On the CFT Operator Spectrum at Large Global Charge (1505.01537v4)

Abstract: We calculate the anomalous dimensions of operators with large global charge $J$ in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a $W = \Phi^3$ superpotential. Working in a $1/J$ expansion, we find that the large-$J$ sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge $J$ is always a scalar operator whose dimension $\Delta_J$ satisfies the sum rule $ J^2 \Delta_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J+1} = 0.035147 $ up to corrections that vanish at large $J$. The spectrum of low-lying excited states is also calculable explcitly: For example, the second-lowest primary operator has spin two and dimension $\Delta\ll J + \sqrt{3}$. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order $J^{1/2}$. The propagation speeds of the Goldstone waves and heavy fermions are $\frac{1}{\sqrt{2}}$ and $\pm \frac{1}{2}$ times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large $J$.

• The paper introduces an analytical framework using an inverse J expansion to study operator dimensions in strongly coupled CFTs.
• It demonstrates that the lowest-energy state is a scalar operator with precise dimensional relationships and vanishing corrections at large J.
• The work highlights supersymmetric effects, including half-integer spin gaps and altered propagation speeds, advancing effective CFT modeling.

An Analysis of the Operator Spectrum in Conformal Field Theories

The paper "On the CFT Operator Spectrum" by Susanne Reffert and Masataka Watanabe et al. presents an in-depth paper of conformal field theory (CFT) dynamics, focusing on the computation of anomalous dimensions associated with operators possessing large global charge $J$. Specifically, the authors investigate these dynamics within certain strongly coupled CFTs, such as the $O(2)$ model and a supersymmetric fixed point described by a chiral superfield with a $W = \Phi^3$ superpotential. By applying an expansion in inverse powers of $J$, the paper argues that the sector with large $J$ is governed by an effective Lagrangian that respects conformal symmetry.

Key Findings and Numerical Insights

In their analysis, the authors find that the lowest-energy state with charge $J$ is consistently a scalar operator with a dimensional relationship governed by the sum rule:

$J\sqd \Delta_J - \left( \tfrac{J^2}{2} + \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J-1} - \left( \tfrac{J^2}{2} - \tfrac{J}{4} + \tfrac{3}{16} \right) \Delta_{J+1} = 0.035147 \ ,$

where corrections vanish at large $J$. Moreover, low-lying excited states can be explicitly calculated; the second-lowest primary operator, for example, has a spin and dimension satisfying $\Delta\ll J + \sqrt{3}$.

In the supersymmetric context, half-integer spin operators are noted to have dimensions exceeding their integer-spin counterparts by a gap proportional to $J^{1/2}$. Furthermore, the dynamics of Goldstone waves and fermions in such models show propagation speeds that are fractions of the speed of light, which is essential for preserving superconformal symmetry in the large $J$ regime.

Theoretical and Practical Implications

The findings have significant implications for understanding CFTs with large global charges. These insights advance the comprehension of analytical tools that might simplify the exploration of operator spectra in strongly coupled theories. Importantly, the authors suggest that their approach reveals structural aspects of CFTs that appear fundamental when viewed through the lens of the conformal bootstrap.

Moreover, the practical applications of these results could lead to a better characterization of quantum field theories that lack simple solvable limits. This, in turn, can provide a basis for the exploration of new theoretical avenues and methodologies that can address previously intractable problems.

Future Directions and Speculations

Given the robustness of the calculations and the significance of the results, further research could extend this analysis to other CFT scenarios, enhancing our understanding of the global symmetry sectors of various quantum field theories. Additionally, the methods and conclusions might be compared against numerical and bootstrap techniques for a more comprehensive understanding of these complex quantum systems.

In conclusion, the paper provides a comprehensive analytical framework to paper CFTs with large global charges, offering novel insights into operator dimensions and their behavior under strong coupling scenarios. Through meticulous calculations and theoretical rigor, the work lays a foundation for future explorations into the multifaceted nature of CFTs and related quantum theories.