A CFT Distance Conjecture

TL;DR

The article proposes a CFT Distance Conjecture that links infinite-distance limits on conformal manifolds to emergent higher-spin symmetry, with an infinite tower of currents whose anomalous dimensions vanish exponentially in the Zamolodchikov distance. It formalizes the framework with Conjectures I–III, defines the spin-conditioned anomalous dimensions $\gamma_J(t)$, and analyzes how the conformal-manifold geometry constrains the operator spectrum, especially in SCFTs in $d>2$. The authors then translate these ideas to AdS gravity, deriving a lower bound on the exponential decay rate $\alpha$ and relating it to central charges and the Planck scale, while comparing to the Swampland Distance Conjecture. They illustrate the framework with explicit AdS/CFT examples—notably ${\cal N}=4$ SYM, SQCD with $N_f=2N$, and $\beta$-deformed theories—showing how HS towers and infinite-distance behavior manifest in holography and imposing bounds on the possible moduli-space diameters. The work thereby connects conformal-manifold geometry, HS symmetry, and holographic duality to swampland constraints, suggesting that infinite-distance limits invariably host HS dynamics and constraining the landscape of SCFTs and their gravity duals.

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.