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This informal High Energy Physics talk explores the convexity properties of scaling dimensions across conformal manifolds in d-dimensional conformal field theories (CFTs). Presented by Nat Levine at NYU Physics, the 75-minute lecture focuses on CFTs with exactly marginal couplings where the finite part of the sphere partition function remains unambiguous. Discover how this property applies to all odd-dimensional CFTs with conformal manifolds and certain supersymmetric even-dimensional ones. Learn about the proven concavity of scaling dimensions for the lightest unprotected scalar operators with fixed quantum numbers, and how this extends to the sum of the lowest n such dimensions. Understand these results in relation to a dressed Riemannian metric on the conformal manifold (the Zamolodchikov metric multiplied by the sphere partition function). Examine how these convexity statements can imply monotonicity in holographic CFTs when unprotected states decouple in weakly coupled gravity limits, with testing demonstrated in N=4 super Yang-Mills theory in both planar and non-planar regimes.
