Universal Topological Classification

• Universal topological classification is a comprehensive framework that organizes physical and mathematical systems using invariants, homotopy, and categorical structures.
• It employs state-space methods and projective representations to derive robust invariants like K-theory and modular matrices, unifying non-interacting systems, quantum orders, and black hole thermodynamics.
• Higher-categorical and operator-based approaches extend the scheme to complex phase transitions and entanglement classifications, offering practical insights for condensed matter and quantum information research.

A universal topological classification scheme provides a systematic, mathematically robust foundation for the organization and distinction of topological phases of physical and mathematical systems, unifying disparate examples and approaches under a single language of invariants, homotopy, and categorical or geometric structures. Theoretical advances over the past decade have yielded such universal frameworks for condensed-matter topological phases, quantum information systems, topological phases of black holes, higher-category orders, and mathematical singularities, revealing a network of approaches linked by invariants defined via homotopy of configuration spaces, projective representations, generalized cohomological data, or geometric covering structures.

1. State-Space-Based Universal Classification for Non-Interacting Systems

The foundational approach for non-interacting topological phases, as articulated by De Nittis (Nittis, 5 Feb 2025), frames the universal classification in terms of the observable algebra $\mathcal{A}$, modeled as a trivial $C^*$-bundle: $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$ where $M$ is a compact Hausdorff “quantum parameter space” (e.g., the Brillouin torus), and $\mathcal{A}_0$ is a reference-fiber $C^*$-algebra (typically $K(\mathcal{H})$, the compact operators on a separable Hilbert space).

Topological phases are represented by continuous “configuration” maps

$f : M \rightarrow P(\mathcal{A}_0),$

with $P(\mathcal{A}_0)$ the pure state space of the fiber algebra. Homotopy classes of these configuration maps,

$[M, P(\mathcal{A}_0)],$

define the universal classification invariant. Homotopy equivalence not only ensures indistinguishability of physical statistics but is stable under all continuous deformations, making the scheme robust.

K-theory emerges as a special case for type A systems: for $C^*$0, one has $C^*$1, so for finite CW-complex $C^*$2,

$C^*$3

and in low dimensions, this recovers the classic K-theory periodic table of topological insulators (Nittis, 5 Feb 2025). This formalism generalizes to crossed-product $C^*$4-algebras, irrational-rotation algebras, and dynamical systems, universally recovering K-theoretic invariants and pointing toward extensions for interacting systems.

2. Universal Classification via Projective Representations: Gapped Phases and Topological Orders

The universal wave function overlap method (Moradi et al., 2014) extracts phase-defining topological data from ground-state subspaces on closed spatial manifolds. For a $C^*$5-dimensional topologically ordered system, one considers the mapping class group $C^*$6 and computes operator overlaps

$C^*$7

with $C^*$8 furnishing a projective unitary representation of $C^*$9. In $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$0D, this formalism recovers modular $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$1 and $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$2 matrices, generating a projective $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$3 representation encoding fusion and braiding data, the chiral central charge, and modular tensor category structure. For higher $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$4, the projective representation generalizes to the mapping class group of $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$5 or other manifolds, determining higher-dimensional mutual statistics.

This framework underpins the computation of universal invariants in both model Hamiltonians and experimental numerical approaches, and is robust under arbitrary local perturbations (Moradi et al., 2014).

The motivic GUT framework (Yamada, 16 Mar 2026), extending far beyond quantum states and projective representations, identifies the correct ambient category for universal classification as the symmetric monoidal $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$6-category of fully dualizable fusion $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$7-categories modulo Morita equivalence. The classification is encoded in the connective Brauer spectrum $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$8 associated with the tangential structure $\mathcal{A} \cong C(M) \otimes \mathcal{A}_0,$9, and lower-dimensional modular tensor category or string-net schemes arise as "shadows" of the full higher-categorical object. This Copernican shift is necessary for the classification of topological orders in $M$0 where genuinely higher-categorical data—capturing extended excitations and exotic statistics—are not visible to any lower-categorical invariant (Yamada, 16 Mar 2026).

3. Universal Thermodynamic Topological Classes in Black Hole Physics

The classification of black hole thermodynamic states as topological defects in parameter space was established by Wei, Liu, and collaborators (Zhu et al., 2024, Wei et al., 2024, Chen et al., 15 Apr 2025, Rizwan et al., 8 Jan 2025), introducing four universal classes—$M$1, $M$2, $M$3, $M$4—labeled by global topological charge $M$5 and the stabilities of the smallest/largest black hole states. The key ingredients are:

• Construction of a two-component defect vector $M$6 from an extended off-shell free energy $M$7,
• Identification of topological “charges” (winding numbers $M$8) at zeros of $M$9 (on-shell thermodynamic states),
• The total topological invariant $\mathcal{A}_0$0, determined solely by the asymptotic behavior of the inverse Hawking temperature,
• The four classes are exhaustively determined by $\mathcal{A}_0$1, directly dictating state stability and the allowed orderings of black hole branches.

For example, all three-dimensional BTZ black holes (neutral, charged, rotating) fall into $\mathcal{A}_0$2, with stable small and large black hole states at zero and infinite temperature, and this classification is robust against variations in charge or angular momentum (Chen et al., 15 Apr 2025). The scheme captures the universality and dimension-dependent dichotomy in black hole thermodynamics and quantum gravity, rigorously extending to black holes with matter couplings, AdS asymptotics, and perfect fluid dark matter backgrounds (Rizwan et al., 8 Jan 2025).

4. Universal Schemes in Quantum Information and Entanglement Classifications

Multipartite entanglement admits a universal topological classification through polynomial SU(2) and SL(2) invariants and balancedness structure (Johansson et al., 2013). Pure $\mathcal{A}_0$3-qubit states are organized by balancedness (c-balanced and a-balanced states) reflected in the structure of integer matrices associated to computational basis decompositions. Each invariant, labeled by its bidegree $\mathcal{A}_0$4 under SU(2) scaling, distinguishes entanglement families and underpins the set of discrete topological phases obtainable from cyclic SU(2) evolutions.

The classification differentiates irreducibly c-balanced (SL-semistable), purely a-balanced (SU-semistable but SL-zero), and unbalanced states, with a full hierarchy for $\mathcal{A}_0$5-qubit families determined by the nonvanishing of a finite collection of such invariants, and correspondingly distinct topological phase spectra (Johansson et al., 2013).

In 2D stabilizer code theory, topological phase equivalence under local unitary transformations collapses all 2D topological stabilizer codes to stacks of Kitaev’s toric code phase—the only invariant being the total quantum dimension—establishing a strict universal phase structure (Bombin et al., 2011).

5. Dynamical and Operator-Based Universal Classifications

Topological phases of quantum matter and their transitions can be dynamically classified via bulk-surface duality: the integer invariant ($\mathcal{A}_0$6D Chern or winding number) is reduced to a $\mathcal{A}_0$7D invariant computed on band inversion surfaces (BISs), which are accessible via post-quench spin dynamics and emergent spin-momentum textures (Zhang et al., 2018). This universal dynamical approach is applicable to arbitrary Clifford-class Hamiltonians and generalizes to multiband systems.

Momentum-space invariants for all symmetry classes and dimensions can also be mapped to real space by universal topological operators, yielding local and non-local "topological markers" defined via projectors and position operators. These markers provide direct, lattice-based spatial diagnostics of phase and a probe of phase transitions through their diverging correlation length (Chen, 2022).

6. Geometric and Covering-Space Universalities in Phase Transitions

Geometric classification of first-order black hole phase transitions is achieved through the analysis of the temperature function $\mathcal{A}_0$8 and its nondegenerate critical points, revealing that the fundamental origin of multivaluedness, swallowtail structures in free energy, and phase coexistence regions is a universal three-sheeted covering of the temperature line by the horizon radius parameter (Zhang et al., 18 Dec 2025). Black holes are classified as class A1 (two nondegenerate extrema, first-order transition, three-sheeted covering), class A2 (one extremum, no swallowtail), or class B (monotonic $\mathcal{A}_0$9, trivially single-valued), complementing the global topological invariants obtained from vector field analysis in thermodynamic space. The classification ties directly to the absence or presence of van der Waals–like first-order transitions and captures phase structure across diverse black hole families (Zhang et al., 18 Dec 2025).

7. Higher Category, Morita, and Homotopy-Theoretic Universality

In the most general scheme, the classification of topological phases in any dimension ($C^*$0) is provided by Morita equivalence classes of Karoubi-complete, fully dualizable fusion $C^*$1-categories with trivial center, as formalized in the higher Morita category framework (Johnson-Freyd, 2020). This structural approach encodes remote detectability and boundary-bulk dualities and is corroborated by the motivic GUT spectrum approach (Yamada, 16 Mar 2026). For $C^*$2, the universal scheme recovers modular tensor categories (Witt group), for $C^*$3 symmetric multifusion 2-categories (finite group gauge theory), and in arbitrary $C^*$4 as anomalous sigma models on finite groupoids, with action in generalized cohomology classes $C^*$5.

• (Nittis, 5 Feb 2025): L. De Nittis, "Topological phases of non-interacting systems: A general approach based on states"
• (Moradi et al., 2014): B. Moradi, X.-G. Wen, "Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States"
• (Yamada, 16 Mar 2026): "Motivic GUT Part I: Grand Unified Theory of Topological Order"
• (Chen et al., 15 Apr 2025): S.-W. Wei et al., "Universal thermodynamic topological classes of three-dimensional BTZ black holes"
• (Wei et al., 2024): Y. Liu et al., "Universal topological classifications of black hole thermodynamics"
• (Zhu et al., 2024): S.-W. Wei, Y.-X. Liu, R. B. Mann, "Universal thermodynamic topological classes of rotating black holes"
• (Rizwan et al., 8 Jan 2025): W. Xu et al., "Universal thermodynamic topological classes of black holes in perfect fluid dark matter background"
• (Zhang et al., 18 Dec 2025): C. Lan et al., "A Universal Geometric Framework for Black Hole Phase Transitions: From Multivaluedness to Classification"
• (Johansson et al., 2013): M. E. Cohen, J. Szulc, "Classification scheme of pure multipartite states based on topological phases"
• (Bombin et al., 2011): S. Bravyi, M. B. Hastings, S. Michalakis, "Universal topological phase of 2D stabilizer codes"
• (Zhang et al., 2018): R. Zhang et al., "Dynamical classification of topological quantum phases"
• (Chen, 2022): D.-L. Deng et al., "Universal topological marker"
• (Johnson-Freyd, 2020): A. Kapustin et al., "On the classification of topological orders"

These works collectively demonstrate the conceptual power and mathematical reach of universal topological classification schemes, providing foundational frameworks across condensed matter, quantum information, black hole thermodynamics, and higher-categorical algebraic topology.