Convexity and Liberation at Large Spin

Abstract: We consider several aspects of unitary higher-dimensional conformal field theories (CFTs). We first study massive deformations that trigger a flow to a gapped phase. Deep inelastic scattering in the gapped phase leads to a convexity property of dimensions of spinning operators of the original CFT. We further investigate the dimensions of spinning operators via the crossing equations in the light-cone limit. We find that, in a sense, CFTs become free at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys additivity: if two twists tau_1, tau_2 appear in the spectrum, there are operators whose twists are arbitrarily close to tau_1+tau_2. We characterize how tau_1+tau_2 is approached at large spin by solving the crossing equations analytically. We find the precise form of the leading correction, including the prefactor. We compare with examples where these observables were computed in perturbation theory, or via gauge-gravity duality, and find complete agreement. The crossing equations show that certain operators have a convex spectrum in twist space. We also observe a connection between convexity and the ratio of dimension to charge. Applications include the 3d Ising model, theories with a gravity dual, SCFTs, and patterns of higher spin symmetry breaking.

• The paper establishes a convexity property among the dimensions of spinning operators in gapped CFT phases via massive deformations.
• The paper demonstrates that CFTs exhibit free theory-like behavior at large spin with 1/s corrections, revealing an additive twist spectrum.
• The paper derives analytical solutions that agree with perturbative and gauge-gravity duality predictions, constraining higher spin currents in various models.

Overview of "Convexity and Liberation at Large Spin"

The paper "Convexity and Liberation at Large Spin" by Zohar Komargodski and Alexander Zhiboedov explores the intricate properties of Conformal Field Theories (CFTs) in higher dimensions, focusing on the behavior of their operator spectra and the implications for various physical theories, including both strongly and weakly coupled systems.

Key Concepts and Findings:

1. Massive Deformations and Convexity: The paper begins by examining massive deformations of unitary CFTs, which flow to a gapped phase. One key result is the establishment of a convexity property among the dimensions of spinning operators in these gapped phases, specifically in the context of deep inelastic scattering.
2. Crossing Equations and Spin Liberation: The authors analyze the dimensions of spinning operators through crossing equations in the light-cone limit. Their results indicate that CFTs exhibit a behavior akin to free theories at large spin, with $1/s$ acting as a weak coupling parameter. This leads to an additivity property in the spectrum of twists, where, if two twists $\tau_1$ and $\tau_2$ exist, there are operators with twists approaching $\tau_1 + \tau_2$ at large spin.
3. Analytical Solutions and Agreement with Dual Theories: By solving the crossing equations analytically, the paper not only determines the form of the leading corrections (including prefactors) but also demonstrates agreement with perturbative computations and gauge-gravity duality predictions. This highlights these analytical solutions' consistency with known CFT examples, such as the 3D Ising model and other theories with gravity duals.
4. Relation to Higher Spin Symmetry: The work further explores the connection between convexity and the ratio of dimension to charge. This aspect finds applications in 3D Ising models, Supersymmetric CFTs (SCFTs), and higher spin symmetry breaking patterns. Notably, the convex twist spectrum implies particular constraints for higher spin currents, vital for understanding symmetry breaking.
5. Concrete Predictions and Applications:
   • The paper proposes that in critical O(N) models, there exist almost conserved currents with twists bounded by the operator with minimal twist, thereby predicting observable constraints in these models.
   • It also speculates on the implications for theories with weak coupling or those possessing gravitational duals, which could inform future studies in quantum gravity scenarios.

Practical and Theoretical Implications:

The implications of this research span both theoretical and practical domains. Theoretically, it enriches the understanding of CFTs' operator spectra and informs the methodology for analyzing high-energy scattering processes. Practically, it provides a framework for predicting the behavior of operators in models relevant to phase transitions.

Speculation on Future Developments:

Future research could extend these findings by further characterizing the additivity property and exploring more generalized settings, potentially under different symmetry constraints or in non-unitary settings. Another avenue is the application of these ideas to various models of higher-dimensional physics, potentially offering insights into the dynamics of quantum gravity and related fields.

In summary, this paper makes significant strides in understanding the interplay between convexity and liberation at large spin within CFTs, with broad implications for both theoretical physics and applications in strongly correlated systems.