[for zoom link, please email s.nagyATqmul.ac.uk] We formulate 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. Our conjectures are related to the Distance Conjecture in the swampland program. We discuss the supporting evidence, their holographic interpretation, and implications for superconformal field theories.

There has been phenomenal recent progress in computing CFT entanglement and Renyi entropies, holographically and otherwise. We extend these investigations to the higher spin regime -- where sensible notions of geometry are sorely lacking -- by computing ground state Renyi entropies in certain classes of holographic CFTs with higher spin symmetry. Our calculations are performed at both classical and one-loop level, from gravity and CFT. Along the way, we establish some new general results about Renyi entropy in any CFT, higher spin symmetry aside.

We present a collection of results supporting holographic dualities between 3D higher spin gravity and 2D higher spin-symmetric minimal model CFTs at large central charge, focusing on those that are not fixed by higher spin symmetry alone. This includes the first bulk-boundary matching of correlators in 3D higher spin gravity whose functional form is not fixed by conformal invariance, namely, 4-point functions of certain scalar primary operators. In the bulk, this involves the study of propagating scalars in higher spin gravitational backgrounds, which will also help clarify what constitutes a higher spin black hole.

We discuss various recent computations that come to bear on the conjectured holographic duality between Vasiliev's theory of 3d higher spin gravity and families of 2d CFTs with W symmetry, at both zero and finite temperature. These rely on novel, purely algebraic methods for computing bulk-boundary propagators for scalar fields in Vasiliev theory. Among other topics to be discussed, our results further clarify the physical content of the term "higher spin black hole" and the nature of the classical limit required for the duality.