The Holographic F Theorem

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 .

Section: Introductionsupporting

confidence: 74%

“…In holography, this can be calculated from the action (1.2), evaluated on the solutions (1.4) or (1.5), supplemented by appropriate counterterms (see e.g. [22]). In our conventions one finds…”

Section: Setup and Summary Of Resultsmentioning

confidence: 99%

Holographic RG flows on curved manifolds and the F-theorem

Ghosh

Kiritsis

Nitti

et al. 2019

“…Again working to quadratic order in the source, the change in the free energy is positive for operators of dimensions 3/2 < ∆ + < 5/2 [12]. Note that such deformations are not equivalent to conformal transformations of the holographic RG flows considered here, which are dual to deformations of the theory on flat space:…”

Section: Jhep08(2016)165mentioning

confidence: 91%

“…We should note however that the corresponding deformations on the three sphere are inhomogeneous and do not therefore correspond to RG flows which respect the SO(4) invariance. Direct calculation of the F quantity for SO (4) invariant RG flows on S 3 driven by such operators also gives an increase in the F quantity to quadratic order in the source, see the companion paper [12]. It would be interesting to understand whether such flows are unphysical or if the strong version of the proposed F theorem is indeed violated.…”

Section: Jhep08(2016)165mentioning

confidence: 99%

See 1 more Smart Citation

Renormalized entanglement entropy

Taylor

Woodhead

2016

Self Cite

We develop a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces. The renormalized entanglement entropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and for entangling surfaces in four dimensional holographic renormalization group flows. The renormalized entanglement entropy for disk regions in AdS 4 spacetimes agrees precisely with the holographically renormalized action for AdS 4 with spherical slicing and hence with the F quantity, in accordance with the Casini-HuertaMyers map. We present a generic class of holographic RG flows associated with deformations by operators of dimension 3/2 < ∆ < 5/2 for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method for the entanglement entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.

Non-conformal entanglement entropy

Taylor

Woodhead

2018

Self Cite

Abstract:We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: non-conformal branes; the Witten model for QCD; UV conformal RG flows driven by explicit and spontaneous symmetry breaking and Schrödinger geometries. Focussing on slab entangling regions, we find that the renormalized entanglement entropy captures features of the previously defined entropic c-function but also captures deep IR behaviour that is not seen by the c-function. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters.