Group Supercohomology in Fermionic SPT Phases

• Group supercohomology is a framework that extends bosonic cohomology with fermionic decorations to classify SPT phases using graded cochains and obstruction theory.
• It employs Z2 and U(1)-valued cochains to capture Majorana chain and complex-fermion decorations in dimensions like 2+1D and 3+1D.
• The framework bridges bosonic shadow models with fermionic physics through lattice constructions, spectral sequences, and higher-category approaches.

Group supercohomology provides a systematic mathematical framework for the classification and construction of certain classes of fermionic symmetry-protected topological (SPT) phases, generalizing the bosonic group cohomology techniques. Developed to address the classification of interacting fermionic SPTs with symmetry group $G_f = G \times \mathbb{Z}_2^f$ (where $\mathbb{Z}_2^f$ represents fermion parity), group supercohomology introduces additional graded cohomological data and obstruction theory. This structure enables the explicit realization and stacking of fermionic SPT phases, particularly in (2+1)D and (3+1)D dimensions, and underpins the interplay between bosonic “shadow” models and fermionic physics via bosonization dualities and spin-structure dependence.

1. Algebraic Structure and Classification Principles

The foundational construction of group supercohomology theory begins by supplementing the data of ordinary cohomology with $\mathbb{Z}_2$-valued “Majorana/Kitaev chain” decorations, reflecting the fundamentally fermionic aspects of the systems of interest. For $d$ spatial dimensions, the relevant cochains are:

• $\mathbb{Z}_2$-valued $(d-1)$-cochain $\tilde{n}_{d-1} \in C^{d-1}(G, \mathbb{Z}_2)$, encoding decorations by 1D Majorana/Kitaev chains on symmetry domain-wall intersections;
• $\mathbb{Z}_2$-valued $d$-cochain $n_d \in C^d(G, \mathbb{Z}_2)$, tracking “complex-fermion decoration” along higher domain-wall intersections;
• $\mathbb{Z}_2^f$0-valued $\mathbb{Z}_2^f$1-cochain $\mathbb{Z}_2^f$2, generalizing the usual bosonic SPT cocycle.

The cocycle conditions are twisted by the Steenrod square, yielding: $\mathbb{Z}_2^f$3 where $\mathbb{Z}_2^f$4 is a degree-$\mathbb{Z}_2^f$5 obstruction term, e.g., $\mathbb{Z}_2^f$6 in $\mathbb{Z}_2^f$7 $\mathbb{Z}_2^f$8 In $\mathbb{Z}_2^f$9D, a triple $\mathbb{Z}_2$0 arises, with $\mathbb{Z}_2$1, $\mathbb{Z}_2$2, and $\mathbb{Z}_2$3 determined by higher obstructions built from the lower layers (Kapustin et al., 2017, Wang et al., 2017).

2. Categorical and Homotopical Perspectives

Group supercohomology admits a homotopy-theoretic definition via generalized cohomology theories. Its spectrum $\mathbb{Z}_2$4 is defined by a Postnikov truncation of connective real $\mathbb{Z}_2$5-theory and encodes the invertible 2-supervector spaces (the Picard 2-groupoid $\mathbb{Z}_2$6). Explicitly,

$\mathbb{Z}_2$7

with explicit low-degree layers corresponding to “Majorana” ($\mathbb{Z}_2$8 in degree 2), “Gu–Wen” ($\mathbb{Z}_2$9 in degree 1), and “Dijkgraaf–Witten” ($d$0 in degree 0) strata (Debray et al., 28 Oct 2025).

There is a natural spectral sequence (“hastened Adams spectral sequence” or HASS) for supercohomology, working with the subalgebra $d$1 of the mod 2 Steenrod algebra. This arises in computational work, connecting group supercohomology to spin-bordism and providing a bridge to invertible TQFTs and categorical anomaly data (Debray et al., 28 Oct 2025).

3. Lattice Models and Spin Structure

The physical realization of group supercohomology SPTs typically proceeds by constructing commuting projector Hamiltonians whose distinctiveness derives from their dependence on spin structure. In $d$2D, constructions often employ a Kasteleyn orientation on a reference trivalent Majorana graph $d$3, encoding all $d$4 inequivalent spin structures for genus $d$5. The Hamiltonian consists of a domain-wall sector (capturing the $d$6-decoration) and fluctuation terms that implement symmetry, all respecting the spin-Kasteleyn matching and global fermion parity (Tarantino et al., 2016). For $d$7, such constructions yield models for the full $d$8 classification, with the “root” phases beyond supercohomology corresponding to symmetry fluxes trapping a Majorana mode.

In $d$9D, finite-depth quantum circuits are constructed to prepare the ground state from an unentangled symmetric state. These circuits are assembled from elementary gates realizing the cocycle phases, domain-wall decorations, and fermionization rules determined by the group supercohomology data and triangulation of the manifold. The stacking law is encoded at the circuit level, implementing the abelian group structure given by

$\mathbb{Z}_2$0

The role of the spin structure remains critical, both for the proper definition of the lattice path integrals and for the condensation of the emergent fermion in shadow models (Chen et al., 2020, Ellison et al., 2018).

4. Relation to Bosonic Shadows and Higher-Group Symmetries

A key conceptual advance in supercohomology theory is the transparent connection to “bosonic shadow” SPTs and the structure of higher-group symmetries. The supercohomology SPTs for $\mathbb{Z}_2$1 are constructed via a two-step process:

1. Auxiliary bosonic SPT with an ordinary or higher-group symmetry, with domain-wall or junctions decorated according to bosonic cohomology;
2. Gauging an appropriate 1-form $\mathbb{Z}_2$2 symmetry to obtain a shadow theory with emergent fermions, followed by fermionic condensation (bosonization duality), which effectively introduces the spin structure and realizes the full fermionic SPT (Ellison et al., 2018, Kapustin et al., 2017).

In $\mathbb{Z}_2$3D, bosonic shadow models are constructed with 3-group global symmetry and nontrivial Postnikov class (e.g., $\mathbb{Z}_2$4). Supercohomology phases arise by gauging both the emergent particle and string excitations, allowing the incorporation of fermion parity through spin-dependent path integrals (Kapustin et al., 2017). The boundary anomalies of such SPTs, encoded categorically, correspond to a generalized supercohomology group $\mathbb{Z}_2$5.

5. Explicit Examples and Group Extensions

The group supercohomology classification is manifest in particular discrete examples. For $\mathbb{Z}_2$6, the $\mathbb{Z}_2$7D supercohomology group yields four classes $\mathbb{Z}_2$8, but the full interacting classification is $\mathbb{Z}_2$9—the extra “root” phases correspond to nontrivial maps $(d-1)$0, i.e., which symmetry flux traps a Majorana (Tarantino et al., 2016). The stacking operation for such phases reflects the group law arising from the cochain data and their cup products.

In higher dimensions, the classification becomes layered. In $(d-1)$1D, the obstruction-free subgroup $(d-1)$2 is defined by the vanishing of $(d-1)$3 and a further obstruction in $(d-1)$4, demarcating which Kitaev-chain decorations can be consistently imposed without violating locality or symmetry (Wang et al., 2017). The exact sequences

$(d-1)$5

and

$(d-1)$6

encode this combined structure: bosonic SPTs, complex-fermion decorations, and Kitaev-chain decorations, subject to obstruction-theoretic constraints (Wang et al., 2017).

6. Categorical Anomalies, Cobordism, and Future Perspectives

Group supercohomology has a deep connection with cobordism theory and fusion 2-categories, especially in the context of topological quantum field theories (TQFTs) with symmetry anomalies. The classification of invertible fermionic TQFTs is naturally formulated in terms of Anderson duals of spin-bordism spectra, with group supercohomology mapping into spin-bordism groups for $(d-1)$7. In the categorical setting, $(d-1)$8-enriched braided fusion 2-categories capture the algebraic and topological data of (3+1)D fermionic SETs, with supercohomology providing a “truncation” encompassing the Dijkgraaf–Witten, Gu–Wen, and Majorana layers of the full categorical $(d-1)$9-anomaly (Debray et al., 28 Oct 2025).

Practical computations of group supercohomology for finite groups and their anomalies are facilitated by spectral sequence methods, particularly the hastened Adams spectral sequence. These calculations are relevant for the Wang–Wen–Witten symmetry extension constructions in fermionic systems and for the design of anomalous TQFTs with specified symmetry properties.

Group supercohomology thus constitutes a central link connecting algebraic topology, categorical quantum mechanics, and the classification of interacting fermionic phases beyond free-fermion or purely bosonic paradigms.

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References:

• (Debray et al., 28 Oct 2025): "How to Build Anomalous (3+1)d Topological Quantum Field Theories"
• (Wang et al., 2017): "Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory"
• (Chen et al., 2020): "Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions"
• (Tarantino et al., 2016): "Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases"
• (Gu et al., 2012): "Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear σ models and a special group supercohomology theory"
• (Ellison et al., 2018): "Disentangling interacting symmetry protected phases of fermions in two dimensions"
• (Kapustin et al., 2017): "Fermionic SPT phases in higher dimensions and bosonization"