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Holographic Renormalization of Einstein-Maxwell-Dilaton Theories" (2016)

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Holographic Renormalization of Einstein-Maxwell-Dilaton Theories University of Kentucky UKnowledge Physics and Astronomy Faculty Publications Physics and Astronomy 11-8-2016 Holographic Renormalization of Einstein-Maxwell- Dilaton Theories Bom Soo Kim University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits oy u. Follow this and additional works at: https://uknowledge.uky.edu/physastron_facpub Part of the Astrophysics and Astronomy Commons, and the Physics Commons Repository Citation Kim, Bom Soo, "Holographic Renormalization of Einstein-Maxwell-Dilaton Theories" (2016). Physics and Astronomy Faculty Publications. 403. https://uknowledge.uky.edu/physastron_facpub/403 This Article is brought to you for free and open access by the Physics and Astronomy at UKnowledge. It has been accepted for inclusion in Physics and Astronomy Faculty Publications by an authorized administrator of UKnowledge. For more information, please contact [email protected]. Holographic Renormalization of Einstein-Maxwell-Dilaton Theories Notes/Citation Information Published in Journal of High Energy Physics, v. 2016, no. 11, article 44, p. 1-43. © The Author(s) 2016 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Digital Object Identifier (DOI) https://doi.org/10.1007/JHEP11(2016)044 This article is available at UKnowledge: https://uknowledge.uky.edu/physastron_facpub/403 Published for SISSA by Springer Received: September 15, 2016 Accepted: November 2, 2016 Published: November 8, 2016 Holographic renormalization of JHEP11(2016)044 Einstein-Maxwell-Dilaton theories Bom Soo Kim Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, U.S.A. E-mail: [email protected] Abstract: We generalize the boundary value problem with a mixed boundary condition that involves the gauge and scalar fields in the context of Einstein-Maxwell-Dilaton theories. In particular, the expectation value of the dual scalar operator can be a function of the expectation value of the current operator. The properties are prevalent in a fixed charge ensemble because the conserved charge is shared by both fields through the dilaton coupling, which is also responsible for non-Fermi liquid properties. We study the on-shell action and the stress energy tensor to note practical importances of the boundary value problem. In the presence of the scalar fields, physical quantities are not fully fixed due to the finite boundary terms that manifest in the massless scalar or the scalar with mass saturating the Breitenlohner-Freedman bound. Keywords: AdS-CFT Correspondence, Black Holes in String Theory, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1608.06252 Open Access, c The Authors. doi:10.1007/JHEP11(2016)044 Article funded by SCOAP3. Contents 1 Introduction & salient features 2 2 Generalities 4 2.1 Scalar or vector variation 5 2.2 Scalar and vector variations: coupled 7 2.3 On-shell action & stress energy tensor 8 2.4 Comparison to minimal coupling 11 JHEP11(2016)044 3 Dilaton coupling W (φ) 11 3.1 Exponential coupling: W (φ) eφ 11 ∼ 3.1.1 Scalar mass above the BF bound 12 3.1.2 Scalar mass saturating the BF bound 13 3.1.3 Massless scalar 14 3.2 Polynomial coupling: W (φ) φk 15 ∼ 3.2.1 Scalar mass above the BF bound 16 3.2.2 Scalar mass saturating the BF bound 17 3.2.3 Massless scalar 19 4 EMD solutions 20 4.1 AdS4 background 21 4.1.1 On-shell action 22 4.1.2 Grand canonical ensemble cF = 0 23 4.1.3 Semi canonical ensemble cF = 1 24 4.2 AdS5 background 25 4.2.1 On-shell action 26 4.2.2 Grand canonical ensemble cF = 0 27 4.2.3 Semi canonical ensemble cF = 1 28 4.3 Interpolating solution 28 4.3.1 On-shell action 29 4.3.2 Case with cF = 0 30 4.3.3 Case with c = 0 31 F 6 5 Two physical implications 31 5.1 Finite boundary terms 32 5.1.1 Examples in holography 32 5.1.2 EMD theories 34 5.2 Non-Fermi liquids & charge splitting 36 6 Conclusion 37 – 1 – 1 Introduction & salient features Einstein-Maxwell-Dilaton (EMD) theories are natural from the dimensional reductions of consistent string theory. They have provided various distinctive physical properties depending on the parameters present in the theory and have been studied extensively. See some earlier literature [1]–[5] and recent ones for condensed matter applications [6]–[12] in the holographic context [13]–[16]. EMD theories have a U(1) gauge field Aµ and a dilaton φ in addition to a metric gµν, where µ, ν run for all the coordinates. Compared to the minimal coupling between the gauge field and a scalar, these theories have a scalar coupling with the form W (φ)F 2 where JHEP11(2016)044 F = dA is the field strength. There is a conserved current J µ that can be evaluated by the variation of the gauge field Aµ. J µ √ gW (φ)F rµ , (1.1) ∝ − when the field Aµ(r) is only a function of a radial coordinate r. g is the determinant of the metric. Upon a close examination, one finds the charge J 0 has contributions not only from the gauge field, but also from the scalar field through the function W (φ). Here we investi- gate the holographic renormalization of EMD theories with emphasis on the role of this cou- pling. We are going to focus on the theories with AdS asymptotics and analytic examples. The variation of the term √ gW (φ)F 2 in an action with respect to the gauge field A − ν provides a boundary contribution of the form √ γW (φ) n F rνδA , (1.2) − r ν where γ is the determinant of a boundary metric, and nr is a unit normal vector for a fixed radius. The notations are systematically explained below. It is consistent to impose the Dirichlet boundary condition δAν = 0 to have a well defined variational problem. This is the case of fixing the constant part of the gauge field, the chemical potential. Holographic renormalization of EMD theories with this Dirichlet boundary condition has been considered in [17]. Now let us consider an alternative quantization that sets δF rν = 0. We are directed to add the boundary term √ γW (φ) n F rνA . (1.3) − r ν One can easily see a new feature for the variation of this term. We produce not only the variation δF rν, but also the scalar variation δφ because of W (φ). As we see below, it is natural to couple the variation of the gauge and scalar fields to have a consistent variational problem. Thus we seek a possibility to impose a generalized mixed boundary condition on both fields. The coupling term √ gW (φ)F 2 brings forth this possibility naturally. The − variation of (1.3) combined with (1.2) gives ∂ log W (φ) √ γW (φ)n A δF rν + F rν δφ . (1.4) − r ν ∂φ – 2 – The first term is expected, and one can impose the variational condition δF rν = 0. As one already anticipated, there is an additional term. It can be simplified, using (1.1), to ∂ log W (φ) (J νA ) δφ . (1.5) ν ∂φ ν Note the combination J Aν is usually finite at the boundary. This term prompts us to consider the variation of the gauge field together with that of the scalar. The canonically normalized scalar kinetic term, upon variation, gives the r boundary contribution √ γn ∂rφ(δφ). We consider two possible boundary terms for − − JHEP11(2016)044 the scalar field with appropriate coefficients following the previously developed variational approaches [18] Λ √ γ φ φ2 + c φnr∂ φ . (1.6) − 2L φ r Here we leave the coefficients Λφ, cφ unfixed. Then putting the variations together, we get Λ ∂W √ γ (c 1)nr∂ φ + φ φ + 4 n F rνA (δφ) (1.7) − φ − r L ∂φ r ν r rν + cφφn δ(∂rφ) + 4W nrAν(δF ) . ∂W rν Note the term ∂φ nrF Aν(δφ) mixes the scalar variation with the gauge field dependent term and plays a crucial role. Investigating this variational problem in more general context is the main task of the paper. The basics of the variational problem are described in two sections, section 2.1 and section 2.2, with some review. The program is carried out systematically for two different forms of the coupling W (φ), the exponential coupling W (φ) eφ in section 3.1 (with ∼ two examples in section 4.1 and section 4.2) and the polynomial coupling W (φ) φk in ∼ section 3.2 (with an example in section 4.3). We contrast the boundary value problem of the EMD theories with that of the theories with a minimal coupling in section 2.4. In parallel, we also carefully examine the on-shell action and the stress energy tensor of the EMD theories. The basics along with some review are presented in the beginning of section 2 and in section 2.3. They are applied to the three examples in section 4. It is pleasant to see that the results of the general variational problem actually fit together nicely with the analysis of the on-shell action and the stress energy tensor. Along the way, our investigations direct us to appreciate two physical implications. One is the finite boundary or counter terms considered in section 5.1. There we attempt to compare the holographic finite boundary terms to the similar notion of finite radiative cor- rections in Quantum field theory `ala Jackiw.
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