Bootstrapping 3D Fermions

Abstract: We study the conformal bootstrap for a 4-point function of fermions $\langle\psi\psi\psi\psi\rangle$ in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the $\psi \times \psi$ OPE, and also on the central charge $C_T$. We observe features in our bounds that coincide with scaling dimensions in the Gross-Neveu models at large $N$. We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.

Overview of "Bootstrapping 3D Fermions"

The paper "Bootstrapping 3D Fermions," authored by Luca Iliesiu and colleagues, explores the conformal bootstrap for four-point functions of fermionic operators in three-dimensional Conformal Field Theories (CFTs). The primary focus is on developing a formalism for addressing fermionic correlators, calculating conformal blocks, and deriving bounds on the spectrum of operators contributing to these correlators.

Embedding Formalism and Conformal Blocks

In an innovative approach, the authors introduce an embedding formalism for three-dimensional fermions. This formalism leverages the correspondence between three-dimensional spinors and the group $\Sp(4, \mathbb{R})$, providing a method to compute the conformal blocks for four-point functions involving fermionic operators. The embedding into a higher-dimensional space allows for a linear realization of conformal transformations, facilitating the bootstrap process for spin-1/2 operators and connecting their conformal blocks to those of scalars through differential operators. The formulation provided is applicable for computing conformal blocks across dimensions greater than two, extending existing methodologies.

Key Results and Numerical Analysis

The paper explores the bounds on operator dimensions in CFTs with fermions. It analyzes the dimensions of parity-even and parity-odd scalars appearing in the Operator Product Expansion (OPE) of fermions. The study reveals significant features, including sharp discontinuities and kinks in the upper bounds for dimensions as functions of the fermionic operator dimension. A notable conjecture is the existence of a fermionic CFT with potentially no relevant scalar operators, a hypothesis inspired by observed anomalies in the numerical results. This conjectured "dead-end" CFT might exemplify self-organized criticality, with implications for RG flow and stability analysis.

Gross-Neveu Models and SUSY Considerations

By imposing gaps in the spectrum of scalar operators, the authors establish connections with known CFTs, such as the Gross-Neveu models at large values of $N$. This procedure not only aligns with expected theoretical results but also elucidates the scaling dimensions for small-$N$ scenarios, including conjectural implications for the $N=1$ case, which may relate to the $\mathcal{N}=1$ supersymmetric Ising model. The study suggests that further efforts involving mixed correlators might isolate the $\mathcal{N}=1$ theory distinctly.

Implications for Central Charge $C_T$

An essential aspect of the work involves bounding the central charge $C_T$, representing the scaling coefficient in the two-point function of the stress tensor. The findings show the bound tightening near free fermion theory values and demonstrate similar behaviors seen in scalar-centric bootstrap analyses, highlighting potential uses in gauging the "strength" of interactions within these theories.

Future Directions and Conclusions

The paper sets a foundation for future work across several domains, including extending the formalism to fractional dimensions for exploratory analysis via $\epsilon$-expansion, examining fermionic theories with global symmetries such as $O(N)$ to probe large-$N$ behaviors, and leveraging mixed correlators to refine bounds and predictions for supersymmetric theories. The authors argue that these steps may advance understanding of nontrivial CFTs, revealing insights into both strongly coupled and self-organized critical points.

Overall, the paper makes substantial contributions by extending the bootstrap methods to fermionic operators, providing new pathways to analyze critical points within 3D CFTs, and suggesting intriguing possibilities for novel fermionic theories with compelling characteristics. The computational and theoretical methodologies laid out hold promise for the ongoing study of quantum field theories and their intricate landscapes.