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.
Highlights
The Distance Conjecture (DC) in particular quantifies how quickly these light states emerge in extreme limits
We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric
Noting that the holographic dictionary maps the bulk moduli space to the space of exactly marginal couplings in CFT, this general intuition encourages a search for a precise conjecture about limits of conformal manifolds phrased purely in the language of CFT
Summary
We are interested in families of d-dimensional CFTs with exactly marginal parameters. For free-field limiting points, this manifests itself in more physical terms in two ways: first, the breakdown of conformal perturbation theory around free CFTs; and second, the number of independent stress tensors changes discontinuously in the strict decoupling limit For these (somewhat formal) reasons, we sometimes refer to limiting points as being at infinite distance from M. The behavior in several class of examples leads us to a yet more quantitative conjecture: the anomalous dimension near a HS point approaches zero exponentially in the distance To motivate this from purely CFT considerations, consider the simple situation of a d = 4, N = 2 gauge theory with a single exactly marginal complexified gauge coupling, τ.
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