Moments in the CFT Landscape

Technical Deep Dive: Moments in the CFT Landscape

Overview

This research develops a novel numerical bootstrap framework for studying unitary, crossing-symmetric conformal field theories (CFTs) through the lens of moment observables. Moments are defined as weighted averages over conformal data, providing a global and coarse-grained probe of the operator spectrum. Using semidefinite programming, this framework yields numerically rigorous bounds on the operator distribution.

The key findings are:

• Moments exhibit a remarkably rich geometric structure across dimensions, with both upper and lower bounds displaying pronounced nonlinear features.
• In the heavy correlator limit, the numerical moment bounds rapidly converge to the analytically derived power laws.
• Specific moments capture the critical Ising CFT, reproducing the familiar bootstrap "kinks".
• Two continuous families of previously unknown kinks are identified, revealing nontrivial spectral reorganizations connected to operator decoupling.

Methodology

• Defined moment variables as weighted averages over conformal data, with the weights determined by squared OPE coefficients and conformal blocks evaluated at the self-dual point.
• Formulated the problem of bounding moments as an infinite-dimensional linear program, which was then transformed into a semidefinite program solvable using SDPB.
• Derived rational approximations of conformal blocks and their derivatives to efficiently compute the relevant crossing constraints.
• Employed maximum entropy reconstruction to systematically study the coarse-grained structure of the operator spectrum from the moment bounds.

Results

Heavy Correlator Limit

• In the heavy correlator regime, the numerical moment bounds show excellent agreement with the analytically derived power-law bounds.
• The moment variables effectively "sense" the collective behavior of the dense operator spectrum, reproducing the correct asymptotic scaling using only a finite number of derivatives.
• Maximum entropy reconstruction confirms the convergence towards generalized free field theory distributions, while capturing systematic finite-\Delta_φ corrections.

The Ising Model

• The critical Ising model is precisely located at the kinks of the upper bounds on the leading \Delta- and \ell-moments.
• These kinks are related to the extremization of the gap and the central charge, as observed in previous bootstrap studies.
• Tight numerical bounds on the Ising moments are obtained under a strong gap assumption, pushing the solution close to the boundary of the allowed theory space.

New Geometric Features

• The lower bounds on moments exhibit two continuous families of previously unobserved kinks, persisting across 2 < d < 6.
• These kinks are closely tied to nontrivial spectral reorganizations, including the decoupling of low-lying scalar operators and the saturation of the scalar unitarity bound.
• In the region between the two kink families, the moment-minimizing solutions probe qualitatively different directions in theory space, with the reconstructed correlators showing unbounded growth.

Interpretation

• Moment variables provide a natural and efficient language for the conformal bootstrap, allowing for the identification of known models and the discovery of new privileged solutions.
• The rich geometric structures revealed by the moment bootstrap suggest the existence of an underlying dynamical mechanism governing the operator spectrum, beyond the traditional focus on individual operator properties.
• The interplay between operator decoupling, anomalous dimension extremization, and the fake-primary effect point to a complex interplay between light and heavy degrees of freedom in CFTs.
• The moment bootstrap unifies a wide range of familiar bootstrap phenomena, while opening up new avenues for further exploration, including the study of mixed correlators and the connections to analytical heavy limit results.

Limitations & Uncertainties

• The precise physical interpretation of the new kink structures and their potential connections to known conformal field theories remain open questions.
• While the numerical moment bounds are rigorously controlled, the spectral reconstruction relies on the maximum entropy principle, which may not fully capture the underlying dynamics.
• The analysis is limited to four-point functions of identical scalar operators; extending the moment bootstrap to mixed correlators and other operator types is an important direction for future work.

Future Directions

• Investigate the relationship between the moment landscape and renormalization group flows of known interacting scalar field theories.
• Explore the connection between the moment bounds and the physics of thermalization and finite-temperature effects in conformal field theories, through the study of heavy-heavy-light-light correlators.
• Apply the moment bootstrap framework to conformal field theories beyond the realm of scalar operators, including fermions and gauge theories.
• Develop analytical tools to better understand the origin and physical interpretation of the newly discovered geometric features in moment space.