Topological Symmetry Theory

• Topological Symmetry Theory is a framework unifying global symmetries, topological invariants, and dualities using higher-dimensional TQFT, applicable to QFT and lattice models.
• The theory employs a SymTFT with distinct topological and physical boundaries to encode abstract symmetry data and microscopic details respectively.
• It extends traditional group symmetries to include non-invertible and fusion-category symmetries, offering analytical tools for classifying topological phases and dualities.

Topological Symmetry Theory unifies and systematizes the relationship between global symmetries, topological invariants, and dualities in quantum field theories, lattice models, and topological phases. At its core, it encodes symmetry data—including internal, spatial, non-invertible, categorical, higher-form, and emergent symmetries—within a higher-dimensional topological field theory (often called a Symmetry Topological Field Theory, or SymTFT). This framework underpins the classification and duality structure of topological phases, subsumes both group-based and fusion-category symmetries, and connects algebraic, geometric, and field-theoretic perspectives across condensed matter and high energy physics.

1. Symmetry Topological Field Theory (SymTFT): Construction and Principle

A central construction is the SymTFT: a (d+1)-dimensional topological quantum field theory assigned to a d-dimensional system with a specified symmetry, designed to “holographically” record the symmetry and anomaly content of the boundary theory. This “bulk–boundary sandwich” is realized by equipping the interval direction with two generalized boundary conditions:

• A topological (symmetry) boundary, encoding abstract symmetry data (for example, a Lagrangian algebra or module category).
• A physical boundary, encoding the microscopic details (such as Boltzmann weights in a lattice model or the local QFT data).

The boundary partition function is interpreted as the inner product between a “topological state” (determined by the symmetry boundary and bulk SymTFT) and the “physical state” (determined by dynamical boundary data): $$\mathcal{Z}(\Sigma) = \langle \Sigma_{\Upsilon}, \mathcal{C}, \mathcal{M} \mid \Sigma_{\Upsilon}, \mathcal{C}; \theta, \vartheta \rangle,$$ where $\Sigma_{\Upsilon}$ is the discretization of the surface $\Sigma$, $\mathcal{C}$ is a fusion category, $\mathcal{M}$ is a module category, and $\theta, \vartheta$ are boundary weights (Delcamp et al., 12 Aug 2024).

This construction successfully realizes:

• Standard group-based symmetries (via fusion categories like $\mathrm{Vect}_G$),
• Generalized non-invertible/categorical symmetries (via more general spherical fusion categories or representation categories of weak Hopf algebras (Jia, 19 Dec 2024)),
• Higher-form and spatial/crystalline symmetries (via appropriate geometric data, as in topological crystalline phases (Thorngren et al., 2016)).

2. From Abelian to Non-Abelian and Fusion Category Symmetries

While the classical Ising model operates with a $\mathbb{Z}_2$ symmetry, topological symmetry theory extends naturally to finite non-Abelian group symmetries and fusion category symmetries. In the non-Abelian case, the following features emerge:

• Local degrees of freedom are $G$-valued, with partition functions of the form

$$\mathcal{Z}^{\mathrm{Vect}_G}(\Sigma^\vee; \theta) = \sum_{\sigma \in G^{\mathrm{Vertices}(\Sigma^\vee)}} \prod_{e} \theta_e(\sigma_{\mathrm{src}}^{-1} \sigma_{\mathrm{tar}}),$$

for even Boltzmann weights $\theta_e : G \rightarrow \mathbb{C}$ (Delcamp et al., 12 Aug 2024).

• Topological sectors correspond to holonomies modulo conjugation, requiring careful handling for non-Abelian $G$.
• Defect lines (“Wilson lines”/“'t Hooft lines”) are fully characterized and the complete set of these is encoded in the Drinfeld center $Z(\mathrm{Vect}_G)$ (or, via Fourier duality, $Z(\mathrm{Rep}(G))$).

A key structural result is that any fusion category symmetry is realized by Rep($H$), the representation category of a (possibly weak) Hopf algebra $H$ (Jia, 19 Dec 2024). The presence of non-invertible, fusion-category symmetries is ubiquitous in spin liquid models and general topological orders.

3. Gauging, Duality, and Fourier Transform: Non-Abelian Kramers–Wannier Dualities

Gauging (sub)symmetries and performing generalized Fourier transforms on local weights are structurally equivalent to changing boundary conditions or module categories in the SymTFT formalism. For example, gauging a subgroup $A \subset G$ in the category $\mathrm{Vect}_G$ amounts to a choice of brane boundary condition labeled by module category $\mathcal{M}(A)$, resulting in a dual theory with representation category $\mathrm{Rep}(G)$ and boundary $\mathrm{Rep}(A)$ (Delcamp et al., 12 Aug 2024). The Fourier transform expresses local weights as

$$\theta^\vee(\rho) = \frac{1}{|G|} \sum_{g \in G} \theta(g) \rho(g)$$

with inverse

$$\theta(g) = \sum_{\rho \in \hat{G}} d_\rho\, \mathrm{Tr}[\theta^\vee(\rho) \rho^\vee(g)]$$

(Delcamp et al., 12 Aug 2024).

Kramers–Wannier duality in this generalized context arises by combining Fourier transformation and gauging. The non-Abelian duality relates the topological sector partition functions of the original and dual models when appropriately matched: $$\left\langle\Sigma_\Delta, \mathrm{Vect}_G, \mathrm{Vect}_G, ([c_1], V_{c_1}) \mid \Sigma_\Delta, \mathrm{Vect}_G; \theta, \vartheta^{\text{triv}} \right\rangle \propto \left\langle\Sigma_\Delta^\vee, \mathrm{Rep}(G), \mathrm{Rep}(G), ([c_1], V_{c_1}) \mid \Sigma_\Delta^\vee, \mathrm{Rep}(G);\, \theta^\vee, \vartheta^{\text{triv}} \right\rangle$$ where the sectors are labeled by conjugacy classes and centralizer representations (Delcamp et al., 12 Aug 2024).

The mathematical underpinning of this duality is the Morita equivalence and the isomorphism of Drinfeld centers: $Z(\mathrm{Vect}_G) \cong Z(\mathrm{Rep}(G)),$ reflecting a deep symmetry between the original and dual theories’ anyon excitation structures. In the abelian case, this reduces to the classical Kramers–Wannier duality of the Ising model.

4. RG Fixed Points and Gapped Phases: Classification via Module Categories

The SymTFT framework provides a concrete route to constructing renormalization group fixed-point models for gapped phases with symmetry. Gapped symmetric phases correspond to choices of module categories $\mathcal{M}$ over the fusion category $\mathcal{C}$, or equivalently, to separable algebra objects $\mathcal{A}$ (e.g., $\mathcal{A} = \mathbb{C}[A]$ for subgroup $A \subset G$) (Delcamp et al., 12 Aug 2024). The Hamiltonian or partition function is then engineered via topologically protected data—multiplication structure and fusion—encoded in the “$\mu$-symbols” associated with the chosen module category. Upon Fourier transform or changing brane boundary conditions, one obtains dual fixed-point models classified up to Morita equivalence.

This approach systematically unifies:

• Symmetry breaking patterns,
• Anyon condensation at the boundary,
• The hierarchy of phases indexed by module categories.

A plausible implication is that the space of topologically distinct symmetric RG fixed points is thus parametrized by the Morita theory of fusion categories and their module categories.

5. Interplay of Symmetry, Topological Order, and Non-Invertibility

Topological symmetry theory encodes not only invertible (group-like) but also non-invertible, categorical, and higher-form symmetries. The generalized formalism encompasses:

• Weak Hopf symmetries, where the phase is protected by both a weak Hopf algebra $H$ and its dual $\hat{H}$, with the resulting symmetry being $H \times \hat{H}$ (Jia, 19 Dec 2024).
• Symmetries corresponding to arbitrary fusion categories, arising on the boundary of a higher-dimensional quantum double model (characterized by $Z(\mathrm{Rep}(H))$).
• Non-invertible duality defects and their fusion structures, naturally encoded in the categorical language of TQFT and SymTFT (Kaidi et al., 2022).

The unified treatment allows explicit solutions (e.g., via weak Hopf tensor network states) and provides exact analytical construction of lattice ground states reflecting the full categorical symmetry structure.

6. Implications and Applications

Key implications and practical applications include:

• The derivation and systematic construction of non-Abelian Kramers–Wannier dualities, enabling new solvable models with nontrivial dualities beyond the abelian paradigm (Delcamp et al., 12 Aug 2024).
• Realization of arbitrary fusion category symmetries in spin liquids and SPT phases, which may exhibit exotic (non-invertible, higher-form) symmetry-protection mechanisms (Jia, 19 Dec 2024).
• Unified description of symmetry-enriched phases and their dualities, including the ability to engineer specific anyon types or symmetry assignments via boundary conditions and module categories.
• Potential for practical algorithms in material prediction, as the categorical classification can be combined with symmetry indicator methods and numerical approaches.

This framework is central for advancing both the mathematical understanding and physical realization of generalized symmetries, topological order, and dualities—including the landscape of gapped and critical phases, the paper of non-invertible and categorical symmetry, and the application to lattice models and quantum computational platforms.