3D Gauged Majorana CFT

• The paper establishes a microscopic realization of an interacting 3D CFT with emergent N=1 supersymmetry using a Gross–Neveu–Yukawa model coupled with a gauged Majorana fermion.
• It employs fuzzy sphere regularization to discretize spatial S² while preserving SO(3) symmetry, enabling precise extraction of operator scaling dimensions and a clear operator-state correspondence.
• The findings reveal distinct supermultiplet structures and continuous RG flow of the operator spectrum, overcoming traditional challenges in lattice and perturbative methods.

The three-dimensional (3D) gauged Majorana conformal field theory (CFT) constitutes a nontrivial example of an interacting, strongly coupled CFT with emergent $\mathcal{N}=1$ supersymmetry. Its microscopic realization involves coupling a 3D Ising CFT to a gauged Majorana fermion via a Yukawa interaction, capturing the Gross–Neveu–Yukawa universality class at criticality. The model exhibits the full structure of superconformal multiplets, including the signatures of spacetime supersymmetry such as precise relations between the scaling dimensions of bosonic and fermionic operators. The fuzzy sphere regularization provides a non-perturbative and numerically exact route to study these phenomena, enabling the extraction of operator scaling dimensions and the renormalization-group (RG) flow at the operator level (Tang et al., 31 Dec 2025).

1. Microscopic Formulation and Hamiltonian Structure

The continuum action of the 3D gauged Majorana CFT is given by the Gross–Neveu–Yukawa (GNY) Lagrangian: $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$ where $\psi$ is a two-component Majorana fermion, $\phi$ is a real $\mathbb{Z}_2$-odd scalar field, $g$ is the Yukawa coupling, and $u$ is the self-interaction of $\phi$.

On the fuzzy sphere, the full model Hamiltonian combines separate bosonic (Ising) and fermionic (gauged Majorana) sectors with a short-range Yukawa coupling: $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$ Here, $H_{\rm Ising}$ and $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$0 are fuzzy-sphere Hamiltonians for the concentric spheres with monopole charge $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$1; $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$2 and $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$3 denote bosonic and fermionic density operators; couplings $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$4 and $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$5 are expanded in Haldane pseudopotentials. The parameter $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$6 tunes the relative "speed of light" for the noninteracting CFTs, ensuring Lorentz invariance at criticality. Setting $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$7 yields the decoupled Ising $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$8 Majorana theory, while tuning to $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$9 brings the system to the GNY ($\psi$0 supersymmetric) critical point (Tang et al., 31 Dec 2025).

2. Fuzzy-Sphere Regularization and Operator-State Correspondence

The fuzzy sphere regularization discretizes the spatial $\psi$1 while preserving exact $\psi$2 symmetry. In this approach:

• The presence of a monopole with charge $\psi$3 results in a lowest Landau level (LLL) of dimension $\psi$4, corresponding to the spin-$\psi$5 representation of $\psi$6. The coordinates $\psi$7 are replaced by rescaled angular-momentum operators satisfying $\psi$8, maintaining the sphere's geometry through $\psi$9.
• Scalar and fermionic fields become $\phi$0 matrices acting on the LLL.
• Through radial quantization, energy eigenstates of the Hamiltonian map directly onto local scaling operators of the CFT. The scaling dimension is extracted as

$\phi$1

with $\phi$2 the speed of light and $\phi$3 the ground-state energy. The normalization is set such that the stress tensor has $\phi$4, or equivalently through the integer spacing of conformal towers predicted by the algebra (Tang et al., 31 Dec 2025).

3. Scaling Dimensions and Operator Spectrum at Fixed Points

The operator spectrum is resolved both at the decoupled and the interacting fixed points:

• Decoupled theory ($\phi$5):
  • Ising primaries: $\phi$6, $\phi$7
  • Free Majorana: $\phi$8, $\phi$9
• Interacting SCFT ($\mathbb{Z}_2$0), using fuzzy-sphere exact diagonalization:
  • Leading supermultiplet: $\mathbb{Z}_2$1, $\mathbb{Z}_2$2, $\mathbb{Z}_2$3
  • Subleading supermultiplet: $\mathbb{Z}_2$4, $\mathbb{Z}_2$5, $\mathbb{Z}_2$6
  • Supercurrent: $\mathbb{Z}_2$7 at $\mathbb{Z}_2$8
  • Stress tensor: $\mathbb{Z}_2$9 (by normalization)
  • Additional spin-2 operator: $g$0

The hallmark of supersymmetry is manifest in the spacing $g$1, $g$2, and in the integer spacing of the corresponding conformal towers (Tang et al., 31 Dec 2025).

4. Multiplet Structure and Supersymmetry Relations

Three main supermultiplets are established, connected by the emergent supercharge $g$3 with scaling dimension $g$4:

1. $g$5, with $g$6 and $g$7.
2. Higher multiplet $g$8 at larger scaling dimensions.
3. Superconformal current multiplet with stress tensor $g$9 ($u$0) and supercurrent $u$1 ($u$2).

Integer-$u$3 sectors reproduce conformal towers predicted by the 3D conformal algebra, with each primary operator succeeded by descendants at $u$4 with the expected degeneracies (Tang et al., 31 Dec 2025).

5. Operator-Level Renormalization Group Flow

Tracking the operator spectrum as the microscopic pseudopotentials $u$5 are varied from $u$6 to $u$7 reveals a continuous RG trajectory:

• $u$8: $u$9 flows from $\phi$0 to $\phi$1
• $\phi$2: $\phi$3 from $\phi$4 to $\phi$5
• $\phi$6: $\phi$7 from $\phi$8 to $\phi$9
• The Yukawa operator $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$0 is initially relevant but becomes irrelevant at the interacting fixed point, reflecting SCFT stability.
• Composite operators $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$1 and $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$2 hybridize along the flow, obeying the critical equation of motion $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$3.

Individual eigenstates evolve smoothly between the decoupled and interacting spectra, making the RG flow at the operator level directly visible in the spectrum (Tang et al., 31 Dec 2025).

6. Evidence for Emergent $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$4 Supersymmetry and Physical Consequences

Several features confirm the emergence of supersymmetry at the critical point:

• Protected gaps: The low-lying spectrum at $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$5 features exact integer spacing characteristic of conformal multiplets and uniform SUSY spacing of $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$6 and $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$7.
• Supercharge algebra: $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$8, with numerical agreement between scaling dimensions at the $$H = H_{\rm Ising}\otimes\mathbb{I} + \mathbb{I}\otimes r H_{\rm Majorana} + \int d\Omega_f d\Omega_b\, n_f^z(\Omega_f) \left[\lambda^{z0}(\Omega_{fb}) n_b^0(\Omega_b) + \lambda^{zx}(\Omega_{fb}) n_b^x(\Omega_b)\right]$$9–$H_{\rm Ising}$0 level.
• Supercurrent and stress tensor: Observation of $H_{\rm Ising}$1 at the unitarity bound ($H_{\rm Ising}$2) and $H_{\rm Ising}$3 at $H_{\rm Ising}$4.
• Central charge $H_{\rm Ising}$5: Not extracted directly, but fixable from the $H_{\rm Ising}$6 gap, with $H_{\rm Ising}$7, using the constraint $H_{\rm Ising}$8.
• Non-perturbative control: The fuzzy sphere approach, in contrast to lattice or perturbative $H_{\rm Ising}$9-expansion, exactly preserves $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$00 symmetry and permits direct measurement of scaling dimensions (Tang et al., 31 Dec 2025).

7. Significance and Methodological Advancements

The fuzzy-sphere regularized Gross–Neveu–Yukawa model provides the first controlled, microscopic realization of a 3D $$\mathcal{L}_{\rm GNY} = \frac{1}{2}(\partial_\mu\phi)^2 + i\,\bar\psi\!\!\not\!\partial\,\psi + g\,\phi\,\bar\psi\psi + \frac{m}{2}\,\phi^2 + \frac{u}{4}\,\phi^4$$01 superconformal Ising CFT. It allows for non-perturbative extraction of operator dimensions, transparent demonstration of operator-level RG flow, and identification of the full supermultiplet and conformal multiplet structure in the spectrum. This approach overcomes the challenges faced by lattice Monte Carlo simulations (notably the fermion sign problem) and perturbative expansions, establishing a pathway for future investigations of strong-coupling phenomena and emergent spacetime supersymmetry in higher-dimensional CFTs (Tang et al., 31 Dec 2025).