Abstract
We study F-functions in the context of field theories on S3 using gauge-gravity duality, with the radius of S3 playing the role of RG scale. We show that the on-shell action, evaluated over a set of holographic RG flow solutions, can be used to define good F-functions, which decrease monotonically along the RG flow from the UV to the IR for a wide range of examples. If the operator perturbing the UV CFT has dimension Δ > 3/2 these F -functions correspond to an appropriately renormalized free energy. If instead the perturbing operator has dimension Δ < 3/2 it is the quantum effective potential, i.e. the Legendre transform of the free energy, which gives rise to good F-functions. We check that these observations hold beyond holography for the case of a free fermion on S3 (Δ = 2) and the free boson on S3 (Δ = 1), resolving a long-standing problem regarding the non-monotonicity of the free energy for the free massive scalar. We also show that for a particular choice of entangling surface, we can define good F-functions from an entanglement entropy, which coincide with certain F-functions obtained from the on-shell action.
Highlights
A fundamental property of Quantum Field Theory is that the number of degrees of freedom decreases under renormalization group (RG) flow
We show that the on-shell action, evaluated over a set of holographic RG flow solutions, can be used to define good F -functions, which decrease monotonically along the RG flow from the UV to the IR for a wide range of examples
If instead the perturbing operator has dimension ∆ < 3/2 it is the quantum effective potential, i.e. the Legendre transform of the free energy, which gives rise to good F -functions. We check that these observations hold beyond holography for the case of a free fermion on S3 (∆ = 2) and the free boson on S3 (∆ = 1), resolving a long-standing problem regarding the non-monotonicity of the free energy for the free massive scalar
Summary
J Comparison with the F -function proposal of Beneventano et al. Introduction and summary. The case of the c-theorem d = 3, referred to as the F -theorem, received particular attention It was suggested in [8,9,10] that the role of the c-quantity can be played by (the finite part of) the free energy of the theory on the 3-sphere, F = − ln |ZS3|. In [24] the behaviour of the REE under renormalization group flow was examined for the theory of a conformally coupled scalar on dS3 In this case the REE fails to exhibit monotonicity and it is not a good F -function on dS3 (see [17]). We will propose candidate F -functions constructed from the free energy We will test their monotonicity both in simple holographic examples and in free theories. The rest of this introduction summarises in a self-contained manner our setup, our results and concluding remarks
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