Universal Bounds on Charged States in 2d CFT and 3d Gravity

TL;DR

This work uses modular invariance in 2d CFTs with a U(1) current to derive universal bounds on the charged spectrum and the charge-to-mass ratio, connecting these bounds to AdS3 gravity and the WGC. A linear-functional bootstrap at the self-dual point yields a bound on the lightest charged state that grows with central charge, with supersymmetry improving the bound to Δ − Δ_vacuum ≤ c/12 + 1. Additional approaches based on asymptotic growth tighten the constraints, and explicit large-gap constructions (extremal lattices) illustrate near-saturation. The results have implications for the necessity and scale of charged states in holographic theories and point to extensions involving non-abelian symmetries and higher-dimensional correlators.

Abstract

We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal field theories with a global abelian symmetry. We find that the bound scales with $c$ and provide examples that parametrically saturate this bound. We also prove than any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.

Figures (1)

• Figure 1: Left: An upper bound on the total gap $\Delta_{\rm gap} + \kappa$ between the vacuum and the lightest charged state, as a function of $\kappa \equiv \frac{c+\bar{c}}{24}$. The slope asymptotes to $2\kappa$ at large $\kappa$, show in red, dashed. Right: The shaded region is where the polynomial $e^{2\pi \Delta} \alpha(F_{\Delta,Q})$ in (\ref{['eq:examplealpha']}) is negative for $\kappa=2$; the right edge asymptotes to a vertical line (shown in blue, dashed) at $\Delta/\kappa = \Delta_{\rm gap}(\kappa)/\kappa$ for large $Q$. In unitary theories, there must be at least one state in the shaded region.