Group Supercohomology Theory

• Group supercohomology theory is a framework that extends group cohomology by incorporating a Z2-graded structure to classify fermionic SPT phases.
• It provides explicit constructions of fixed-point lattice Hamiltonians and finite-depth circuits for realizing interacting topological phases.
• The theory unifies bosonic and intrinsically fermionic states, enabling deeper insight into anomalies and associated topological quantum field theories.

Group supercohomology theory is a mathematical framework that extends group cohomology to classify short-range entangled interacting fermionic symmetry-protected topological (SPT) phases in arbitrary spatial dimension. It systematically generalizes the standard group cohomology description of bosonic SPTs by encoding not only the group-theoretic data of the symmetry but also a $\mathbb{Z}_2$-graded “fermionic” structure. Group supercohomology yields both an explicit construction of fixed-point lattice Hamiltonians and a classification of fermionic SPT phases—including intrinsically fermionic states that cannot be realized by free-fermion systems or by bosonic cohomology (Gu et al., 2012, Wang et al., 2017, Ellison et al., 2018, Chen et al., 2020, Debray et al., 28 Oct 2025).

1. Mathematical Structure of Group Supercohomology

Fundamentally, group supercohomology is a generalization of ordinary group cohomology $H^{d+1}(G_b, U_T(1))$ for a bosonic symmetry group $G_b$, adapted to the case where the full symmetry is $G_f = G_b \times \mathbb{Z}_2^f$, with $\mathbb{Z}_2^f$ denoting fermion parity. The supercohomology group, denoted $\mathfrak{H}^{d+1}[G_f, U_T(1)]$ or $$H_{\text{super}}^{d+1}(G_b \times \mathbb{Z}_2^f, U(1))$$, classifies fermionic SPT phases in $(d+1)$ spacetime dimensions (Gu et al., 2012). These groups fit into a short or, in higher dimensions, multi-step exact sequence, expressing how the fermionic theory extends bosonic group cohomology by graded $\mathbb{Z}_2$-valued data and higher obstruction constraints:

$$0 \to H^{d+1}(G_b, U_T(1))/\Gamma \to \mathfrak{H}^{d+1}[G_f, U_T(1)] \to B H^d(G_b,\mathbb{Z}_2) \to 0$$

Here $B H^d$ is the “obstruction-free” subgroup of $H^d(G_b, \mathbb{Z}_2)$ defined by certain secondary conditions, typically associated with Steenrod squares $Sq^2$, and $\Gamma$ is generated by bosonic SPTs trivialized in the presence of fundamental fermions (Gu et al., 2012).

The group supercohomology data consists of cochain layers: for instance, in 3+1 dimensions,

• a “Kitaev chain layer” $\tilde n_2\in Z^2(G_b,\mathbb{Z}_2)$,
• a “Gu–Wen layer” $n_3\in C^3(G_b,\mathbb{Z}_2)$ obeying $dn_3 = \tilde n_2 \cup \tilde n_2$,
• a “Dijkgraaf–Witten layer” $\nu_4\in C^4(G_b, U_T(1))$ subject to a twisted cocycle equation involving secondary cohomological operations (Wang et al., 2017).

Equivalence classes are defined modulo “fermionic gauge transformations,” intertwining the cochain data via appropriate coboundary and cup product shifts (Wang et al., 2017, Ellison et al., 2018, Chen et al., 2020).

2. Cocycle Data and Supercocycle Equations

A $(d+1)$-dimensional fermionic SPT is encoded by a supercocycle $(\nu_{d+1}, n_d)$, where $n_d\in Z^d(G_b, \mathbb{Z}_2)$ encodes the “graded structure” corresponding to the placement of fundamental fermions, and $\nu_{d+1}\in C^{d+1}(G_b, U_T(1))$ encodes the $U(1)$ phase data (Gu et al., 2012, Wang et al., 2017, Ellison et al., 2018).

The defining supercocycle equations are:

• $n_d$ is $G_b$-invariant and satisfies $$\sum_{i=0}^{d+1} n_d(g_0,\dots,\widehat{g}_i,\dots,g_{d+1}) \equiv 0 \pmod{2}$$,
• the “supercocycle” constraint,

$$\delta \nu_{d+1}(g_0,\ldots,g_{d+2}) \cdot (-1)^{f_{d+2}(g_0,\ldots,g_{d+2})} = 1,$$

where $f_{d+2}$ is a function of the lower-degree $n_d$, typically given via Steenrod squares or cup products, e.g., $$f_{4}(g_0,g_1,g_2,g_3,g_4) = n_3(g_0,g_1,g_2)\, n_3(g_2,g_3,g_4)$$ in $d=3$ (Gu et al., 2012, Ellison et al., 2018).

Coboundary equivalence classes group cocycles related by bosonic (trivial) or fermionic (graded, parity-involving) redefinitions, rendering the cohomology a $\mathbb{Z}_2$-extension of bosonic SPT classification (Ellison et al., 2018).

3. Explicit Lattice Constructions and Finite-Depth Circuits

Group supercohomology provides an explicit prescription for constructing exactly solvable lattice Hamiltonians and corresponding quantum circuits for fermionic SPT phases (Gu et al., 2012, Ellison et al., 2018, Chen et al., 2020). The construction proceeds via these principal steps:

1. Assign $G_b$-valued spins to lattice sites or simplicial vertices.
2. Decorate $(d-1)$-cells with $\mathbb{Z}_2$ Kitaev chains according to $n_d$.
3. Assign $U(1)$ phase factors $\nu_{d+1}$ to $(d+1)$-simplices.
4. Form commuting projector Hamiltonians using local unitary circuits generated from the supercohomology data.

For instance, in 2+1D, wavefunctions and Hamiltonians can be constructed by triangulating the manifold and using the data $(n, \nu)$ to define weightings to simplex configurations, where the $\mathbb{Z}_2$ data determines Majorana chains across domain walls and the $U(1)$ data fixes amplitudes (Gu et al., 2012, Ellison et al., 2018). Fermion parity is ungauged via bosonization dualities, entangling the fermionic and bosonic sectors in the auxiliary construction (Ellison et al., 2018).

A similar but higher-dimensional prescription builds the fixed-point ground states in 3+1D (Chen et al., 2020), employing a finite-depth quantum circuit $U^{(\rho,\nu)}$ acting on an atomic insulator state to prepare the SPT phase characterized by $(\rho, \nu)$. The circuit composition rules directly encode the stacking law for SPT phases (Chen et al., 2020).

4. Supercohomology Classification and Stacking Laws

The classification of fermionic SPTs by group supercohomology yields an abelian group structure, describing stacking of SPT phases via supercohomology cocycles. In 2+1D, the data $(n, \nu)$ stack as

$$(n, \nu) * (n', \nu') = (n+n',\, \nu \nu' (-1)^{n \cup_1 n'}),$$

where $\cup_1$ is the first higher cup product (Ellison et al., 2018). Equivalent relations hold in higher dimensions using higher cup-$i$ products ($\cup_2$, etc.) (Chen et al., 2020).

The group supercohomology classes fit into an exact sequence: $$0 \to H^{d+1}(G_b, U(1)) \to SH^{d+1}(G_b) \to H^d(G_b, \mathbb{Z}_2) \to 0$$ with extension classes governed by Steenrod operations (e.g., $Sq^2$) (Gu et al., 2012, Chen et al., 2020). This correctly captures both “bosonic descent” SPTs and intrinsically fermionic SPTs.

Generalizations to twisted cases and to arbitrary spatial dimension yield similar abelian extensions, with higher supercohomology groups $SH^n(BG, s, \omega)$ classified via spectral sequence machinery such as the Atiyah–Hirzebruch and an Adams-type “hastened” spectral sequence (Debray et al., 28 Oct 2025).

5. Physical Interpretation and Applications

Supercohomology theory supplies both a classification and a constructive approach for SPT phases of fermions in condensed matter and topological phases of quantum field theory:

• SPT Classification: It fully captures a large subclass of interacting fermionic SPTs, including phases with nontrivial symmetry action on fermion parity fluxes (fractionalization), anomalous boundary symmetries, and short-range entangled ground states not obtainable from free-fermion or bosonic cohomological constructions (Gu et al., 2012, Ellison et al., 2018).
• Lattice Models: Exactly solvable commuting-projector Hamiltonians are explicitly constructed using the cocycle data on triangulations, showing how the SPT phase emerges via local symmetric unitaries (Ellison et al., 2018, Chen et al., 2020).
• Boundary Anomalies: Every nontrivial supercohomology bulk phase has a protected boundary anomaly; the induced symmetry on the edge is necessarily non-onsite (i.e., Wess–Zumino–Witten-like), forbidding a symmetric gapped boundary unless topological order is present (Gu et al., 2012, Chen et al., 2020).
• Anomalies and TQFTs: In 3+1D, the extension to topological quantum field theories (TQFTs) with specified anomalies proceeds systematically via generalized group supercohomology, cobordism, and higher-categorical machinery (Debray et al., 28 Oct 2025).

Examples include:

• 2D SPTs for $G_f = \mathbb{Z}_2 \times \mathbb{Z}_2^f$: With four supercohomology classes $\mathbb{Z}_4$, two intrinsically fermionic, each yielding an edge mode of central charge $c=1$ (Gu et al., 2012, Ellison et al., 2018).
• 3D SPTs for $G_f = \mathbb{Z}_2^T \times \mathbb{Z}_2^f$: An intrinsically fermionic SPT phase not realized in free-fermion or bosonic frameworks, whose surface symmetry squares to $(-1)^F$ (Gu et al., 2012).

6. Extensions, Generalizations, and Connections

Recent work situates group supercohomology within a broader context of generalized cohomology, cobordism, and categorical TQFTs:

• Generalized Supercohomology: The cochain model may be cast in terms of AHSS and Adams-type spectral sequences, and group supercohomology is seen as dual to a Postnikov truncation of connective $KO$-theory, capturing the low-dimensional part of the invertible spin-TQFT spectrum (Debray et al., 28 Oct 2025).
• Twisted and Categorical Extensions: Incorporating twisted coefficients and anti-unitary (e.g., time-reversal) symmetries, and relating to fusion 2-categories, further extends the reach of supercohomology to a wider class of anomalous TQFTs (Debray et al., 28 Oct 2025).
• Beyond Supercohomology: Certain higher-dimensional fermionic SPTs (e.g., requiring “Kitaev chain” or $p+ip$ layer decorations) are not captured by group supercohomology alone, necessitating more general spin cobordism invariants and novel state-sum models (Chen et al., 2020, Debray et al., 28 Oct 2025).

In all dimensions, the key feature is the explicit realization of the ground state via symmetric finite-depth circuits, with the supercohomology data directly dictating the entanglement and symmetry structure (Gu et al., 2012, Chen et al., 2020).

7. Open Problems and Research Directions

Several outstanding problems remain:

• Full classification: Determining the complete SPT classification, including “beyond supercohomology” phases in all dimensions, especially in the presence of time-reversal or crystalline symmetries (Chen et al., 2020, Wang et al., 2017).
• Explicit Computation: Efficient algorithms for cohomology and spectral sequence computations for arbitrary finite groups and generalizations (Debray et al., 28 Oct 2025).
• Support Varieties and Module Theory: In the algebraic context, extending the support variety theory for finite supergroup schemes, and computing full cohomology rings for classical supergroups and quantum analogues (Drupieski, 2014).
• Categorical and Cobordism Links: Further elucidation of the interplay between supercohomology TQFTs, invertible spin-cobordism phases, and fusion 2-category structures (Debray et al., 28 Oct 2025).
• Physical realization and detection: Experimental and numerical probes to distinguish fermionic SPTs classified by supercohomology from bosonic and free-fermion counterparts.

Group supercohomology thus serves as a foundational tool in the understanding and classification of fermionic quantum phases of matter, combining the algebraic structure of group extensions, cohomology operations, and explicit lattice model constructions (Gu et al., 2012, Wang et al., 2017, Ellison et al., 2018, Chen et al., 2020, Debray et al., 28 Oct 2025).