit time-reversal invariance: the conditions on $\varphi$ and $s$ (i.e., compatibility with the pivotal structure and sign) are often nontrivial, and in some cases preclude the existence of time-reversal-invariant phases altogether, reflecting a mixed anomaly between duality and time-reversal. In the SymTFT picture, such mixed anomalies correspond to obstructions in the boundary data—certain dualities cannot be “gauged” in the presence of time-reversal symmetry. Explicitly, this controls which boundary theories realize duality as actual symmetry defects.

5. Mathematical Characterization and Consistency Conditions

The algebraic consistency and cohomology constraints derived from the fusion category, the fiber functor structure, and the orientation reversal data guarantee that the partition functions on all oriented and non-orientable surfaces are well defined. The involution condition on $\varphi$ and the sign $s$ ensure that the unoriented TQFT satisfies the required reflection positivity and compatibility with defect fusions. This classification is both necessary and sufficient for the existence of consistent unoriented gapped phases with fusion category symmetries; any “missing” data corresponds to an anomalous symmetry not realizable as a boundary condition.

6. Applications and Physical Implications

The classification by $(Z, M, i, s, \varphi)$ has direct consequences for constructing exactly soluble models of SPT phases with noninvertible symmetries, including those arising in statistical mechanics (self-dual points in lattice models), quantum spin chains, and topological order with categorical symmetries. The explicit identification of mixed anomalies provides guidance in building models that realize or forbid certain symmetries—a key consideration in numerical and experimental proposals for novel phases of matter. Additionally, this framework ensures that lattice regularizations and their corresponding defect algebras faithfully reproduce the continuum symmetry structures, including time-reversal and duality anomalies.

7. Integration With Broader Developments in SymTFTs

The approach developed for (1+1)d fusion category SPTs with time-reversal symmetry forms the foundation for higher-dimensional generalizations and connects to a growing body of work on SymTFTs for noninvertible and categorical symmetries, both in condensed matter and in high-energy theoretical contexts (e.g., geometric engineering, anomaly inflow, and holography). The precise algebraic input data—fiber functors, orientation-reversing isomorphisms, and sign invariants—are now integrated as part of the standard toolkit for constructing consistent boundary theories and understanding the bulk-boundary correspondence in the presence of generalized global and higher categorical symmetries.

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For classification and analysis of 1+1d bosonic SPT phases with fusion category and time-reversal symmetry, the full data of a quintuple $(Z, M, i, s, \varphi)$ is both necessary and sufficient. This ensures that the boundary conditions assigned to a SymTFT exactly capture the allowed gapped phases, their fusion rules, and the presence or absence of anomalies, including those arising from noninvertible duality structures (Inamura, 2021).