  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.
