Fermionic Gauging in Topological Phases

• Fermionic gauging is the process of promoting fermion parity (Z2^f) to a local gauge redundancy in systems ranging from superconductors to lattice models.
• It utilizes methods such as Majorana-based implementations, dual bosonization, tensor networks, and categorical constructions to reveal anomalous constraints and topological features.
• This approach converts inherent fermionic data into bosonic or spin-dependent topological theories, as demonstrated by the emergence of Z16 structures and gravito-magnetic anomalies.

Searching arXiv for the papers on arXiv to ground the article in current arXiv sources. arXiv Search Query: "id:(Berg et al., 20 Apr 2026) OR id:(Catterall et al., 2024) OR id:(Su, 23 Oct 2025) OR id:(Ando, 2024) OR id:(Borla et al., 2020) OR id:(Bruillard et al., 2016) OR id:(Bhardwaj et al., 2016) OR id:(Guo et al., 2018) OR id:(Wang et al., 2023) OR id:(Ambrosino et al., 2024) OR id:(Shirley, 2020) OR id:(Roose et al., 2024) OR id:(Sun et al., 23 Jun 2026) OR id:(Smith et al., 2024) OR id:(McKeon, 2012) OR id:(McKeon, 2020) OR id:(Katsura et al., 2022)" Fermionic gauging denotes a family of procedures in which a fermionic symmetry, most commonly fermion parity $\mathbb Z_2^f$ generated by $(-1)^F$, is promoted to gauge redundancy or incorporated into a gauge-theoretic, topological, categorical, or lattice construction. In condensed-matter settings it includes gauging fermion parity in superconductors and electronic matter, where the dynamical gauge field becomes a spin$_c$ connection and the low-energy theory can reduce to BF theory with a residual gravito-magnetic anomaly (Berg et al., 20 Apr 2026). In lattice systems it includes Majorana-based implementations of $\mathbb Z_2$ gauge structure, exact Gauss-law constraints, and bosonization maps between fermionic and bosonic codes (Su, 23 Oct 2025). In topological field theory and tensor-category language it refers to passing from fermionic or super-modular data to a bosonic “shadow” theory or to a spin modular extension, often described as gauging fermion parity (Bhardwaj et al., 2016). Other usages concern gauging fermionic higher-form, subsystem, or shift symmetries, where anomalies and Stückelberg-like mechanisms play a central role (Wang et al., 2023).

1. Terminological scope and basic structures

Across current arXiv usage, “fermionic gauging” does not refer to a single universal construction. The phrase is used for at least four closely related operations: gauging fermion parity in electronic systems; implementing $\mathbb Z_2^f$ gauging on lattices by auxiliary Majoranas or qubits; promoting fermionic shift symmetries to local symmetries with fermionic gauge fields; and passing from fermionic phases to bosonic shadows or spin-TQFTs by gauging parity or summing over appropriate structures (Berg et al., 20 Apr 2026).

Context	Symmetry or structure gauged	Typical outcome
Electronic matter	$(-1)^F$ in a dynamical $U(1)$ system	spin$_c$ connection, BF theory, gravito-magnetic anomaly
Lattice models	$\mathbb Z_2^f$ or discrete fermionic shifts	Gauss law, Majorana stabilizers, toric-code-like theories
Generalized symmetries	fermionic shift or fermionic higher-form symmetry	fermionic gauge fields, anomaly obstruction or Stückelberg mass generation
Spin-TQFT and category theory	fermion parity or fermionic 1-form symmetry	bosonic shadow, spin modular extension, gauged spin-TQFT

A recurring structural feature is that gauging fermionic data does not simply reproduce ordinary bosonic gauging. In electronic systems the gauge field must satisfy the modified flux quantization

$$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$

so $(-1)^F$0 is a spin$(-1)^F$1 connection (Berg et al., 20 Apr 2026). In 2+1-dimensional spin-TQFT constructions, gauging a fermionic 1-form symmetry is obstructed unless a spin-structure-dependent counterterm is included (Bhardwaj et al., 2016). In 1+1 dimensions, gauging $(-1)^F$2 can differ from summing over spin structures, and the result can remain fermionic rather than bosonic when the gravitational anomaly vanishes only mod $(-1)^F$3 (Smith et al., 2024).

2. Fermion-parity gauging in superconductors and electronic systems

For electronic matter, the starting point is that the microscopic local spacetime symmetry group is

$(-1)^F$4

with $(-1)^F$5 identified with the $(-1)^F$6-torsion in $(-1)^F$7 (Berg et al., 20 Apr 2026). As a consequence, the dynamical electromagnetic gauge field is not an ordinary $(-1)^F$8 connection but a spin$(-1)^F$9 connection. One Lagrangian implementation introduces a $_c$0-form Lagrange multiplier $_c$1 and writes

$_c$2

whose equation of motion imposes the spin$_c$3 condition (Berg et al., 20 Apr 2026).

In the s-wave Higgs phase, the relativistic Abelian-Higgs model with pairing charge $_c$4 reduces at very low energy to BF theory. Dualizing the compact phase and integrating out the massive mode gives

$_c$5

Here $_c$6 is the $_c$7 gauge field, Wilson loops $_c$8 carry the $_c$9-fold charge, and the dual $\mathbb Z_2$0-form field $\mathbb Z_2$1 couples to vortex surfaces $\mathbb Z_2$2. Their correlators obey the Wilson–’t Hooft algebra

$\mathbb Z_2$3

which exhibits the topological order of the Higgs phase (Berg et al., 20 Apr 2026).

After gauging $\mathbb Z_2$4, the low-energy theory is bosonic in the sense emphasized in the superconductors analysis, yet it retains a gravito-magnetic anomaly that records its fermionic origin. With magnetic $\mathbb Z_2$5-form background $\mathbb Z_2$6, the relevant inflow term is

$\mathbb Z_2$7

and the same anomaly appears from the Gaiotto–Kapustin–Thorngren bosonization viewpoint as

$\mathbb Z_2$8

upon identifying $\mathbb Z_2$9 (Berg et al., 20 Apr 2026). The paper states that this anomaly characterizes gauged electronic matter in three and four spacetime dimensions, holds beyond the validity of the Higgs model, and forbids trivial massive phases at low energy (Berg et al., 20 Apr 2026).

The same framework extends to more general pairing patterns. For pairing charge $\mathbb Z_2^f$0, Higgsing gives $\mathbb Z_2^f$1 and the low-energy theory becomes a level-$\mathbb Z_2^f$2 twisted BF theory,

$\mathbb Z_2^f$3

In 2+1 dimensions, any $\mathbb Z_2^f$4 pairing with even $\mathbb Z_2^f$5 reduces to a $\mathbb Z_2^f$6 gauge theory that already contains a fermionic anyon; for $\mathbb Z_2^f$7 the example is the toric code, and for general $\mathbb Z_2^f$8 it is level-$\mathbb Z_2^f$9 BF theory (Berg et al., 20 Apr 2026). The explicit conclusion drawn there is that no purely trivial gapped phase is possible once $(-1)^F$0 is gauged.

3. Lattice realizations, dualities, and stabilizer formulations

In one spatial dimension, gauging the fermion-parity symmetry of the Kitaev chain introduces a $(-1)^F$1 gauge field $(-1)^F$2 on each link and imposes the local Gauss law

$(-1)^F$3

Minimal coupling inserts $(-1)^F$4 in every hopping or pairing term, and in the gauge-invariant subspace the model maps locally to an Ising chain in both transverse and longitudinal field,

$(-1)^F$5

At $(-1)^F$6, the deconfined regime has gauge-invariant fermionic domain-wall operators, whereas the Higgs regime is described as a non-trivial fermionic SPT phase protected by the magnetic symmetry together with the surviving global fermion parity acting only at the edges (Borla et al., 2020). The same work studies “gentle gauging” by an energetic penalty $(-1)^F$7, obtaining a phase diagram with four regions separated by the lines $(-1)^F$8 and $(-1)^F$9, and an intrinsically gapless SPT phase with a $U(1)$0 Luttinger liquid in the particle-conserving limit (Borla et al., 2020).

A two-dimensional lattice implementation of fermion-parity gauging is given by inserting a pair of Majorana modes on each edge and imposing a generalized Gauss law

$U(1)$1

together with the flatness condition

$U(1)$2

After disentangling, the gauge-invariant Hamiltonian becomes a commuting-projector Majorana stabilizer code,

$U(1)$3

with $U(1)$4 and $U(1)$5 (Su, 23 Oct 2025). The paper identifies this as a “fermionic toric code” whose ground-state manifold is four-fold degenerate on the torus. It also proves a linear-depth local unitary equivalence between this fermionic code and the conventional bosonic toric code, while emphasizing that flux statistics are redistributed in a direction-dependent way: $U(1)$6-fluxes become fermions when braided in the vertical direction but remain bosons in the horizontal direction (Su, 23 Oct 2025).

A broader stabilizer framework is provided by Majorana-Pauli stabilizer codes. There, gauging $U(1)$7 introduces an auxiliary qubit on each edge and imposes a vertex Gauss law $U(1)$8; after restriction to the gauge-invariant subspace one can fix the gauge and recover the bosonic shadow theory (Sun et al., 23 Jun 2026). The paper gives an exact bosonization map in which, for each edge $U(1)$9,

$_c$0

and generalizes the construction to $_c$1 symmetries for even $_c$2 through a fermionic clock-shift algebra (Sun et al., 23 Jun 2026). It places the fermionic toric code in a duality web generated by anyon condensation and by gauging bosonic or fermion-parity symmetries.

Tensor-network implementations fit the same general pattern. Gauged Gaussian fermionic PEPS are obtained by replacing each local fiducial tensor by a gauge-covariant version, integrating over link variables with Haar measure, and exploiting the fact that for each fixed gauge configuration the fermionic sector remains Gaussian (Roose et al., 2024). The resulting norm defines a positive probability density on gauge fields, so gauge-invariant observables can be evaluated by Monte Carlo sampling without a sign problem (Roose et al., 2024).

4. Effective gauging, anomaly matching, and refermionization

A distinct but related construction enforces Gauss law only energetically. For a bosonic lattice theory with a finite symmetry $_c$3 fitting into a central extension

$_c$4

the effectively gauged or effectively fermionized Hamiltonian is

$_c$5

with $_c$6 (Ando, 2024). For finite $_c$7, the full Hilbert space still carries the original $_c$8 symmetry, and one additionally has the dual abelian symmetry $_c$9; in the fermionic case the UV symmetry is $\mathbb Z_2^f$0 (Ando, 2024). In the low-energy subspace, the penalty term enforces the strict Gauss law and recovers the gauged model.

The same work gives a general topological response formula. In the bosonic setting,

$\mathbb Z_2^f$1

while in the fermionic case

$\mathbb Z_2^f$2

with $\mathbb Z_2^f$3 a quadratic refinement or generalized Arf phase (Ando, 2024). The same paper derives ’t Hooft anomalies in terms of a $\mathbb Z_2^f$4-cocycle in the bosonic case and a Gu–Wen pair $\mathbb Z_2^f$5 in the fermionic case, and it constructs both gapped and gapless SPT examples, including a fermionic gapless SPT with a non-trivial $\mathbb Z_2^f$6 Gu–Wen anomaly (Ando, 2024).

In 1+1 dimensions, “Backfiring Bosonisation” distinguishes summing over spin structures from gauging $\mathbb Z_2^f$7. Bosonization exists only when the gravitational anomaly vanishes mod $\mathbb Z_2^f$8, whereas gauging $\mathbb Z_2^f$9 is unobstructed when the relevant anomaly vanishes mod $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$0 (Smith et al., 2024). When the anomaly is $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$1, summing over spin structures is ill-defined but gauging $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$2 remains allowed and yields a fermionic theory rather than a bosonic one (Smith et al., 2024). This is the regime the paper calls refermionization. The torus formulas show explicitly that the spin-structure label need not disappear after gauging, and the resulting duality web is organized by an $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$3 structure (Smith et al., 2024).

These constructions make anomaly matching central rather than incidental. In the superconducting setting the anomaly forbids a trivial massive infrared phase after gauging fermion parity (Berg et al., 20 Apr 2026); in effective gauging, emergent anomalies protect gapped or intrinsically gapless SPT phases (Ando, 2024); and in 1+1 dimensions the anomaly class determines whether gauging parity produces a bosonic or fermionic theory (Smith et al., 2024).

5. Fermionic higher-form, subsystem, and shift gauging

Fermionic higher-form symmetries are generated by topological operators with fermionic parameter acting on fermionic extended objects. Their gauging introduces a fermionic $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$4-form gauge field $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$5 with transformations

$$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$6

and the naive minimal coupling replaces $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$7 by $$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$8 (Wang et al., 2023). In simple free theories this fails because the current is not gauge invariant, producing a non-zero variation and a ’t Hooft anomaly. The paper shows that anomaly-free gauging is possible in derivative-only theories or in BF-type mixed systems where one gauges only one of two fermionic higher-form symmetries (Wang et al., 2023).

A related continuum construction starts from a free massless Dirac field with global fermionic shift

$$\int_\Sigma \frac{da}{2\pi}+\frac12\,w_2(TX)\in\mathbb Z,$$9

and gauges it with a vector-spinor field $(-1)^F$00 (Ambrosino et al., 2024). Gauge invariance requires a modified transformation law,

$(-1)^F$01

and leads to a unique two-derivative action containing both a Rarita–Schwinger kinetic term and gauge-invariant mass terms (Ambrosino et al., 2024). In this construction the Dirac spinor plays the role of a Stückelberg field: in unitary gauge it is removed, leaving a massive Rarita–Schwinger field (Ambrosino et al., 2024). The same paper also identifies magnetic fermionic symmetries, disorder operators, and a dual description in four dimensions using a fermionic $(-1)^F$02-form.

Subsystem fermion-parity symmetries supply another major arena. Gauging planar or fractal subsystem fermion parity in three spatial dimensions yields exactly solvable spin models with fractonic order and emergent fermionic gauge theory (Shirley, 2020). The construction introduces a $(-1)^F$03 gauge qubit for each minimal-coupling operator and imposes

$(-1)^F$04

After a local unitary circuit, the fermions freeze and one obtains a bosonic stabilizer code whose gauge charges inherit fermionic statistics from the original fermion parity (Shirley, 2020). Examples include a fermionic $(-1)^F$05-foliated lineon code, a fermionic X-cube model, and a fermionic Fibonacci-prism model; the paper states that lineons, fractons, and fractal excitations can carry emergent fermionic statistics (Shirley, 2020).

By contrast, gauging fermionic subsystem shift symmetry can be obstructed. In a supersymmetric fracton-like model with row-by-row fermionic shifts, introducing fermionic gauge fields $(-1)^F$06 and $(-1)^F$07 and covariant derivatives $(-1)^F$08, $(-1)^F$09 leaves a pure anomaly term

$(-1)^F$10

which cannot be canceled by any local counterterm (Katsura et al., 2022). The authors therefore do not obtain a fully consistent gauged Hamiltonian, but they identify a candidate fermionic defect line

$(-1)^F$11

whose absence of an $(-1)^F$12-component implies immobility in the $(-1)^F$13-direction (Katsura et al., 2022).

On Euclidean hypercubic lattices, staggered-fermion shift symmetries can be partially gauged by promoting the $(-1)^F$14 shift parameters to local fields and introducing a hierarchy of $(-1)^F$15 higher-form gauge fields on links, plaquettes, cubes, and hypercubes (Catterall et al., 2024). The resulting dressed kinetic term remains exactly invariant under local single, double, triple, and quadruple shifts (Catterall et al., 2024). The paper proposes that at a suitable continuous phase transition the $(-1)^F$16 higher-form symmetries might enlarge into a continuous $(-1)^F$17 gauge symmetry, although it also states that this has yet to be demonstrated (Catterall et al., 2024).

6. Local fermionic gauge symmetry in continuum field theories

Some papers use “fermionic gauging” in a stricter field-theoretic sense: the gauge symmetry itself has fermionic generators and acts locally on bosonic and fermionic fields. One example is a $(-1)^F$18-dimensional model with a non-Abelian gauge field with topological action coupled to a spin-$(-1)^F$19 Majorana field. Canonical analysis yields bosonic and fermionic first-class constraints, and the Dirac brackets of the remaining first-class constraints satisfy

$(-1)^F$20

so the fermionic constraints close onto the bosonic Gauss law (McKeon, 2012). The corresponding gauge transformations are

$(-1)^F$21

and the action is invariant without auxiliary fields (McKeon, 2012).

A related $(-1)^F$22-dimensional model constructs a gauge theory from a superalgebra with two bosonic generators $(-1)^F$23 and one fermionic generator $(-1)^F$24 (McKeon, 2020). The gauge connection

$(-1)^F$25

has curvature components that mix bosonic and fermionic fields, and the gauge transformations contain bosonic parameters $(-1)^F$26 and a fermionic Majorana parameter $(-1)^F$27 (McKeon, 2020). The closure of two transformations produces new parameters $(-1)^F$28 of the same form, and the paper stresses that the algebra closes off shell without auxiliary fields (McKeon, 2020).

These models are not gauging fermion parity. Instead, they realize local fermionic gauge generators directly in the sense of the Dirac constraint formalism. This suggests a second major meaning of fermionic gauging: not only gauging a symmetry of fermions, but gauging by means of fermionic gauge transformations.

7. Categorical, state-sum, and cobordism formulations

In tensor-category language, fermionic gauging is the passage from a super-modular category $(-1)^F$29 with transparent fermion $(-1)^F$30 to a minimal modular extension or spin modular closure $(-1)^F$31 (Bruillard et al., 2016). A super-modular category has Müger center

$(-1)^F$32

with $(-1)^F$33, $(-1)^F$34, and $(-1)^F$35 (Bruillard et al., 2016). Gauging fermion parity means adjoining the defect or twisted sector so that $(-1)^F$36 becomes modular and $(-1)^F$37. The paper formulates the $(-1)^F$38-fold way conjecture: every super-modular category admits exactly $(-1)^F$39 inequivalent minimal modular extensions, reflecting the $(-1)^F$40 structure of invertible spin TQFTs (Bruillard et al., 2016).

In state-sum and string-net constructions, a $(-1)^F$41-dimensional fermionic phase is represented by a bosonic “shadow” theory containing an Abelian anyon $(-1)^F$42 of order $(-1)^F$43 and topological spin $(-1)^F$44 (Bhardwaj et al., 2016). Coupling the shadow to a background $(-1)^F$45 produces a partition function with the universal fermionic $(-1)^F$46-form anomaly

$(-1)^F$47

The spin theory is recovered by a fermionic-anyon-condensation formula

$(-1)^F$48

where the Gu–Wen Grassmann integral $(-1)^F$49 cancels the anomaly in a spin-structure-dependent way (Bhardwaj et al., 2016). In Hamiltonian form the same procedure dresses $(-1)^F$50-form symmetry generators by Majorana bilinears to obtain strictly commuting projectors (Bhardwaj et al., 2016).

Cobordism gives a global formulation. For a finite group $(-1)^F$51, reflection-positive invertible $(-1)^F$52-equivariant spin-TQFTs are classified by

$(-1)^F$53

and dynamical gauging sums the invertible action over all isomorphism classes of $(-1)^F$54-bundles,

$(-1)^F$55

This produces generally noninvertible spin-TQFTs and provides explicit partition functions and ’t Hooft anomalies in dimensions $(-1)^F$56, $(-1)^F$57, and others (Guo et al., 2018). The same framework extends to unoriented pin$(-1)^F$58 theories and to crystalline symmetries through modified tangential structures (Guo et al., 2018).

Viewed together, the categorical and cobordism approaches formalize a persistent theme already visible in electronic and lattice constructions: fermionic gauging converts local fermionic data into bosonic or spin-dependent topological data, but the conversion is controlled by anomalies, spin structure, and the existence of a distinguished fermionic line or parity symmetry.