Holographic RG flows on curved manifolds and the F-theorem

Abstract

We study F -functions in the context of field theories on S 3 using gaugegravity duality, with the radius of S 3 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 ∆ > 3/2 these F -functions correspond to an appropriately renormalized free energy. 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 S 3 (∆ = 2) and the free boson on S 3 (∆ = 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. Appendix 55 A Perturbative expansion near the maximum of the potential 55 B Calculation of the on-shell action 56 C Calculation of the entanglement entropy 57 -i -D Analytical results for large and small boundary curvature 59 D.1 Large curvature expansion 59 D.2 Small curvature expansion 62 E Holographic entanglement entropy of a spherical region in flat space 68 F De Sitter entanglement entropy and thermodynamics 69 F.1 The de Sitter static patch: thermal entropy, and the ADM mass 69 F.2 Identities from thermodynamic relations 71 G Comments on the renormalization scheme 72 H Zeta-function renormalization vs. covariant counterterms 74 I Further monotonic functions 76 References 78 5 The operator D 3/2 eliminates any dependence on Cct as it only appears in the combination Cct R −3/2 .

Strong-coupling dynamics and entanglement in de Sitter space

“…The divergent piece of the entanglement entropy satisfies an area law, which prohibits a direct extraction of the IR part of the entropy (see also [46]). It is hence difficult to have a direct interpretation of the entropy of de Sitter itself (see however [62], where a thermodynamic interpretation with a suitable regulated free energy is proposed), but we note that in a theory with dynamical gravity it is conjectured that this entropy is limited by a Bekenstein-Hawking term of A/4G 4 . This entropy, or part thereof, can then potentially be identified with the entanglement entropy whereby 1/G 4 plays the role of the UV cut-off [63].…”

The Self‐Tuning of the Cosmological Constant and the Holographic Relaxion

“…Moreover, thanks to the last property in the list above, a DGP‐like mechanism of gravity quasi‐localisation [ 12 ] allows the four‐dimensional observers on the brane to experience ordinary four‐dimensional gravity in a range of scales. In [18] it was shown that the self‐tuning mechanism is robust, in the sense that stabilized solutions with curved branes require a modification of the boundary conditions at the AdS boundary, and therefore belong to a different superselection sector than flat solutions. A dynamical study of this model in the cosmological setting was initiated in [19].…”

“…The expansions for for small φ can be found using similar techniques as in [46,47]. The leading terms in this expansion are universal.…”

Information flows in strongly coupled ABJM theory

“…It would be useful to develop more such examples. Finally, it would be interesting to put the Chern-Simons field theory studied in this paper on a 3-sphere and test the recent proposal for the c-function [38].…”