Carving Out the Space of 4D CFTs (1109.5176v1)

Abstract: We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N=1 superconformal theories, we place strong bounds on dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of N=1 theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.

• The paper introduces a semidefinite programming algorithm that computes rigorous bounds on operator dimensions, central charges, and OPE coefficients in 4D CFTs.
• It significantly improves constraints on operator dimensions for theories with global symmetries, effectively narrowing down flavor-generic scenarios in conformal technicolor models.
• For N=1 SCFTs, the study establishes strong bounds on chiral operator dimensions and reveals insights into the structure of protected operator interactions.

Carving Out the Space of 4D CFTs

The paper in question presents a comprehensive exploration of the space of four-dimensional conformal field theories (CFTs) using innovative numerical techniques and theoretical insights. Authored by David Poland, David Simmons-Duffin, and Alessandro Vichi, the paper introduces a new numerical algorithm based on semidefinite programming to compute bounds on various quantities of interest in 4D CFTs and $N=1$ superconformal field theories (SCFTs). This approach allows for significant improvements over previous methods in determining constraints on operator dimensions, central charges, and operator product expansion (OPE) coefficients, especially in theories with global symmetries.

Key Contributions and Results

1. Semidefinite Programming Algorithm: The authors develop a new numerical technique utilizing semidefinite programming to efficiently calculate bounds on CFT quantities. This method avoids the numerical challenges associated with previous discretization techniques and can handle theories with global symmetries more effectively.
2. Operator Dimensions: The paper dramatically improves existing bounds on operator dimensions, particularly for theories with $\SO(4)$ and $\SU(2)$ global symmetries. In the context of conformal technicolor models, these enhanced bounds impose strong constraints, excluding certain flavor-generic scenarios.
3. $N=1$ Superconformal Field Theories: The research places robust bounds on the dimension of chiral operators and their composites in SCFTs. The bounds are particularly strong near certain dimensions, implying the exclusion of positive anomalous dimensions in specific regions.
4. OPE Coefficients: A novel aspect of the paper is the derivation of upper and lower bounds on the OPE coefficients of protected operators in SCFTs. This is made possible by leveraging the dimension gaps enforced by unitarity, providing insights into the allowed structure of theories.
5. Central Charges and Flavored Currents: The paper finds lower bounds on central charges and flavor current two-point functions, which scale with the size of global symmetry representations. This has implications for the AdS/CFT correspondence as it relates to the strength of interactions in bulk theories.

Implications and Future Directions

The theoretical and practical implications of these results are manifold. The improved bounds have direct applications in constraining beyond-the-Standard-Model physics, particularly in the areas related to the hierarchy problem and the search for new physics at colliders. Moreover, the semidefinite programming approach offers a versatile tool that can handle more complex symmetry structures and potentially higher-dimensional CFTs.

The paper invites further exploration into several promising directions. One immediate avenue is the extension of these methods to theories with more intricate global symmetry patterns, possibly illuminating new classes of CFTs. Additionally, pushing the exploration into lower and higher spacetime dimensions could provide critical insights into the universality of these constraints.

In conclusion, this paper marks significant progress in the landscape of 4D CFTs and SCFTs. The use of semidefinite programming not only refines existing bounds but also opens up new vistas for future exploration, providing both a foundation and a pathway for further theoretical advancements in the field of quantum field theories.