A CFT Distance Conjecture

TL;DR

This paper proposes a conjecture linking the geometry of conformal manifolds in higher-dimensional CFTs to the emergence of higher-spin symmetries at infinite distance, with implications for holography and the swampland.

Contribution

It introduces a new conjecture connecting infinite-distance points in conformal manifolds to emergent higher-spin symmetries and their spectral properties.

Findings

Theories at infinite distance exhibit an emergent higher-spin symmetry.

The diameter of conformal manifolds diverges logarithmically with the higher-spin gap.

Higher-spin fields become massless exponentially fast at infinite distance.

Abstract

We formulate a series of conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in $d>2$ spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Stated geometrically, the diameter of a non-compact conformal manifold must diverge logarithmically in the higher-spin gap. In the holographic context our conjectures are related to the Distance Conjecture in the swampland program. Interpreted gravitationally, they imply that approaching infinite distance in moduli space at fixed AdS radius, a tower of higher-spin fields becomes massless at an exponential rate that is bounded from below in Planck units. We discuss further implications for conformal manifolds of superconformal field theories in three and four dimensions.