3D Gauged Majorana CFT

• 3D Gauged Majorana CFT is a three-dimensional quantum field theory featuring gauge-coupled Majorana fermions with emergent conformal symmetry.
• It unifies dual descriptions from bosonic and fermionic constructions through orbifold and regularization techniques to precisely resolve operator spectra and topological defects.
• The theory underpins topological phase transitions and holographic dualities, providing insights into criticality, symmetry breaking, and non-Abelian statistics in quantum matter.

A 3D Gauged Majorana Conformal Field Theory (CFT) is a three-dimensional quantum field theory whose low-energy sector is governed by critical, gauge-coupled Majorana fermions and exhibits emergent conformal symmetry. These theories sit at the intersection of non-perturbative field theory, topological quantum matter, and algebraic conformal methods, offering a unified language for criticality, topological orders, and symmetry-enriched phases in systems ranging from topological quantum Hall effects to supergravity backgrounds. Central to their analysis are dualities inherited from the two-dimensional Ising CFT, orbifold and gauging constructions, and novel regularization techniques such as the fuzzy sphere, which allow for the exact resolution of operator spectra and topological defects in genuinely fermionic settings.

1. Bosonic–Fermionic Duality and Correlators

The foundational insight for 3D Gauged Majorana CFTs is the exact duality between bosonic (vertex operator) and fermionic (Pfaffian) constructions of correlators exhibited in the 2D Ising CFT (Ardonne et al., 2010). For primary fields such as the spin field σ and the Majorana fermion ψ, multi-point correlators admit both bosonized and fermionic representations, leading to powerful functional identities: $$[\mathrm{Pf}(1/(z_i-z_j))]^2 = \mathrm{Hf}(1/(z_i-z_j)^2)$$ This relation, and extensions thereof, generalize to more complicated settings where, after gauging (for instance, a ℤ₂ symmetry), physical correlators of Majorana degrees of freedom can be equivalently computed using bosonic building blocks (e.g., vertex operators of underlying chiral bosons) or explicit Majorana Pfaffians. The su(2)₂ WZW model, as the product of Ising and free boson sectors, embeds the non-Abelian structure of Majorana topological properties, suggesting that 3D gauged variants inherit similar algebraic structures and fusion rules. The dual correlator structures provide powerful tools for calculating topological properties such as braiding, modular data, and non-Abelian statistics in 3D settings.

2. Orbifold Constructions and Gauging Procedures

Interpreting a 3D CFT as a boundary or edge theory of a (3+1)D or (2+1)D bulk topological phase, gauging is achieved via orbifold constructions that project onto symmetry-invariant sectors and sum over twisted boundary conditions (Chen et al., 2017). The sector partition functions,

$$Z(g, h) = \operatorname{Tr}_{\text{twisted by } h}[g\, e^{2\pi i \tau_1 P_0 - 2\pi \tau_2 H_0}]$$

incorporate the effect of discrete symmetry fluxes introduced by gauging, with modular invariants encoding fusion and braiding data of emergent anyonic or Majorana excitations. The Verlinde formula and modular S, T matrices, derived from characters built out of such twisted partition functions, yield the full fusion algebra of these excitations. In fermionic orbifold settings, care must be taken to sum over spin structures, especially to account for Ramond and Neveu–Schwarz sectors and the action of discrete gauge symmetries on fermion parity.

The universality of this construction extends to three-dimensional gauged Majorana CFTs, where continuous or discrete gauge symmetries (such as a ℤ₂ fermion parity gauge field) are incorporated. This produces nontrivial modular data and topological defect structures (such as twist defects), robustly capturing the non-Abelian exchange statistics characteristic of Majorana zero modes in 3D.

3. Fuzzy Sphere Regularization and Operator Spectrum

A significant technical advance is the application of the fuzzy sphere regularization to fermionic theories in 3D (Voinea et al., 9 Sep 2025, Zhou et al., 9 Sep 2025). By projecting the system onto a finite-dimensional Hilbert space with exact SO(3) symmetry, the fuzzy sphere allows the numerical extraction of conformal operator spectra in both fermionic and bosonic sectors. For instance, in quantum Hall bilayers at the Halperin–Pfaffian transition, the fuzzy sphere resolves energy levels into angular momentum multiplets:

• ℤ₂-even (integer spin) sector: primaries such as $\bar\psi\psi$ (Δ=2, L=0) and the stress-energy tensor $T^{\mu\nu}$ (Δ=3, L=2), with descendants up to higher angular momentum.
• ℤ₂-odd (half-integer spin) sector: the Majorana field $\psi$ (Δ=1, L=1/2) and its derivatives as descendants.

The closing of the neutral fermion gap marks the critical point, and the operator towers match precisely the expectations from a free, gauged Majorana theory,

$$\mathcal{L} = \bar\psi (i\gamma^\mu \partial_\mu)\psi$$

The method also enables studies of correlation functions and topological (Wilson line) defect operators, confirming conformal invariance through scaling of two-point functions and identification of endpoint operators associated with nonlocal (e.g., Ising σ) fields in the gauged theory.

4. Phase Structure and Topological Transitions

3D gauged Majorana CFTs arise naturally as critical points separating topologically distinct gapped phases (Zhou et al., 9 Sep 2025). In boson–fermion mixtures on the fuzzy sphere, the phase diagram supports three phases:

• Fermionic integer quantum Hall (fIQH), characterized by filled fermion orbitals,
• f-wave chiral topological superconductor (TSC, ν_K=3),
• Bosonic Pfaffian (bPf), a non-Abelian topological order.

Continuous phase transitions between these are governed by:

• a free Majorana CFT at the fIQH–TSC boundary,
• a gauged Ising CFT (“Ising*”, with the Z₂ symmetry coupled dynamically) at the TSC–bPf boundary.

Topological defects, such as Wilson lines corresponding to gauge flux insertions (or odd particle number), are precisely resolved as nonlocal operators with scaling dimensions matching those of twist fields in the Ising CFT. The appearance of such endpoints in the spectrum validates the identification of the critical points with gauged Majorana and gauged Ising universality classes.

5. Holographic Duality and Conformal Manifolds

Within the context of the AdS/CFT correspondence, 3D gauged Majorana CFTs are provided by the boundary duals of AdS₄ backgrounds in four-dimensional maximal gauged supergravity (Bobev et al., 2021) and their string/M-theory uplifts. Exactly marginal couplings correspond to geometric moduli (axionic scalars χ in the bulk), producing compact conformal manifolds (e.g., T² topology) in the field theory. The operator spectrum is organized in multiplets, with scaling dimensions computed via the holographic dictionary: $$\Delta = \tfrac{3}{2} \pm \sqrt{\tfrac{9}{4} + m^2L^2}$$ Marginal operators reside in L₂[2,0] multiplets, and flavor symmetry enhancements at special loci are reflected in recombination of long into short multiplets. Supersymmetric and non-supersymmetric RG flows in truncations of D=3, N=(2,0) gauged supergravity (with scalar target space SU(1,1)/U(1)) further illuminate the structure of irreducible boundary conformal field theories and, via exact dynamical systems, yield nontrivial behaviors such as bouncing RG flows and multitrace deformations (Arkhipova et al., 18 Feb 2024).

6. Interacting Lattice Gauge Realizations and Topological Order

Majorana lattice gauge theories constructed purely from interacting Majorana fermions realize the coexistence of spontaneous symmetry breaking and gauge topological order, serving as exact lattice analogs for continuum 3D gauged Majorana theories (Miao, 2022). The ground state exhibits both Z₂ symmetry breaking (in the matter sector, with intertwined antiferromagnetic and η-pairing orders) and a Z₂ quantum spin liquid (topologically ordered Wen plaquette model) in the gauge sector. Such models highlight the infrared decoupling of order and topological sectors, offering a platform for potential experimental realizations of quantum phases with combined conventional and topological quantum order.

7. Generalizations, Modular Data, and Algebraic Correspondence

Topologically twisted 3D N=4 SCFTs of rank zero, when analyzed through their partition functions and half-indices, reproduce modular data (characters, S and T matrices) for 2D minimal models, including those relevant for Majorana descriptions (Gang et al., 2023). Fermionic sum representations for minimal model characters arise naturally from gauge theory calculations, underscoring a deep algebraic correspondence between 3D gauged Majorana CFTs and 2D non-unitary rational CFTs. This structure enables richer classification and analytic control over modular invariants, boundary degrees of freedom, and fusion/braiding properties in the presence of dynamical gauge fields or topological twists.

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A 3D Gauged Majorana Conformal Field Theory thus constitutes a mathematically and physically robust framework for the paper of criticality in gauge-coupled Majorana systems, leveraging dualities between bosonic and fermionic formalisms, orbifold/gauging procedures, state–operator correspondence in curved regularizations, and holographic as well as algebraic tools. Applications span from the microscopic modeling of topological phase transitions in quantum Hall bilayers to the exploration of new universality classes in interacting lattice gauge theories and the nonperturbative landscape of supersymmetric and non-supersymmetric quantum field theories.