Holographic Interpolation between $a$ and $F$

Abstract

An interpolating function $\tilde{F}$ between the $a$-anomaly coefficient in even dimensions and the free energy on an odd-dimensional sphere has been proposed recently and is conjectured to monotonically decrease along any renormalization group flow in continuous dimension $d$. We examine $\tilde{F}$ in the large-$N$ CFT's in $d$ dimensions holographically described by the Einstein-Hilbert gravity in the AdS${}_{d+1}$ space. We show that $\tilde{F}$ is a smooth function of $d$ and correctly interpolates the $a$ coefficients and the free energies. The monotonicity of $\tilde{F}$ along an RG flow follows from the analytic continuation of the holographic $c$-theorem to continuous $d$, which completes the proof of the conjecture.

arXiv comment3 pages, v2: a reference added, minor changes

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• Holographic and Localization Calculations of Boundary F for $\mathcal{N}=4$ SUSY Yang-Mills Theory
• Holographic RG flows on curved manifolds and the $F$-theorem
• Renormalization group flow of entanglement entropy on spheres
• Einstein-AdS action, renormalized volume/area and holographic Renyi entropies
• Topological terms, AdS_2n gravity and renormalized Entanglement Entropy of holographic CFTs
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