Condensable Algebras in symTO

• Condensable algebras in symTO are algebraic structures composed of mutually local bosonic excitations that satisfy precise fusion, quantum dimension, and cyclotomic constraints.
• They classify global phase structures by encoding gapped boundaries, domain walls, and symmetry transitions, linking symmetry-protected with symmetry-broken phases.
• These algebras enable anomaly resolution and the identification of dualities through holographic modular bootstrap methods and higher categorical frameworks.

A condensable algebra in a symmetry topological order (symTO) is a distinguished algebraic structure built from mutually local, bosonic excitations ("anyons" or higher-dimensional analogues) in a braided fusion category or its higher categorical generalizations. The identification and classification of condensable algebras—often formalized as Lagrangian algebras or connected étale algebras—are foundational for understanding the global phase structure, gapped and gapless boundaries, and the landscape of symmetry-protected and symmetry-broken states in systems described by a symTO. Condensable algebras underpin the symmetry/topological order correspondence, encode generalized symmetry breaking, and determine emergent and anomaly-constrained dynamics across a wide range of quantum systems.

1. Definition and Algebraic Criteria

A condensable algebra $\mathcal{A}$ in a symTO (typically realized as a modular tensor category $\mathcal{M}$ or a higher categorical structure) is a finite direct sum

$$\mathcal{A} = \bigoplus_{a \in \mathcal{M}} A^a \cdot a$$

with $A^a \in \mathbb{Z}_{\geq 0}$, $A^1=1$ (the “trivial” object appears with multiplicity 1), obeying the following constraints:

• Bosonic closure: All $a$ with $A^a > 0$ satisfy $s_a = 0$ mod $1$ (trivial topological spin).
• Closure under fusion: For all $a, b$ with $A^a, A^b > 0$, there exists $c$ with $A^c > 0$ and fusion coefficient $N_{ab}^c > 0$.
• Fusion inequalities: $\sum_c N^{ab}_c A^c \geq A^a A^b$.
• Quantum dimension bounds: $A^a \leq d_a - \delta(d_a)$ with $\delta(d_a) = 0$ if $d_a$ integer, $1$ otherwise.
• Number-theoretic (cyclotomic) constraints: For example,

$\frac{ \sum_b S_{M}^{ab} A^b }{ \sum_b d_b A^b }$

must be a cyclotomic integer for compatibility with modular transformations.

Such an $\mathcal{A}$ is called Lagrangian if every nontrivial bulk excitation has trivial mutual statistics with (and/or is in the ideal generated by) $\mathcal{A}$. In the context of higher categories ($E_k$-algebras in looped defect categories), the condensable algebra carries additional coherence (e.g., associators, 2-associators) and Morita-equivalence becomes a central organizing concept.

2. Physical Interpretation and Domain Walls

Condensable algebras label all gapped (Lagrangian) boundaries and symmetry-broken phases of the symTO in one lower spatial dimension:

• Gapped boundary condition: Specified by a Lagrangian condensable algebra $\mathcal{A}$. Condensing $\mathcal{A}$ trivializes the topological order near the boundary.
• Phase transition/domain wall: Condensing a non-Lagrangian $\mathcal{A}$ generically realizes a gapless (critical) state protected by an effective, residual symmetry (possibly anomalous, higher-form, or non-invertible). The reduced topological order $M_{/\mathcal{A}}$ (after modding out by $\mathcal{A}$) classifies these critical/IR fixed points (Chatterjee et al., 2022).
• Condensable subalgebras and Galois correspondence: There exists a categorical Galois correspondence between fusion subcategories of the module category $\mathcal{C}_A$ (all $A$-modules) and condensable subalgebras of $A$, with invariants classifying intermediate phases (Dong et al., 18 Aug 2025).

In higher dimensions, condensable E$_k$-algebras (e.g., in modular 2-categories for 3+1D) govern the condensation of string-like or membrane-like extended objects and topologically nontrivial domain walls, as elaborated for 3+1D toric code models (Zhao et al., 2022). Algebraic data (associators, braiding isomorphisms, projective group actions) capture refined boundary anomaly structure and topological defect fusion.

3. Global Phase Structure and Dualities

The collection of all condensable algebras in a given symTO provides a complete algebraic skeleton for the phase diagram:

• Each phase is indexed by a unique (up to automorphism) condensable algebra.
• Continuous phase transitions (critical points) correspond to "intersections" or inclusions among condensable algebras ($$\mathcal{A}_c \subseteq \mathcal{A}_1, \mathcal{A}_2$$ for adjacent phases).
• Automorphisms of the symTO induce dualities—finite-depth quantum circuits or categorical autoequivalences—that relate condensable algebras and hence identify SPT or symmetry breaking phases up to "decorations" (e.g., SPT/pSPT/aSPT classes). For non-invertible symmetry, automorphism classification extends this phase structure far beyond traditional group cohomology (Aksoy et al., 27 Mar 2025).
• 2-Morita equivalence organizes E$_2$ condensable algebras and gapped domain walls, connecting boundaries to central (lagrangian) algebras in Drinfeld centers (see trinity below) (Xu et al., 28 Mar 2024).

Object type	Algebraic structure	Physical interpretation
Condensable algebra ($\mathcal{A}$)	Direct sum of bosonic anyons/objects with fusion, dimension, and number-theoretic constraints	Catalogues gapped and gapless boundaries, phase transitions
Lagrangian algebra	Maximal condensable algebra	Fully gapped, trivial topological order
Non-Lagrangian condensable algebra	Subalgebra with residual topological order ($M_{/\mathcal{A}}$ nontrivial)	Gapless phase or "anomaly-protected" criticality

4. Holographic Modular Bootstrap and Operator Content

The low-energy properties (central charges, scaling dimensions, operator content) of an $A$-state (boundary or critical point) can be systematically calculated from the reduced topological order $M_{/A}$ using holographic modular bootstrap methods. This approach uses the "bulk-boundary" correspondence enforced by the $S$ and $T$ modular data (Chatterjee et al., 2022). For example,

$$S_M \cdot A = A \cdot S_{M/A}, \qquad T_M \cdot A = A \cdot T_{M/A}$$

together with cyclotomic constraints, completely determine the admissible partition functions and corresponding low-energy boundary CFTs.

This method classifies not only gapped and symmetry-breaking phases but also protected gapless points and phase diagrams in systems exhibiting anomalous, higher-form, or non-invertible symmetries.

5. Mathematical Generalization: Higher Category and Enriched Settings

Recent advances have elevated condensation theory to the level of enriched $\infty$-categories and higher fusion $n$-categories (Stockall, 30 Jun 2025, Kong et al., 12 Mar 2024). Iterated condensation of $E_k$-algebras within multi-fusion $n$-categories realizes symmetries of arbitrary (co)dimension and codimension (including continuous, non-semisimple, and derived symmetries).

• An E$_k$-condensable algebra is a separable $E_k$-algebra object typically in a $k$-fold "looped" defect category. Its condensation is performed in $k$ steps, each reducing the codimension, and results in a new phase and an associated (higher) Morita equivalence.
• Truncated condensation and monoidal enrichment (Day convolution, Eilenberg–Moore functors) provide a general framework for functorial and monoidal properties required for physical compatibility.

6. Applications to Symmetry, Anomaly, and Non-Invertible Structures

Condensable algebras in symTOs have far-reaching consequences:

• Classification of group and categorical SPT phases: Lagrangian algebras correspond to gapped SPT and symmetry-broken phases; non-Lagrangian ones classify intrinsically gapless (igSPT) phases, including those with anomaly constraints that cannot be gapped (Robbins et al., 5 Sep 2025).
• Resolution of anomalies: Embedding an anomalous symmetry into a larger fusion category and condensing a suitable algebra can resolve global anomalies by extending symmetries, introducing "trivial" extra degrees of freedom that absorb the 't Hooft anomaly (Robbins et al., 5 Sep 2025).
• Bulk-boundary correspondence for gapless states: Minimal (often non-Lagrangian) condensable algebras (e.g., those whose local module category has no intrinsic topological order) capture gapless SPT (gSPT) structure and cohomological data, often linked—through the holographic dictionary—to the LHS spectral sequence and obstruction theory (Wen, 17 Mar 2025, Wen, 11 Aug 2024).
• Generalized symmetry and emergent structures: In lattice models with non-invertible symmetries (e.g., those implemented by categorical or fusion actions), condensable algebra structures determine which phases are accessible and how dualities and self-dual anomalies manifest in the low-energy theory (Chatterjee et al., 8 May 2024, Chen et al., 10 Sep 2025).

7. Practical Computation and Example Classification

The explicit computation of condensable algebras and their physical consequences proceeds through:

• Algebraic enumeration: Solving the system of cyclotomic, fusion, and modular equations for $A^a$ to enumerate all admissible patterns.
• Mechanistic correspondence: Matching allowed Lagrangian algebras with physical boundary conditions (dimer, valence bond solid, SPT, SSB, etc.) and gapless boundaries with residual topological order ($M_{/A}$ nontrivial).
• Holographic and lattice realization: Bridging categorical data with Hamiltonian/lattice model constructions; for example, string-net models and strange correlator constructions use Frobenius algebras and condensation methods to realize critical models and to exhaustively scan possible CFT boundaries as in the "CFT factory" approach (Hung et al., 5 Jun 2025).

Summary Table: Core Role of Condensable Algebras in symTO

Function	Algebraic Structure	Physical Interpretation
Boundary/phase classification	(Lagrangian) condensable algebra	Distinguishes gapped and gapless states
Duality and SPT order differentiation	Automorphism group/2-Morita equivalence	Classifies SPT/pSPT/aSPT phases and dualities
Criticality and anomaly protection	Non-Lagrangian condensable algebra	Signals protected gaplessness and emergence of gSPT/igSPT
Anomaly resolution and reduction	Embedding in larger fusion category, condensation sequence	Realizes anomaly cancellation and extension of symmetry

Condensable algebras in symmetry topological orders thus serve as central organizing objects, unifying the algebraic, categorical, and physical structure of quantum phases and transitions across all dimensions and symmetry settings, from conventional Landau symmetry breaking, through SPT and beyond-Landau criticality, to the rich domain of non-invertible and higher-form symmetries and generalized anomaly phenomena.