Analytic Bounds and Emergence of $\textrm{AdS}_2$ Physics from the Conformal Bootstrap (1611.10060v1)

Abstract: We study analytically the constraints of the conformal bootstrap on the low-lying spectrum of operators in field theories with global conformal symmetry in one and two spacetime dimensions. We introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the external scaling dimensions are large, our functionals provide a direct point of contact between crossing in a 1D CFT and scattering of massive particles in large $\textrm{AdS}_2$. In particular, CFT crossing can be shown to imply that appropriate OPE coefficients exhibit an exponential suppression characteristic of massive bound states, and that the 2D flat-space S-matrix should be analytic away from the real axis.

An Overview of Analytic Bounds and Emergence of AdS₂ Physics from the Conformal Bootstrap

The paper under consideration, authored by Dalimil Maz, makes significant advancements in understanding the applications of conformal bootstrap to field theories in lower spacetime dimensions. Specifically, it explores analytic bounds in one- and two-dimensional field theories exhibiting global conformal symmetry, elucidating connections to AdS₂ physics.

The conformal bootstrap method, recognizing the constraints imposed by unitarity and associativity in conformal field theories (CFTs), has proven powerful across various dimensions. Yet, in lower-dimensional field theories, the analytic origins of these constraints have remained less explored, particularly in 1D and 2D scenarios. This paper addresses the gap, focusing on the low-lying operator spectrum constrained by bootstrap, providing a thorough analytical investigation.

Summary of Key Findings

1. Introduction of Linear Functionals: The paper introduces a novel set of linear functionals acting on the conformal bootstrap equation, improving our understanding of the operator spectrum in field theories of lower dimensionality.
2. 1D Analytic Bounds: In one-dimensional CFTs, extremal functionals are constructed to achieve optimal bounds on gaps above the identity operator within the operator product expansion (OPE). This powerfully bridges the 1D conformal bootstrap constraints to large AdS₂ scattering physics, highlighting an exponential suppression of appropriate OPE coefficients, akin to the mass-suppressed bound states encountered in physical scenarios.
3. 2D Upper Bounds: Moving to the two-dimensional context, the research establishes upper bounds on twist gaps, unveiling a theoretical framework that juxtaposes CFT behavior against large AdS₂ spaces.
4. Connection to Massive Particle Scattering: The established framework draws tangible connections to the scattering mechanics of massive particles in expansive AdS₂ spaces. The exponential suppression of certain OPE coefficients shows close correlation with boundary four-point functions representing massive (1+1)D quantum field theory (QFT) scattering.

Implications and Future Directions

These findings have pivotal implications for theoretical and practical understanding. By clearly establishing bounds and drawing connections to physicality in AdS₂ scenarios, this research lays groundwork for future explorations into bootstrap methods in higher dimensions. Additionally, the elucidation of extremal functional behaviors relative to large scaling dimensions promises improvements in developing dual theories and computational techniques within AdS/CFT correspondences.

Further exploration is likely to consider how the framework can be extended to implement numerical methods for large spacetime dimensional applications. Continued refinement of the new basis for linear functionals, alongside evaluating their efficacy in modular bootstrap scenarios, holds promise for broadening the analytical toolsets available to physics researchers. Through these efforts, the overarching understanding of both bootstrap dynamics and AdS physics can gain new dimensions of clarity, enriching the theoretical landscape.

In summary, this paper pushes the boundaries of what is theoretically achievable with the conformal bootstrap in lower dimensions, illustrating potent connections to AdS physics and paving avenues for future research characterized by enhanced analytic tools and numerical integration. The implications extend beyond theoretical contours, brushing on the foundational aspects of quantum mechanics and field theory dualities.