Lessons for Gravity from Entanglement

Lessons for gravity from entanglement

A thesis submitted for the degree of Doctor of Philosophy in the Faculty of Sciences

Arpan Bhattacharyya

Centre for High Energy Physics Indian Institute of Science Bangalore - 560012. India.

June 2015 ii Declaration

I hereby declare that the work presented in this thesis “Lessons for gravity from entanglement” is based on the research work by me under the supervision of Prof. Aninda Sinha and with my collaborators at the Centre for High Energy Physics, Indian Institute of Science, Bangalore, India. It has not been submitted elsewhere as a requirement for any degree or diploma of any other Institute or University. Proper acknowledgements and citations have been made in appropriate places while borrowing research materials from other investigations.

Date : Arpan Bhattacharyya

Certiﬁed by :

Prof. Aninda. Sinha Centre for High Energy Physics Indian Institute of Science Bangalore - 560012 India

iii iv List of publications

This thesis is based on the following publications :

1. “ Entanglement entropy in higher derivative holography ” A. Bhattacharyya, A. Kaviraj and A. Sinha. arXiv:1305.6694 [hep-th] JHEP 1308, 012 (2013)

2. “ On generalized gravitational entropy, squashed cones and holography ” A. Bhattacharyya, M. Sharma and A. Sinha. arXiv:1308.5748 [hep-th] JHEP 1401, 021 (2014)

3. “ Constraining gravity using entanglement in AdS/CFT ” S. Banerjee, A. Bhattacharyya, A Kaviraj, K. Sen and A. Sinha. arXiv:1401.5089 [hep-th] JHEP 1405, 029 (2014)

4. “ On entanglement entropy functionals in higher derivative gravity theories” A. Bhattacharyya and M. Sharma. arXiv:1405.3511 [hep-th] JHEP 1410, 130 (2014)

5. “ Renormalized Entanglement Entropy for BPS Black Branes ” A. Bhattacharyya, S. S. Haque and A. Veliz-Osorio. arXiv:1412.2568 [hep-th] Phy. Rev. D 91, 045026 (2015)

The following works were done during my PhD but is not included in this thesis :

1. “ On c-theorems in arbitrary dimensions ” A. Bhattacharyya, L. Y. Hung, K. Sen and A. Sinha. arXiv:1207.2333 [hep-th] Phy. Rev. D 86, 106006 (2012)

2. “ Entanglement entropy from the holographic stress tensor ” A. Bhattacharyya and A. Sinha. arXiv:1303.1884 [hep-th] Class. Quantum Grav 30, 235032 (2013)

3. “ Entanglement entropy from surface terms in general relativity ” A. Bhattacharyya and A. Sinha. arXiv:1305.3448 [hep-th] IJMPD 22 12, 1342020 (2013)

4. “ Attractive holographic c-functions ” A. Bhattacharyya, S. S. Haque, V. Jejjala, S. Nampuri and A. Veliz-Osoio. arXiv:1407.0469 [hep-th] JHEP 1411, 138 (2014)

v vi

5. “ Viscosity bound for anisotropic superﬂuids in higher derivative gravity ” A. Bhattacharyya and D. Roychowdhury. arXiv:1410.3222 [hep-th] JHEP 1503, 063 (2015)

6. “ Lifshitz Hydrodynamics And New Massive Gravity ” A. Bhattacharyya and D. Roychowdhury. arXiv:1503.03254 Synopsis

One of the recent fundamental developments in theoretical high energy physics is the AdS/CFT correspondence [1,2,3,4] which posits a relationship between Quantum Field Theories (QFT) in a given dimension and String Theory on a higher dimensional anti- de Sitter (AdS) spacetime. This has revolutionised our understanding of QFTs (more specifically conformal field theories (CFTs)) and string theory/gravity, and has far reaching consequences for explo- rations into a vast array of physical phenomena. Using the elegant formalism provided by this powerful duality, often called “holography”, one can now use fundamental physical observables in QFT to better understand the nature of quantum gravity. The theoretical tools provide a translation of calculable field theoretic observables into the language of gravity thereby leading to the construction of holographic models for several interesting QFTs. Entanglement is a fundamental physical property of all quantum systems. From models of various condensed matter systems to its application as a tool for secure and fast communi- cation in quantum information theory [5], it serves as an intersection point between different subfields of physics [6]. From the AdS/CFT point of view quantum entanglement connects geometry with quantum information, providing a window to understand how the bulk gravity physics emerges from the holographic field theoretic viewpoint. Probing various aspects of this connection in detail will be the broad theme of this thesis. For extended, many-body systems, the most well known measure of quantum entanglement is the “Entanglement Entropy” (EE) which is also the best understood measure within the holographic framework. In early 2006, Ryu and Takayanagi (RT) gave a simple and elegant prescription for computing this quantity using AdS/CFT duality within Einstein gravity [7,8]. They proposed that EE for a subsystem within an extended system (QFT), is computed by the (proper) area of a static, codimension- 2, “extremal” surface inside the dual AdS spacetime. The RT proposal has passed several non-trivial consistency checks, for example strong sub-additivity, area law to name a few [9]. A remarkable aspect of the proposal is the ease with which EE can now be calculated, while it is well known that obtaining EE from first principles in QFT presents several technical challenges which have so far been surmounted only in some 2d field theories using the “replica method” [10, 11, 12]. The most intriguing aspect of the RT proposal is its striking similarity to Bekenstein- Hawking (BH) entropy which is proportional to the area of a black hole horizon, further confirming an intimate relationship between entropy and geometry [13, 14, 15, 16]. This leads to the natural question: what is the connection between EE and BH entropy? This question has been sharpened recently by Lewkowycz and Maldacena (LM) via the concept of Generalized Gravitational Entropy which extends the QFT replica trick to a replica symmetry

vii for the dual space-time [17]. This was used to prove the RT conjecture successfully by deriving the correct extremal surface equation for two derivative gravity theories. In this thesis I have studied the generalization of LM method for higher derivative gravity theories [18, 19, 20, 21, 22, 23] describing holographic duals (of QFT’s with finite number of colours) and finite ’t Hooft coupling which takes the AdS/CFT correspondence beyond the usual supergravity limit. If one wants to use AdS/CFT to study real life systems then it is absolutely necessary to incorporate the finite coupling effect into the theory and hence the study of higher derivative effects becomes very important. In these two papers [21, 22] I have formulated a proof for the existence of the entropy functionals for certain higher derivative theories extending LM method. We have shown that the for a certain special class of higher derivative theories there exist well defined entropy functionals. To extend this proof for more general theories of gravity opens up a possibility of breaking replica symmetry in the bulk space-time [24]. For higher derivative gravity, black hole entropy for a large class of stationary black holes with bifurcate killing horizon is given by the well known Wald prescription [25, 26, 27] which relates the concept of the Noether charge with the black hole entropy. Iyer and Wald proposed a generalization for dynamical horizons. This throws up the question whether there is a relation between these EE functionals and the Noether charge, and whether we can derive them using the approach of Iyer and Wald. For a certain class of theories I have shown that there exists a relation between these two [28] but a more rigorous proof is needed. This somewhat firms up the area-entropy relation for arbitrary surfaces and proves the existence of holographic EE functionals for higher curvature theories thereby extending the applicability of Iyer-Wald formalism beyond the bifurcation surface. Apart from this, it is well known that there exist several measures of quantum entanglement, each satisfying a variety of mathematical inequalities and conditions [5]. Translating these into the language of holography constrains the dual gravity theory and will lead to general statements about the consistency of the theory. In this thesis I have discussed one such measure namely Relative entropy [29], the positivity of which has led to constraints on the underlying gravity theory [30]. Also entanglement entropy is a very useful tools for probing renormalization group (RG) flow from the holographic point of view [34, 31, 32, 35, 33]. We end with exploring the concept of renormalized entanglement entropy [36, 37] and its application in probing RG flow in the context of N = 2 gauged supergravity [38].

References

[1] J. M. Maldacena, “The large N limit of superconformal ﬁeld theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998), [arXiv:hep-th/9711200] [2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B428 (1998) 105, [arXiv:hep-th/9802109]

[3] E. Witten, “ Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998) 253, [arXiv:hep-th/97802150]

[4] O. Aharony, S. S. Gubser, J. Maldacena H. Ooguri and Y. Oz, ” Large N ﬁeld theories, String theory and gravity,” Phys. Rept, 323 (2000) 183-386, [arXiv:hep-th/ 9905111].

[5] Michael A. Nielsen and Isaac L. Chuang, “ Quantum Computation and Quantum Information”, Cambridge University Press, 23-Oct-2000

[6] J. Eisert, M. Cramer and M. B. Plenio, “Area laws for the entanglement entropy - a review,” Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773 [quant-ph]] and the references there in.

[7] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001].

[8] T. Nishioka, S. Ryu and T. Takayanagi, “Holographic Entanglement Entropy: An Overview,” J. Phys. A 42 (2009) 504008 [arXiv:0905.0932 [hep-th]].

[9] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP 0608 (2006) 045 [hep-th/0605073] and the references there in.

[10] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108].

[11] P. Calabrese and J. L. Cardy, “ Entanglement entropy and conformal ﬁeld theory”, Journal of Physics A: Mathematical and Theoretical, Volume 42, Issue 50, article id. 504005, 36 pp. (2009).

[12] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum ﬁeld theory,” J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152].

[13] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin, “Quantum source of entropy for black holes”, Phys. Rev. D 34, 373

[14] M. Srednicki, “Entropy and area,” Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048].

[15] S. N. Solodukhin, “Entanglement entropy of black holes,” Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712 [hep-th]] and the references there in. [16] E. Bianchi and R. C. Myers, “On the Architecture of Spacetime Geometry,” Class. Quant. Grav. 31 (2014) 21, 214002 [arXiv:1212.5183 [hep-th]].

[17] A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP 1308 (2013) 090 [arXiv:1304.4926 [hep-th]].

[18] L. Y. Hung, R. C. Myers and M. Smolkin, “On Holographic Entanglement Entropy and Higher Curvature Gravity,” JHEP 1104 (2011) 025 [arXiv:1101.5813 [hep-th]].

[19] X. Dong, “Holographic Entanglement Entropy for General Higher Derivative Gravity,” JHEP 1401 (2014) 044 [arXiv:1310.5713 [hep-th], arXiv:1310.5713].

[20] J. Camps, “Generalized entropy and higher derivative Gravity,” JHEP 1403 (2014) 070 [arXiv:1310.6659 [hep-th]].

[21] A. Bhattacharyya, A. Kaviraj and A. Sinha, “Entanglement entropy in higher derivative holography,” JHEP 1308 (2013) 012 [arXiv:1305.6694 [hep-th]].

[22] A. Bhattacharyya and M. Sharma, “On entanglement entropy functionals in higher derivative gravity theories,” JHEP 1410 (2014) 130 [arXiv:1405.3511 [hep-th]].

[23] R. X. Miao and W. z. Guo, “Holographic Entanglement Entropy for the Most General Higher Derivative Gravity,” arXiv:1411.5579 [hep-th].

[24] J. Camps and W. R. Kelly, “Generalized gravitational entropy without replica symmetry,” JHEP 1503 (2015) 061 [arXiv:1412.4093 [hep-th]].

[25] R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038].

[26] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50 (1994) 846 [gr-qc/9403028].

[27] V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes,” Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052].

[28] A. Bhattacharyya, M. Sharma and A. Sinha, “On generalized gravitational entropy, squashed cones and holography,” JHEP 1401 (2014) 021 [arXiv:1308.5748 [hep-th]].

[29] D. D. Blanco, H. Casini, L. Y. Hung and R. C. Myers, “Relative Entropy and Hologra- phy,” JHEP 1308 (2013) 060 [arXiv:1305.3182 [hep-th]]. T. Faulkner, M. Guica, T. Hartman, R. C. Myers and M. Van Raamsdonk, “Gravitation from Entanglement in Holographic CFTs,” JHEP 1403 (2014) 051 [arXiv:1312.7856 [hep-th]].

[30] S. Banerjee, A. Bhattacharyya, A. Kaviraj, K. Sen and A. Sinha, “Constraining gravity using entanglement in AdS/CFT,” JHEP 1405 (2014) 029 [arXiv:1401.5089 [hep-th]].

[31] H. Casini and M. Huerta, “A Finite entanglement entropy and the c-theorem,” Phys. Lett. B 600 (2004) 142 [hep-th/0405111].

[32] H. Casini and M. Huerta, “A c-theorem for the entanglement entropy,” J. Phys. A 40 (2007) 7031 [cond-mat/0610375].

[33] R. C. Myers and A. Sinha, “Holographic c-theorems in arbitrary dimensions,” JHEP 1101 (2011) 125 [arXiv:1011.5819 [hep-th]].

[34] H. Casini and M. Huerta, “On the RG running of the entanglement entropy of a circle,” Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650 [hep-th]].

[35] H. Casini and M. Huerta, “Positivity, entanglement entropy, and minimal surfaces,” JHEP 1211 (2012) 087 [arXiv:1203.4007 [hep-th]].

[36] H. Liu and M. Mezei, “A Reﬁnement of entanglement entropy and the number of degrees of freedom,” JHEP 1304 (2013) 162 [arXiv:1202.2070 [hep-th]].

[37] H. Liu and M. Mezei, “Probing renormalization group ﬂows using entanglement entropy,” JHEP 1401 (2014) 098 [arXiv:1309.6935 [hep-th], arXiv:1309.6935].

[38] A. Bhattacharyya, S. Shajidul Haque and A. Veliz-Osorio, “Renormalized Entanglement Entropy for BPS Black Branes,” Phys. Rev. D 91 (2015) 4, 045026 [arXiv:1412.2568 [hep-th]].

Acknowledgements

First and foremost, I would like to convey my sincere thanks to my advisor, Aninda Sinha for his generous support as well as outstanding guidance during the entire tenure of my doctoral research. Being his first PhD student was always a stimulating experience. I thank him for guiding me throughout my doctoral work and helping me to complete my PhD in just 3 years. Apart from learning a great deal of physics from him, he has helped me a lot to improve my soft skills. I am also thankful to Menika Sharma, Ling-Yan (Janet) Hung, Shajid Haque, Alvaro Veliz Osorio, Vishnu Jejjala, Suresh Nampuri and Dibakar Roychowdhury for useful collaborations and numerous productive discussions which has added a lot towards my understanding of the subject itself. I would also like to thank Rajesh Gopakumar, Jose Edelstein, Axel Kleinschmidt, Johanna Erdmenger, Tadashi Takayanagi, Heng-Yu Chen, Janet Hung, Shamik Banerjee, Vishnu Jejjala for inviting me to give seminars. I am also thankful to them for numerous stimulating discussions in various occasions. I am also thankful to Ashoke Sen, Rob Myers and Joan Camps for valuable discussions. I would also like to thank all the professors of the Department of Physics and the Centre for High Energy Physics for providing beautiful courses. In particular, I would like to thank Prof. Justin David for providing a beautiful course on QFT. I thank the Chairman of CHEP, B Ananthanarayan for striving to maintain a vibrant and simulating atmosphere in the department. I am indeed grateful to my Integrated PhD batchmates and colleagues at the Centre for High Energy Physics for creating friendly and competitive atmosphere. During my stay at IISc, I found various departmental activities like the weekly math-phys meets, journal club sessions, seminars and colloquia etc. as quite stimulating and in particular playing a very crucial role in developing the scientific mind. Finally, I would specially like to thank Apratim and Shouvik for helping me enormously with all the diagrams and the Latex. I thank the Indian Institute of Science for their generous financial support for attending numerous conferences and visiting other research institutes in India and abroad during my tenure. Finally, I’m very thankful to my parents for giving me constant support and never giving up hope on me.

Arpan Bhattacharyya Bangalore, June 2015.

xiii xiv xv

To my parents. xvi xvii

“Somewhere, something incredible is waiting to be known”. – Carl Sagan xviii Contents

1 Introduction1 1.1 Introduction...... 1 1.1.1 Quantum Entanglement...... 3 1.1.2 Entanglement entropy and Holography...... 9

2 Holographic entanglement entropy functionals: A derivation 29 2.1 Introduction...... 29 2.2 Entropy functional for general theories of gravity...... 32 2.3 Test of the entropy functional for R2 theory...... 34 2.3.1 Minimal surface condition from the entropy functional...... 36 2.3.2 Minimal surface condition from the Lewkowycz-Maldacena method. 41 2.3.3 The stress-energy tensor from the brane interpretation...... 49 2.4 Quasi-topological gravity...... 51 2.4.1 The entropy functional...... 52 2.4.2 Universal terms...... 53 2.4.3 Minimal surface condition...... 54 2.5 Discussion...... 55

3 Entanglement entropy from generalized entropy 69 3.1 Introduction...... 69 3.2 Generalized entropy and Feﬀerman-Graham expansion...... 69 3.2.1 Four derivative theory...... 72 3.2.2 New Massive Gravity...... 74 3.2.3 Quasi-Topological Gravity...... 75 3.2.4 α03 IIB supergravity...... 76 3.3 Comment about singularities in the metric...... 76 3.4 Discussion...... 77

4 Connection between entanglement entropy and Wald entropy 81

xix xx CONTENTS

4.1 Introduction...... 81 4.2 Wald Entropy...... 82 4.3 Four derivative theory...... 83 4.3.1 Cylinder...... 83 4.3.2 Sphere...... 84 4.4 Quasi-Topological gravity...... 86 4.5 α03 IIB supergravity...... 87 4.6 Connection with Ryu-Takayanagi...... 87 4.7 Comments on the connection with the Iyer-Wald prescription...... 89 4.8 Universality in Renyi entropy...... 90 4.9 Discussion...... 91

5 Constraining gravity using entanglement entropy 95 5.1 Introduction...... 95 5.2 Smoothness of entangling surface...... 96 5.3 Discussion...... 98

6 Relative entropy 101 6.1 Introduction...... 101 6.2 Relative entropy considerations...... 103 6.3 Relative entropy in Gauss-Bonnet holography...... 109 6.3.1 Linear order calculations...... 111 6.3.2 Quadratic corrections...... 113

6.3.3 Constant Tµν ...... 113 6.3.4 Shockwave background...... 115 6.3.5 Correction from additional operators...... 117 6.4 Relative entropy for an anisotropic plasma...... 118 6.5 Discussion...... 122

7 Coding holographic RG ﬂow using entanglement entropy 131 7.1 Introduction...... 131 7.2 Renormalized Entanglement Entropy...... 132

7.3 BPS black objects in AdS4 ...... 134 CONTENTS xxi

7.4 REE for BPS black branes...... 136 7.5 Discussion...... 138

8 Conclusions 145 xxii 1 Introduction

1.1 Introduction

The concept of entanglement is a very old one and dates back to early 1930’s when quantum mechanics was born. One of the measures of quantum entanglement is the entanglement entropy. Although it has played a crucial role in understanding some aspects of quantum mechanics, its application remains rather limited mainly due to its non local behaviour until late 90’s.

In early 1970’s Bekenstein proposed that the black hole entropy (SBH ) follows an area law [1]. A SBH = (1.1) 4GN where A is the area of the horizon and GN is the Newton constant. This formula is quite counter-intuitive as entropy is usually an extrinsic quantity, depends on the volume of the system. Later Stephen Hawking showed that a black hole emits radiation with a well defined temperature [2], thereby establishing the concept of black hole entropy. It was observed that the calculation of the entropy associated with the radiation emitted from the black hole is plagued by the presence of ultraviolet divergences. These divergences can be associated with the particles close to the horizon [3] and one has to regulate them to get a finite answer for the entropy. Then in early 1980’s Bombelli, Koul, Lee and Sorkin in their seminal work [4] showed that one can possibly understand that black hole entropy using the concept of ‘entanglement’. To the observers outside the black hole horizon there is no information about the spacetime inside the horizon. They considered scalar fields in the black hole background and traced out the spacetime inside the horizon, thereby defining a “reduced density” matrix for the system. Using this, they computed the von-Neumann entropy and it was shown that the entropy follows an area law. Later it was generalized by Srednicki [5] for massless scalar fields in flat spacetime. He also showed that if one divides the spacetime in two parts, then the entropy associate with the reduce density matrix for one of these two parts is proportional to the area of the boundary between these two halves. Later this concept of “ entanglement entropy”(EE) was made concrete by Callan, Holzhey, Larsen and Wilczek [6] and separately by Susskind and Uglum [7]. From their work it is evident that the EE exhibits a universal

1 2 1.1. INTRODUCTION behaviour, it goes as a logarithm of correlation length in 1 + 1 dimensions after suitably regulating the ultraviolet divergences. But still its application remain rather limited as in general it is very hard to compute this quantity for generic field theories. But in early 2000, Calabrese and Cardy used the “ Replica Trick” formulated by Callan, Holzey, Larson and Wilczek successfully to compute the EE for many cases in the context of 1+1 dimensional conformal field theory (CFT) [8], thereby increasing its physical importance. After that EE has been calculated extensively not only for 1+1 dimensional CFT but also for various other simple quantum field theories both analytically and numerically. Also in recent times it has been successfully computed numerically using the lattice technique for quantum many body systems [9]. Although the techniques employed for computing EE is very hard and yet to be developed fully, still in recent times, we have lots of data regarding quantum entanglement and EE coming from both analytical and numerical approaches [10, 11]. Recently EE has found many applications in various branches of physics like, quantum information, Figure 1.1: Entanglement and its diverese applications condensed matter system, statis- in physics. (Picture courtesy- From the talk given by tical physics and in AdS/ CFT Prof. Robert. C. Myers in the Conference “Entangle- [9, 10, 11, 12, 13, 14]. It serves as an ment from gravity”, ICTS, 2014, Bangalore, India) intersecting point between various subfields of physics. In recent times it has played a crucial role in understanding the nature of holography (AdS/ CFT correspondence). AdS/ CFT correspondence, commonly known as “ Holography” is one of the most important dualities in physics. It postulates that gravity emerges from a certain class of field theories. It is still not known rigorously how holography works from first principles. EE has the merit to shed light on this problem as it connects quantum information of the system with geometry. There exists an interesting connection between geometry and EE and our goal in this thesis will be to explore some aspects of this connection. Recently apart from EE, CHAPTER 1. INTRODUCTION 3 many different tools of quantum entanglement like, entanglement negativity, differential entropy, quantum error coding, relative entropy, cMERA (continuous multiscale entanglement renormalization ansatz) etc, have been used in an attempt to build geometry from the field theory data, is some sense trying to prove the holographic principle [15, 16, 17, 18, 19, 20]. But still EE plays the central character in this program and we will use this to learn many lessons of gravity in the holographic set up.

1.1.1 Quantum Entanglement

1 Let us ﬁrst consider a quantum mechanical system consists of two spin 2 particles. We denote the total Hilbert space by H . Now HA and HB denote the Hilbert spaces of the two individual particles . Also,

H = HA ⊗ HB. Now consider the following state belonging to H, 1 |ψ1 >= √ (| ↑ + ↓>)A ⊗ (| ↓ + ↑>)B . (1.2) 2 This state is not an entangled state as there is no correlation between the two particles in this state. On the other hand, if we consider the following state 1 |ψ2 >= √ (| ↑A↓B> −| ↑B↓A>), (1.3) 2 then one cannot factorize this state in terms of the individual particle states. So this state is an example of “ entangled state”. In other words, entanglement is a property of a quantum mechanical system that tells us, that one cannot describe the underlying pair of particles belonging to this particular state independently. All the physical properties of the two individual particles are correlated with each other. This argument can be extended for any number of particles and interesting things shows up when one consider many body systems due to the non local nature of the entanglement. One of the measures of entanglement is the entanglement entropy (EE). Let us first see how we can define this. First step is to define a “ density matrix” for the full system. If |ψ > is the wavefunction characterizing the total system then the density matrix can be defined in the following way, ρ = |ψ >< ψ|. (1.4) Next step is to define the “ reduced density matrix”. Suppose we want to compute the EE for the sub-system A. Then we first trace out the degrees of freedom corresponding the system B.

ρA = T rBρ =< b|ρ|b >, (1.5) 4 1.1. INTRODUCTION

where ρA is the reduced density matrix for the subsystem A. Now we deﬁne EE by deﬁning the “von-Neumann entropy” which is,

SEE(A) = −T rAρA ln ρA . (1.6)

Here we have completed the trace over the subsystem A. Now this is a good time to cook up some examples and elucidate the process of computing EE. First consider the state as mentioned in Eq. (1.2). We trace out B and the corresponding reduced density matrix is , ! 1 1 1 ρA = . (1.7) 2 1 1

Now to compute the EE as defined in Eq. (1.6) one first diagonalize this and find the eigenvalues. In terms of the eigenvalues Eq. (1.6) becomes

2 X SEE = − λi ln λi. (1.8) i=1 This in turn gives,

SEE = 0. (1.9) So the entropy is zero and hence the state is not entangled. Now consider the state as shown in Eq. (1.3). Corresponding reduced density matrix is, ! 1 1 0 ρA = . (1.10) 2 0 1

The entropy is,

SEE = ln 2. (1.11)

It is an entangled state, in fact it is a maximally entangled state. So whenever SEE is nonzero the state is entangled. Now let us consider a more complicated system, a system of two coupled oscillators. This type of systems are considered in [5] and we will review their calculation here to demonstrate the increasing diﬃculty of computing this quantity when one consider quantum many body systems. Let us start by writing the hamiltonian that describes the two coupled oscillators. 1h i H = p2 + p2 + k (x2 + x2) + k (x − x )2 , (1.12) 2 1 2 1 1 2 2 1 2 1 and 2 respectively denote the two oscillators. Then one deﬁnes the canonical coordinate

x1 + x2 x1 + x2 xA = √ , xB = √ . (1.13) 2 2 CHAPTER 1. INTRODUCTION 5

In terms of these coordinates the ground state wave function can be written as

ωAωB 1/4 −(ω x2 +ω x2 )/2 ψ (x , x ) = ( ) e A A B B , (1.14) 0 1 2 π2

1/2 1/2 where ωA = k1 and ωB = (k1 + 2k2) are the two frequencies corresponding to the two normal modes. Now suppose we integrate out the oscillator 2. The reduced density matrix is deﬁned as, Z ∞ 0 0 ∗ ρ(x1, x1) = dx2ψ0(x1, x2)ψ0(x1, x2) . (1.15) −∞ This gives

0 δ 1/2 −α(x2+x02)/2+βx x0 ρ(x , x ) = ( ) e 1 1 1 1 , (1.16) 1 1 π

2 1 (ωA−ωB ) 2ωAωB where β = and δ = α − β = . Then we have to just compute the SEE 4 (ωA+ωB ) (ωA+ωB ) as deﬁned in Eq. (1.6). To do that we have to ﬁnd the eigenvalues of this reduced density matrix. In this case we are fortunate, as one can easily solve this problem and eigenvalues are given as, β β n λ = 1 − . (1.17) n α + (α2 − β2)1/2 α + (α2 − β2)1/2 Then X SEE = − λn ln λn. (1.18) n After performing this sum, which is somewhat tedious, we get

k1 β β β SEE( ) = − ln 1 − 2 2 1/2 − 2 2 1/2 ln 2 2 1/2 . k2 α + (α − β ) α − β + (α − β ) α + (α − β ) (1.19)

Ultimately SEE is just a function of the ratio of k1 and k2. Now the stage is prepared for us to generalize this concept for field theory. In the field theory the problem becomes much more difficult and subtle. One key issue is to factorize the Hilbert space. One way is to discretize the system over a lattice.1 However one can still use the von-Neumann formula as defined in Eq. (1.6), but one has to deal with the ultraviolet divergences that are present in the field theory. As shown in the Fig. (1.2), we can consider a particular region in the field theory denoted by A. To compute the EE for this region we trace out the remaining portion. The system is discretized over the full space. The SEE(A) for the subsystem A is roughly proportional to the number of links cut by the boundary of the region A. So it is telling us that, indeed EE

1 Several ambiguities might enter in the calculation because of the discretization, specially for the gauge theory. But still one can extract a meaningful answer for EE in the ﬁeld theory. 6 1.1. INTRODUCTION

P Figure 1.2: System is discretized on a lattice and H = i Hi is proportional to the area of the boundary dividing the two regions. If the total system is in the pure state, then one can show

SEE(A) = SEE(B), (1.20) where B denotes the remaining portion of the spacetime. From this one can intuitively guess that SEE is proportional to the number of degrees of freedom that live at the boundary between the two regions. Although there is no formal proof of area law, it has been checked for many instances. For almost all the cases when one considers a ground state of a local hamiltonian, one indeed gets the area law. It is more or less robust, although the violation of it has been observed for the excited states and also for non-local hamiltonians [21]. Let us close this section by briefly sketching an argument for the area law. We will follow [5] and consider scalar field theory. To show this let us go back to the oscillator case as almost all the field theory can be described effectively using the coupled oscillators model. To start with let us write down the Hamiltonian, 1 Z H = d3x[π2 + |∇φ(x)|2]. (1.21) 2 Here φ(x) denotes the scalar field and π is the canonical momentum. After this we express this Hamiltonian in terms of the partial wave expansion of the scalar field, Z φlm = x dΩZlmφ(x). (1.22)

Zlm are the spherical harmonics. The above relation stems from the fact that we can expand the scalar ﬁeld in the basis of spherical harmonics. A similar relation can be written for the CHAPTER 1. INTRODUCTION 7 conjugate momentum. We impose the canonical quantization relation,

0 0 [φlm(x), πl0m0 (x )] = iδll0 δmm0 δ(x − x ). (1.23)

In terms of this partial wave, X H = Hlm. (1.24) l,m Now, Z ∞ 1 2 2 φlm 2 l(l + 1) Hlm = dx πlm(x) + x [∂x( )] + 2 φlm(x) . (1.25) 2 0 x x 1 Then we discretize the system. We put it on a lattice with a lattice spacing M . The M plays the role of the uv cutoﬀ. The boundary condition imposed on φlm(x) is such that it vanishes when x ≥ L where L is the length of the box in which the system is placed. Also 1 L = (N + 1) , (1.26) M where N is a integer and this relation shows that the system is discretized. So the Hamiltonian becomes N 2 M X h 1 φlm,j φlm,j+1 l(l + 1) i H = π2 + (j + )2 − + φ2 . (1.27) lm 2 lm 2 j j + 1 j2 lm,j j=1 Now this looks exactly the same as the N coupled oscillators hamiltonian. We can proceed as before extending the result of 2 coupled oscillators. We trace over the ﬁrst n number of sites to obtain the EE. Finally we get X SEE(n, N) = al(n)[− ln al(n) + 1], (1.28) l

n(n+1)(2n+1)2 1 where al(n) = 64l2(l+1)2 + O( l6 ). At this level we are only interested in the leading result in l, as that will give after summing all values of l’s, the area like term. We perform the sum over l numerically. We deﬁne a radius R midway between the outermost point which was traced over and the innermost point which was not as, 1 1 R = (n + ) . (1.29) 2 M Then it can be shown that the leading term of the EE is,

2 2 SEE = 0.30M R . (1.30)

From this it is clearly evident, that the EE corresponding to the ground state wavefunction of a local hamiltonian of the scalar field satisfies the area law. A more intuitive way to understand the area law [22] is to consider a particular entangling region A as shown in Fig. (1.3). For simplicity let us stick to the massless scalar field model 8 1.1. INTRODUCTION

Figure 1.3: Modes straddling the boundary ∂A is responsible for SEE in 3 + 1 dimensions. We expand the scalar ﬁeld in terms of its modes. These modes are quasi-localized and each has an momentum ~k. Now we know, 2π |k| = , (1.31) λ where λ is the usual wavelength. The total number of modes inside the region A is given by,

Z kmax Z kmax V d3k N = dN = 3 , (1.32) kmin kmin (2π)

2π 2π where V is the volume of A. Now kmin = 2R and kmin = . As all the wavelengths are localized inside A, maximum wavelength can atmost be equivalent to 2R and the minimum wavelength is 0. But then kmin will be divergent, hence we have to put a uv cut-oﬀ . From this the necessity of a cut-oﬀ becomes quite clear. Now we count the fraction of the modes which resides at the boundary of A, responsible for the EE.

Z kmax αAdN˜ Ns ≈ , (1.33) kmin V where Ns denotes the number of modes straddling the boundary and it is a fraction of the total number of modes living inside A. α denotes the thickness of the boundary (α << R) and A˜ denotes its area. Now only the 2π mode localized near the boundary is responsible for SEE, so we can approximate α by k . Also d3k = 4πk2dk and αd3k = 8π2kdk. We next perform the integration and the entropy is proportional to Ns upto some phase-space factors. We get,

2πA˜ S ≈ . (1.34) EE 2 So EE is proportional to the area, hence proportional to number of degrees of freedom residing at the boundary between the two region. It can be also shown in the same way that for 2 + 1 dimensions it is proportional to the circumference and in 1 + 1 it goes as logarithm. CHAPTER 1. INTRODUCTION 9

Lastly, SEE satisﬁes one more important property, namely strong subadditivity. For a bipartite system it tells us that,

SEE(A) + SEE(B) ≥ SEE(A ∪ B) + SEE(A ∩ B). (1.35)

This result can be extended for any arbitrary number of subsystems. This inequality provides a non trivial constraints on SEE and any consistent holographic proposal should pass this test.

1.1.2 Entanglement entropy and Holography

In this section we will review various facts about holographic entanglement entropy. The main goal of this thesis is to understand the connection between EE and geometry, thereby learning important lessons about underlying gravity theory in the holographic set up. By holography we will mean AdS/ CFT correspondence. Two main character of this play is Anti-de Sitter space (AdS) and conformal ﬁeld theory (CFT). So before proceeding further let us brieﬂy comment on the structure of the conformal group and AdS spacetime [23].

Conformal group and structure of AdS

Conformal isometries keep the metric invariant upto a scale transformation. The conformal transformations form a group by themselves. Poincar´e group comes as a subgroup under the broad structure of the conformal group. The conformal transformation preserves the angle between the two curves. These transformations consist of the following four kinds of transformations.

Translation→ x0µ = xµ + aµ.

0µ µ ν Lorentz → x = Rν x , where inﬁnitesimal matrix Rµν is antisymmetric.

Dilatation → x0µ = cxµ.

0µ xµ−cµx2 Special Conformal Transformation(SCT) → x = 1−2 c.x+c2x2 .

For SCT the conformal factor is (1−2 c.x+c2x2)2. SCT is nothing but a translation preceded and followed by an inversion. The corresponding generators for the inﬁnitesimal transformations are listed below. For a generic ﬁeld

Translation(P µ) → −i∂µ.

Rotation (J µν)→i(xµ∂ν − xν∂µ) + Sµν. 10 1.1. INTRODUCTION

Dilatation(D)→−i(d + (x.∂)).

µ µ ν µν 2 µ µν SCT(K )→−i((2x x − 2g x )∂ν + 2d.x ) + 2xνS ,

where Sµν is an anti-symmetric spin matrix for a given field and satisfies the Lorentz algebra. d is a real number that depends on the nature of the fields that are present in the underlying theory.2 These generators satisfy the following commutation relations among themselves. [D,D] = 0 , [P a,P b] = 0 , [D,P a] = iP a, [J ab,P c] = −i(gacP b − gbcP a) , [J ab,J cd] = −i(gadJ bc + gbcJ ad − gacJ bd − gbdJ ac) , (1.36) [J ab,D] = 0 , [D,Ka] = −iKa , [iKa,Kb] = 0 , [Ka,P b] = 2i(gabD − J ab) . For example, in 2 + 1 dimensional flat spacetime we have the following 10 conformal generators,

J1 = ∂a = iPa ,

J2 = xb∂a − xa∂b = −iJab , (1.37) a J3 = −(x ∂a) = −iD , d d J4 = (2xa(x ∂d) − (x xd)∂a) = iKa. a, b runs from 1 to 3. These generators satisfy the usual conformal commutation rules. Now we will see what are the corresponding isometry generators of AdS4. We ﬁrst write the AdS4 metric in poincare coordinates. This is the coordinate system we will often use throughout this thesis. L2(dz2 + dx2 + dy2 + dt2) ds2 = . (1.38) z2 t denotes the Euclidean time. Then we do the following substitution L r = . (1.39) z It gives L2 r2 ds2 = dr2 + (dt2 + dx2 + dy2). (1.40) r2 L2 2 3 e.g for Fermion d = 2 and for Boson d = 1 . CHAPTER 1. INTRODUCTION 11

Next we list all the 10 generators.

J1 = ∂t ,

J2 = ∂x ,

J3 = ∂y ,

J4 = x∂t − t∂x ,

J5 = y∂x − x∂y ,

J6 = t∂y − y∂t , (1.41)

J7 = r∂r − t∂t − x∂x − y∂y , 1 J = rt∂ − t2∂ − tx∂ − ty∂ , 8 r 2 t x y 1 J = rx∂ − tx∂ − x2∂ − xy∂ , 9 r t 2 x y 1 J = ry∂ − ty∂ − xy∂ − y2∂ . 10 r t x 2 y We make suitable identiﬁcations and t → it, such that the generators satisfy the usual

SO(3, 2) algebra [Jab,Jcd] = i[gadJbc +gbcJad −gacJbd −gbdJac] , where a, b, c, d ∈ {0, 1, 2, 3, 4}, So basically they satisfy the same algebra as the CFT generators in one lower dimensions.

AdS/CFT

Now we describe what exactly this correspondence is. There are many dualities that exist in the physics [24]. Among them AdS/CFT connects a strongly coupled field theory with a weakly coupled gravity in Anti-de Sitter (AdS) space time [25]. It is a strong weak duality. It has been observed that there exists an equivalence between a strongly coupled N = 4 5 supersymmetric SU(N) Yang-Mills (SYM) theory and Type IIB string theory on AdS5 × S in the large N limit. Now consider a stack of N D3-branes. Open strings describe the excitations of the D3-branes and the low energy dynamics is governed by N = 4 SYM gauge 2 theory. For this theory one can define a ’t-Hooft coupling λ = gYM N = gsN. We can do a perturbation theory when λ << 1 (also gs << 1). On the other hand we have closed string excitations in the vacuum. This gives rise to the gravity multiplate in 10 dimensions, low energy description of which is effectively given by Type IIB supergravity. One can construct a metric solution for this theory for which the near horizon geometry looks like,

r2 dr 2 0 √ 2 2 2 2 p p 2 ds = α [ (−dt + dx1 + dx2 + dx3) + 4πgsN 2 + 4πgsNdΩ5] (1.42) 4πgsN r

We have assumed that α0 → 0 so that we can neglect stringy eﬀects and work in the su- 2 0√ pergravity regime. Identifying L = α 4πgsN, where L is the AdS radius we can see that 12 1.1. INTRODUCTION

√ 0 metric deﬁned in Eq. (1.42) describes AdS5 × S5 geometry. Also string length ls = α and this description is valid when,

L4 = 4πgsN >> 1. (1.43) ls

This means that classical gravity description is valid when the AdS length scale is much bigger than the string length and one can use this supergravity language when the ’t-Hooft 2 coupling becomes very large. Also we know gYM N = 4πgsN. All these things point to the fact that we have a classical gravity description when L >> ls in the bulk and it is equivalent to a strongly coupled gauge theory at the boundary. Although this conjecture has not been proved yet, it passes several important checks, for eg, matching of the spectrum of chiral operators, correlation functions, supersymmetric indices etc. We obtain a precise dictionary between field theory correlators and correlators of fields living inside the AdS space time. One example is that, currents in the conformal field theory (CFT) side correspond to a gauge field living inside the bulk spacetime.

One can easily see that the isometry group of AdS5 is SO(4, 2) and isometry group for S5 is SO(6). On the other hand the gauge theory remains invariant under the action of SO(4, 2) conformal group and also possess an SO(6) R-symmetry. So we have obtained a geometric realization of the ﬁeld theory degrees of freedom. Based on this, one can study systems described by strongly coupled ﬁeld theories by using equivalent classical gravity description. Holography is being used to study hydrodynamic transports of quark gluon plasma, phase transitions in condensed matter systems etc [26]. 3

Holographic entanglement entropy

Although there exist several evidences supporting holographic principle, but it is still not clear how gravity emerges from ﬁeld theory. To understand this several tools have been employed, EE is one of them. In AdS/CFT set up we will investigate EE and will see that it will provide us with a nice geometrical problem. We will see that we can extract important information about the underlying geometry, hence the gravity theory using this quantity. Now as the AdS/CFT is a two way street, let us start by reviewing some basics about EE in the CFT side of the story.

3 3 An analogous duality has been observed between the near horizon geometry of AdS3 × S × M and that of the low energy description on the branes in D1-D5 system in terms of 1+1 dimensional CFT. Also in recent times holographic principles are being applied for other spacetimes, for eg, Lifshitz, de-Sitter etc [27]. CHAPTER 1. INTRODUCTION 13

Entanglement entropy in CFT

Most general structure of EE for a CFT in even spacetime dimensions is , Rd−2 R S = c + ··· + c ln + ··· (1.44) EE 1 d−2 2 First term is the “ area term”. R denotes the radius of the entangling surface i.e the surface for which the EE is computed. is the uv-cutoff. In even dimensions one gets a “logarithmic term” known as universal term in EE. The coefficient proportional to this term is cut-off independent and carries the information of the central charges of the underlying CFT. So this term is also known as “ universal term” and we will be interested in computing this term throughout this thesis for various theories of gravity. Also we will be considering time independent scenarios and choose a particular time slice t = 0. Let us take an example. In d = 4 dimensions, c2 takes the following form [28], Z Z 1 c = A d2xR + C d2x(W abcdh h − K2 + K Ksab). (1.45) 2 ac bd s 2 sab We have chosen t = 0 slice and rest of the 3 dimensional space is divided into two halves. So the boundary between the entangling region and the rest of the space is a two dimensional space. We will call it a “codimension-2” surface. The integration defined in Eq. (1.45) is essentially over this boundary. c2 depends on the geometrical property of this codimension-2 surface. R is the Ricci scalar, Wabcd is the Weyl tensor and Ksab is the extrinsic curvature of this surface. hac is the projection operator to the surface from the 4 dimensional spacetime.

s hab = ηab − nanbs. (1.46) s denotes the two transverse directions and a, b are the surface indices. The extrinsic curvature can be defined as, α β µ ν Ksab = ea eb hαhβ∇µnsν. (1.47) It has also an index (s) corresponding to the two transverse directions and two normals are µ α defined for that. Ks is the trace of Ksab. hα is the bulk to surface projection operator and ea is the tangent vector 4. A and C are related to the two anomaly coefficient that are present for the 4 dimensional CFT. A C A = ,C = . (1.48) 16π2 16π2 A is known as Euler anomaly and C is know as Weyl anomaly. They show up in the non vanishing part of trace of the stress energy tensor. C A < T i >= W 2 − E (1.49) i 16π2 16π2 4

4 α ∂Xα ea = ∂Xa , where µ and a denote respectively bulk and surface indices. 14 1.1. INTRODUCTION

ijkl ij 2 2 where, E4 = RijklR − 4RijR + R is the 4 dimensional Euler tensor and W is the square of 4 dimensional Weyl tensor. i, j, k, l denote the 4 dimensional indices. In general to compute EE form ﬁeld theory one uses “ Replica Trick” [6,8]. For that,

ﬁrst step is to deﬁne Renyi-entropy (Sn), 1 S = − ln tr ρn . (1.50) n n − 1 A A

Then one has to evaluate this quantity on a “ Replica space ”. First we have to write down the Figure 1.4: Path integral formulation of reduced density matrix in path integral formalism density matrix (“Aspects of holographic [6,8]. For simplicity consider a 1 + 1 dimensional entanglement entropy”, S. Ryu and space. We will closely follow the notations and con- T. Takayanagi, arxiv:-:hep-th/0605073.) ventions of [13]. We choose a tE = 0 slice and consider an interval (A) as our entangling region as shown in the figure. We denote all the dynamical fields that are present collectively as φ(tE, x) where tE is the Euclidean time. We also impose the following boundary condition on the fields living inside A.

φ0(tE = 0+, x) = φ+(x). (1.51) and for other ﬁelds living outside A we demand that, 0 φ0(x) = φ (x). (1.52) The ground state wave function can be written as,

Z φ0(tE =0+,x)=φ+(x) −S(φ(x)) Ψ(φ0(x)) = Dφe . (1.53) tE =−∞ Complex conjugate of this is deﬁned as,

Z tE =∞ ∗ −S(φ(x)) Ψ(φ0(x)) = Dφe . (1.54) φ0(tE =0−,x)=φ−(x) From this one can construct the density matrix easily,

Z tE =∞ 1 −S(φ(x)) ρAφ+φ− = Dφe Πx ∈Aδ(φ(tE+ = 0, x − φ+(x))δ(φ(tE− = 0, x − φ−(x)). Z1 tE =−∞ (1.55) CHAPTER 1. INTRODUCTION 15

We have taken care of the fact, that the boundary conditions imposed are obeyed by inserting the delta functions. Then to evaluate the n’th Renyi-entropy we have to ﬁnd the product of the n copies of this density matrix glued with each other by making suitable identiﬁcations which we do in the next step.

n T rAρA = ρAφ1+φ1− ··· ρAφn+φn− . (1.56)

Also we have imposed φi− = φi+1,+ for all i = 1, ··· n. So this quantity gives rise to the “Replica space ” which is similar to an n sheeted Riemann surface and has a discrete Z(n) symmetry coming from permuting the n copies of the replica. So ﬁnally we have to compute the path integral on this replicated manifold which is denoted by Zn. Z n −n −S(φ) Zn T rAρ = (Z1) Dφe = n . (1.57) replicaspace (Z1) This is an important formula for computing EE from ﬁeld theory using the replica trick. We will see in the later sections, what is its implication in the context of holography. Replica method has been employed successfully for computing EE for 1 + 1 dimensional CFT ’s [8], but for higher dimensions it is hard to apply this method [29].

Entanglement entropy in Holography

Now let us turn our attention to the holography, main hero of our story. Importance of EE in this context is profound. We will see that it geometrize the problem in the bulk space time. Before going into the details let us take an example, which will show the connection between EE and geometry. We start by drawing an analogy with the quantum mechanics. We consider CFTs on two spatially disconnected regions [30]. Next we consider a wavefunction for this system of the form,

|Ψ >= |ΨA > ×|ΨB > (1.58) where A and B denote the individual wavefunctions of the two CFTs. So it is evident that the state is not an entangled state as it is written as a direct product of two states. This kind of state in holography corresponds to a disconnected geometry. Now let us consider two d disconnected CFTs placed on S .Ei is the energy corresponding to the ith eigenstate. Now let us consider the following state,

X − βEi |ψ >= e 2 |Ei × |Ei > . (1.59) i This state is not a direct product state. So it is an “ entangled state”. From holography we know that this corresponds to a thermoﬁeld double state and the dual geometry is an eternal 16 1.1. INTRODUCTION

Figure 1.5: On the left hand side, we have shown a thermoﬁeld double state and its holographic dual eternal black hole is shown on the right hand side. (Picture courtesy- “ Building up spacetime with quantum entanglement ” by Mark Van Raamsdonk, arXiv:-1005.3035) black hole [31]. So this shows that quantum superposition of two states of two classically disconnected CFTs corresponds to a classically connected geometry. This analogy makes things more interesting as one can possibly hope to understand geometry using EE. In the context of AdS/CFT one can ask, whether it is possible to associate a concept of entropy for any arbitrary region of ﬁeld theory sitting at the boundary of AdS? If so, then the next obvious thing is to ask which portion of the bulk spacetime capture that information? The answer comes in the form of Ryu-Takayanagi (RT) proposal for Einstein gravity [32]. The proposal is very simple, it says that one can attach a notion of entanglement entropy for any arbitrary region at the boundary of AdS at a constant time slice and the corresponding entropy is given by the area of some special codimension 2 surface (γA) inside the bulk spacetime. So consider a d dimensional bulk space time. Then the EE associated with a region A at the boundary is given by, 2π SEE(A) = d−2 Area(γA). (1.60) `p

`p is the planck length. To remind ourselves , this proposal is made for Einstein gravity. Z 1 d+1 S = − d−1 d x[R − 2Λ]. (1.61) 2`p The AdS metric is a solution of the action mentioned in Eq. (1.61) with a negative cosmological constant d(d − 1) Λ = − . 2L2 CHAPTER 1. INTRODUCTION 17

Now let us demystify this proposal and see how it works. Let us for simplicity Figure 1.6: holographic prescription for EE: consider a AdS5 metric and a spherical re- γA is the extremal surface gion (A) at the boundary as shown in the Fig. (1.6).

L˜2 ds2 = (dz2+dt2+dr2+r2dθ2+r2 sin(θ)2dφ2). z2 (1.62) z is the radial coordinate of the AdS space. We consider a constant time slice and set the Euclidean time t = 0. Now to evaluate EE associated with the region A we will employ RT method which tells us to ﬁnd a particular surface (γA) extending inside the bulk spacetime which minimizes the area en- closed. Area of that particular surface will give us the EE and γA is known as the “ minimal surface”. To elucidate this further, we put t = 0 and r = f(z) in the (1.62) to obtain an induce metric for the minimal surface. Then we evaluate the area for this. 8π2L˜3 Z f(z)2p1 + f 0(z)2 SEE = 3 dz 3 . (1.63) `p z

We minimize SEE deﬁned in Eq. (1.63), thereby obtaining an equation for f(z). Solving that equation with the following boundary condition ,

f(z = 0) = f 0(z = 0) = 0, (1.64) where the prime denotes the derivative with respect to z, we get 5 √ f(z) = R2 − z2. (1.65)

R is the radius of the spherical region (A) at the boundary. We plug this into Eq. (1.63) and expand the resulting expression around z=0 which is divergent. We introduce a uv-cutoﬀ and ﬁnally we get, 4π2L˜3 hR2 R i SEE = 3 2 − ln( ) . (1.66) `p The leading term follows the area law and also we get an universal term. One can consider more general surfaces and still get the expected results for the universal term (1.45). RT proposal passes several consistency checks, it produces correct universal

5For any generic entangling surface, the extremization condition in the context of Einstein gravity can be written as Ks = 0 where, Ks is the trace of the extrinsic curvature of this codimension-2 surface. 18 1.1. INTRODUCTION

term for EE, which in the context of AdS3/CF T2 reproduces many know results coming from CFT computations [13]. In higher dimensions all these holographic results give us some insights about the field theory. Also RT method has been successfully applied for computing EE in excited state [33, 34], for time dependent cases (covariant RT proposal [35]) and for many other interesting cases 6. It also satisfies the strong subadditivity relation [36]. More importantly it has provide us with a geometrical interpretation of boundary data which is the first step towards understanding how gravity (geometry) emerges form the field theory. We will also consider various higher derivative corrections to Einstein gravity. As we are considering effective theories of gravity, these corrections naturally arise as we integrate out higher momentum modes in Wilsonian RG flow. These higher derivative corrections describe holographic duals of field theory with finite ’t-Hooft coupling ( they correspond to QFTs with 1 finite number of colour) and can arise either as N corrections or when one considers string loop corrections [38]. But these terms take the AdS/CFT description beyond supergravity limit, thereby providing us with a good platform to understand the effect of finite coupling on the underlying field theory. We will explore EE in this context of higher derivative gravity, hoping to understand some important lessons about these effective theories. For eg, one can add an Gauss-Bonnet term to the Einstein Lagrangian [39, 40]. Z 1 5 h 2 µναβ αβ 2 i SEGB = − 3 d x R − 2Λ + λL (RµναβR − 4RαβR + R ) (1.67) 2`p where λ is the Gauss -Bonnet coupling. AdS is still a solution for this theory provided the following relations hold, L = f 2 (1.68) L˜ ∞ and 2 7 1 − f∞ + λf∞ = 0. (1.69) To compute EE for this theory one has to start with a suitbale entropy functional. For this case we have famous Jacobson-Myers functional (JM) [41]. Z √ 2π 3 h 2 i SJM = 3 d x h 1 + λL R . (1.70) `p √ h is the induce metric for the extremal surface and R is Ricci scalar of the extremal surface. Now we will consider two different type of entangling region - sphere and cylinder. We know that there are two types of anomaly coefficients in 3 + 1 dimensional CFT. For generic higher curvature theories, unlike Einstein gravity, these two anomaly coefficients are

6Interested readers are referred to this thesis [37] 7We choose the particular root which smoothly goes to one when λ → 0. CHAPTER 1. INTRODUCTION 19 numerically different from each other. From Eq. (1.45) it can be shown that, for a cylindrical entangling surface the first term of Eq. (1.45) vanishes and one picks out the contribution of Weyl anomaly in the universal part of the EE. On the other hand if one chooses a spherical entangling surface the second term in Eq. (1.45) vanishes and one picks out the contribution of Euler anomaly in the universal part. So we will consider these two different types of entangling surface to explore the nature of the universal terms in the context of the higher derivative theories. Extremizing the JM functional we get the following results for the universal parts [41]. CH R R Scylinder = − ln( ) ,Ssphere = −4A ln( ), (1.71) EE 2R EE where, C = π2L3 (1 − 2f λ) and A = π2L3 (1 − 6f λ) are respectively Weyl and Euler 3/2 3 ∞ 3/2 3 ∞ f∞ `p f∞ `p anomaly. So JM functional produces the expected universal terms. For any arbitrary higher curvature theories one can construct an entropy functional which produces the correct universal part. But the question remains whether one can derive them or not. We will try to formulate such a derivation and that will teach us some important lessons about underlying gravity theories. So we have now introduced all the characters of the story. We are now ready to explore the interplay between gravity and entanglement and hoping to uncover some interesting physics.

Summary of the thesis

Before ending this section let us summarize the key points of the thesis at this stage.

1. In Chapter 2 we will try to derive the proposed entropy functionals for general theories of gravity using the generalized gravitational entropy method proposed in [42]. We will show that it is possible to derive the entropy functional only for a certain class of gravity theories and discuss the implications of that.

2. In Chapter 3 we will demonstrate how to compute the universal terms of EE using the generalized entropy method.

3. Identifying EE with the generalized entropy opens up the possibility of connecting EE with the black hole entropy. For general theories of gravity black hole entropy is given by the Wald formula [43]. We will explore the possibility of connecting EE with the Wald entropy in Chapter 4, thereby opening up the possibility of deriving EE by Noether charge method.

4. In Chapter 5 we will see how put constraints on the couplings of the higher derivative terms using extremal surfaces. 20 REFERENCES

5. In Chapter 6 we will took a small step towards understanding how bulk equation of motion arises from EE, which will give us some intuition about the emergence of gravity from ﬁeld theory. We will use the concept of relative entropy to understand this, which is nothing but the change of entanglement entropy between vacuum and excited states. Using relative entropy we will demonstrate how one can constrain underlying gravity theory.

6. After that we will discuss brieﬂy how to code holographic RG ﬂow using EE in the Chapter 7 and end with summarizing the main results and some open questions.

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2.1 Introduction

It is has been observed that there exists a striking similarity between black hole entropy and entanglement entropy [1]. In the context of AdS/CFT, the entanglement entropy1 for a boundary ﬁeld theory which is dual to Einstein gravity can be calculated using the well- known Ryu-Takayanagi proposal (RT) [18, 19]. This proposal states that the entanglement entropy SEE of any region on the boundary of AdS can be calculated by evaluating the area of a minimal surface in the bulk which is homologous to this boundary region: Area SEE = . (2.1) 4GN Building upon earlier attempts [20, 21, 22, 23], this proposal was recently proved in [24], for a general entangling surface. The entanglement entropy formula in Eq. (2.1) is of the same form as the formula for calculating the entropy of a black hole. In the black hole case, there exists a simple generalization of this area law for calculating the entropy of a black hole in any general higher-derivative gravity theory, known as the Wald entropy [25, 26, 27]. It is natural to ask then if one can generalize the Ryu-Takayanagi prescription to higher-derivative gravity theories by simply replacing the RHS of Eq. (2.1) with the Wald entropy. However, this is known not to be the case [28, 29]. Recently, a general formula for calculating the holographic entanglement entropy (HEE) in higher-derivative gravity theories was proposed in [30, 31]. It was also conjectured that the minimal entangling surface can be determined by interpreting this formula as the entropy functional for the higher derivative gravity theory and extremizing it. At present there exists no general proof of this proposal. Main objective of this chapter is to carry out various tests to determine the validity of this conjecture and formulate a general proof. Lewkowycz and Maldacena (LM) [24] have proposed a derivation of the Ryu-Takayangi (RT) prescription [18] for computing entanglement entropy (EE) [2] in holography [19]. A

1There exists a huge literature on entanglement entropy. For background and interesting applications see [2,3,4,5,6,7,8,9, 10, 11, 12, 13, 14, 15, 16, 17].

29 30 2.1. INTRODUCTION generalization of black hole entropy is proposed in the context where there is no U(1) symmetry in the bulk. In the Euclidean theory, although there is no U(1) symmetry, one imposes a periodicity condition of 2πn with n being an integer on the Euclidean time direction at the boundary. This time direction shrinks to zero at the boundary. By suitably choosing boundary conditions on the fields, LM propose to identify the on-shell Euclidean action with a generalized gravitational entropy. In calculations of entanglement entropy in quantum field theories, one frequently uses the replica trick which entails introducing a conical singularity in the theory 2. An earlier attempt to prove the RT formula was made by Fursaev [20]. In recent times, in the context of AdS3/CFT2 there have been further developments in [8] towards a proof. In the context of holography, this corresponds to taking the n → 1 limit. In this case, LM suggest that the time direction shrinks to zero on a special surface. The equation for this surface is derived in Einstein gravity by showing that there is no singularity in the bulk equations of motion. This surface has vanishing trace of the extrinsic curvature and corresponds to a minimal surface–which is precisely what comes from minimizing the RT area functional. Next we will try to generalize this for various higher derivative gravity theories. We will first work with general four-derivative theory. It is also sufficient for our purpose to consider only four-derivative theory as that will capture all the essential issues that we like to bring up. The conjectured form of the holographic entropy functional for general R2 theory first appeared in [33]. The formula of [30, 31] also reduces to this functional for general R2 theory. For the purpose of this thesis, we will refer to this functional as the FPS (Fursaev-Patrushev-Solodukhin) functional after the authors of the paper where it was first proposed. In [34] it was shown that this entropy functional leads to the expected universal terms in the entanglement entropy for cylindrical and spherical entangling surfaces, so the FPS functional passes this basic first test. The obvious next step is to determine whether the surface equation of motion derived from extremizing this functional is the same as that derived using the generalized gravitational entropy method (which we will refer to as the LM method) of [24]. 2 General R theory depends on three parameters: λ1, λ2 and λ3. Gauss-Bonnet gravity is a special point in this parameter space [35] and the FPS functional reduces to the Jacobson- Myers functional at this point. For Gauss-Bonnet gravity, the question whether the surface equation of motion one gets from the Jacobson-Myers functional matches with the surface equation of motion derived using the LM method was addressed in [34, 36, 37]. We will look

2The only example where a derivation of EE exists without using the replica trick is for the spherical entangling surface [4, 22] although in [32] it has been explained how this procedure is connected with the replica trick. A proposal has been made in [7] for the equation for the entangling surface which does not depend on the replica trick. CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 31 at the Gauss-Bonnet case first to emphasize several interesting points for this theory. For this theory, the surface equation of motion that one gets from the Jacobson-Myers functional matches with what one gets from the LM method, provided that terms cubic in the extrinsic curvature are suppressed. We will find that for general R2 theory using a procedure similar to the Gauss-Bonnet case leads to a match in the leading-order terms on both sides, where we designate terms cubic in the extrinsic curvature as sub-leading. However, as we will show, in the case of R2 theory, the LM method also yields an extra condition that cannot be satisfied at arbitrary points of the parameter space. The conclusion is, therefore, that for a general R2 theory the conditions that follow from the LM method do not correspond exactly to the surface equation of motion derived from the FPS functional. An alternative method to demonstrate that the FPS functional is the correct entropy functional for R2 theory is to show that it can be interpreted as the action of a cosmic brane. This method was employed in [30], where it was referred to as the cosmic brane method. We will re-examine this procedure for R2 theory and show that the result we get is consistent with what we get using the LM method. What happens when we go to a six-derivative gravity theory? In this case, we consider quasi-topological gravity [38] which is again a special point in the parameter space of R3 theories. We first construct the entropy functional for quasi-topological gravity using the formula proposed in [30, 31]. We then show that this functional reproduces the expected universal terms for this theory for the cylindrical and spherical entangling surfaces. This is in agreement with the result of [39] that the entropy functional proposed in [30, 31] leads to the correct universal terms for a general higher-derivative theory. We also find the minimal surface condition for this theory using the LM method and show that it deviates from what is expected from the HEE functional. This chapter is organized as follows. In Sec. (2.2) we review the general entropy functional proposed in [30, 31]. Our main focus in this chapter is general four-derivative gravity theory, for which the entropy functional is the FPS functional. In Sec. (2.3) we find the surface equation of motion for R2 theory by extremizing the FPS functional on the codimension-2 surface. We then compare it with what we obtain using LM prescription. We also make some remarks on the Gauss-Bonnet case. We then investigate the cosmic-brane method of [30]. In Sec. (2.4), we repeat our analysis for quasi-topological gravity. Lastly, in Sec. (2.5) we summarize our findings and discuss their implications. 32 2.2. ENTROPY FUNCTIONAL FOR GENERAL THEORIES OF GRAVITY

2.2 Entropy functional for general theories of gravity

In this section we will review the general entropy formula proposed in [30, 31]. First we summarize the argument leading up to this proposal, following [30]. For details the reader is referred to [24, 30, 31, 33]. Some applications of this entropy formula are in [40].

In field theory, the entanglement entropy SEE = −Tr[ρ log ρ] can be calculated as the n → 1 limit of the Rényi entropy. The Rényi entropy in turn can be computed as 1 S = − (log Z − n log Z ) . (2.2) n n − 1 n 1

Here Zn is the partition function of the field theory on the manifold Mn which is the n- fold cover of the original spacetime manifold M1. In the holographic dual theory one can construct a suitable bulk solution Bn with boundary Mn. The manifold Mn at integer n has a Zn symmetry that cyclically permutes the n replicas. In [24] it was proposed that this replica symmetry extends to the bulk Bn. Orbifolding the bulk by this symmetry results in ˆ a space Bn = Bn/Zn , that is regular except at the fixed points of the Zn action. The fixed points form a codimension 2 surface with a conical defect in the bulk. This is the surface that is ultimately identified with the minimal entangling surface in the n → 1 limit. Further, one can use gauge-gravity duality [41] to identify the field theory partition function on Mn with the on-shell bulk action on Bn in the large-N limit

−S[Bn] Zn ≡ Z[Mn] = e . (2.3)

It is now straightforward to calculate the entanglement entropy. By construction, one can identify ˆ S[Bn] = nS[Bn] (2.4) ˆ ˆ at integer n, where S[Bn] is the classical action for the bulk conﬁguration Bn not including ˆ any contribution from the conical defect. By analytically continuing Bn to non-integer n,

Eq. (2.4) can be used to deﬁne S[Bn]. Using Eqs. (2.2) and (2.3) and expanding around n = 1, one gets n ˆ ˆ SEE = lim S[Bn] − S[B1] = ∂S[Bn] (2.5) n→1 n − 1 =0 ˆ where ≡ n−1. The quantity S[Bn] can be calculated for the bulk theory by writing the bulk metric locally around the surface in gaussian normal coordinates and introducing a conical ˆ defect. It can be shown [24, 30] that ∂S[Bn] gets a contribution entirely from the tip of the cone. To compute it, therefore, one employs a metric regularized at the tip of the cone. This calculation is similar to that employed in [42] for calculating the Wald entropy ˆ from a regularized cone metric. Indeed, evaluating S[Bn] for a bulk theory with the cone metric to linear order in and using Eq. (2.5) will result in two terms. The ﬁrst is SWald: CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 33 the Wald entropy for the theory. However, as was noted in [33], there is a second way for a linear contribution to arise. A term in the bulk lagrangian that is of order 2 can get enhanced to order after integrating over the transverse directions. Following [30], we label the contribution of such terms as SAnomaly. At this point, the calculation of the form of

SEE is basically finished. Eq. (2.5) can be used to find the entanglement entropy for any higher-derivative theory, including ones whose lagrangians involve derivatives of the Riemann tensor. However, for a general higher-derivative theory it can be computationally difficult to compute SAnomaly directly using Eq. (2.5). In [30, 31] a simpler prescription for calculating the holographic entanglement entropy for higher-derivative theories of gravity in d + 1 dimensions, for which the lagrangian (L) contains only contractions of the Riemann tensor, was given. The formula is:

Z √ ( 2 ) 2π ∂L X ∂ L 8KzijKzkl¯ S = ddy h + . (2.6) EE `d ∂R ∂R ∂R q + 1 p zzz¯ z¯ α zizj zk¯ zl¯ α α

The notation used in the above equation and also in the rest of the chapter is as follows: We use Greek Letters µ, ν, ρ, σ, ··· to label the bulk indices. We use the Latin letters a, b, ...... m, n to label the indices of the codimension 2 surface, while reserving the letters p, q, r, s to denote the indices of the transverse directions. In these directions, we use the complex coordinates z andz ¯. The bulk metric is denoted by gµν.The metric on the codimension-2 entangling surface is denoted by hij while the surface itself is denoted by Σ. The bulk Rie- mann tensor is denoted by Rµνρσ while the intrinsic Riemann tensor of the surface is denoted by Rikjl. The extrinsic curvatures of the surface are denoted by Krij, where the ﬁrst index labels the extrinsic curvature in the transverse directions. We follow the curvature conventions in [43]. The ﬁrst term in Eq. (2.6) is the Wald entropy. The second term is the correction to the Wald entropy and is evaluated in the following way: The second derivative of the lagrangian L is a polynomial in components of the Riemann tensor. We expand the components Rpqij,Rpiqj and Rikjl using

˜ k k Rpqij = Rpqij + KpjkKqi − KpikKqj , ˜ k Rpiqj = Rpiqj + KpjkKqi − Qpqij ,

Rikjl = Rikjl + KpilKpjk − KpijKpkl . (2.7)

1 ˜ ˜ Here, Qpqij ≡ 2 ∂p∂qgij|Σ. Rpqij and Rpiqj can also be defined in terms of metric variables, but the exact definition is not needed here. The variable α is used to label the terms in the expansion. For each term labelled by α, qα is defined as the number of Qzzij and Qz¯zij¯ , plus one half of the number of Kpij, Rpqri, and Rpijk. The final step is to sum over α with weights 34 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY

˜ ˜ 1/(1 + qα). The quantities Rpqij, Rpiqj, and Rikjl can then be eliminated using Eq. (2.7), resulting in an expression involving only components of Rµνρσ, Kpij and Qpqij. To yield the entanglement entropy, the formula in Eq. (2.6) should be evaluated on the minimal entangling surface. This surface is supposed to be determined following the LM method. [30, 31] also contain the proposal that the minimal surface can be determined by extremizing SEE as given in Eq. (2.6)— SEE therefore being the entanglement entropy functional for a general theory of gravity. Rest of this chapter is mainly based on the work done with Prof. Aninda Sinha and Dr. Menika Sharma [34, 44].

2.3 Test of the entropy functional for R2 theory

In this section we consider general R2 theory in ﬁve dimensions. The lagrangian for this theory is

L = L1 + L2 , (2.8) where 12 L = R + (2.9) 1 L2 is the usual Einstein-Hilbert lagrangian with the cosmological constant appropriate for ﬁve- dimensional AdS space and

L2 L = λ R Rαβγδ + λ R Rαβ + λ R2 (2.10) 2 2 1 αβγδ 2 αβ 3 is the R2 lagrangian. The proposed entropy functional for this theory is

SEE, R2 = SWald, R2 + SAnomaly, R2 , (2.11) where Z √ 2π 3 L2 ν rµ µ ν rρ sσ SWald, R2 = 3 d x h 1 + 2 (2λ3R + λ2Rµνnr n + 2λ1Rµνρσnr ns n n ) , `P (2.12) Z √ 2π 3 L2 1 r sij and SAnomaly, R2 = 3 d x h 2 − 2 λ2KrK − 2λ1KsijK . (2.13) `P As mentioned earlier, this entropy functional leads to the correct universal terms. To demonstrate this, we will write down ﬁrst the bulk AdS metric as,

L˜2 ds2 = (dz2 + dτ 2 + h dxidxj) (2.14) z2 ij CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 35

˜ where, L is the AdS radius and hij is a three dimensional metric given below. We will use Greek letters for the bulk indices and Latin letters for the three dimensional indices. For the calculation of EE for a spherical entangling surface we write the boundary hij in spherical polar coordinates as, sphere i j 2 2 2 hijdx dx = dρ + ρ dΩ2 , (2.15)

2 2 2 2 where dΩ2 = dθ + sin θdφ is the metric of a unit two-sphere and θ ∈ [0, π] and φ ∈ [0, 2π]. For a cylindrical entangling surface,

cylinder i j 2 2 2 2 hijdx dx = du + dρ + ρ dφ . (2.16) u is the coordinate along the direction of the length of the cylinder. For a cylinder of length H, u ∈ [0,H]. Here L˜ = √L . f∞ We put ρ = f(z), τ = 0 in the metric and minimize (2.11) on this codimension 2 surface and ﬁnd the Euler-Lagrange equation for f(z). Using the solution for f(z) we evaluate (2.11) to get the EE.

p 2 2 For the sphere, we get f(z) = f0 − z which gives the universal log term, f S = −4a ln( 0 ) . (2.17) EE δ

z2 For the cylinder, f(z) = f0 − + ... which gives, 4f0 cH f S = − ln( 0 ) . (2.18) EE 2R δ π2L3 π2L3 a = (1 − 2f∞(λ1 + 2λ2 + 10λ3)) and c = (1 + 2f∞(λ1 − 2λ2 − 10λ3)) . (2.19) 3/2 3 3/2 3 f∞ `P f∞ `P and δ is the UV cut-off comes from the lower limit of the z integral. f0 is the radius of the entangling surface. These are the expected results [28, 35]. In this section, we will further test this entropy functional by determining whether the surface equation of motion one gets from extremizing this functional is the same as the surface equation of motion one gets following the LM method. In Sec. (2.3.1), we extremize the functional for R2 theory. In this particular section, we will first find the surface equation of motion for this functional in a general spacetime background. However, the Ryu-Takayanagi proposal and its generalizations are most precisely formulated in the AdS/CFT context, so eventually we will specialize to the AdS background. In Sec. (2.3.2) we find the surface equation of motion using the LM method. In this case, we will always assume that the bulk is AdS space. Since a variation of the LM method – called the cosmic-brane method – was used in [30] to formulate a proof that the FPS functional is the correct entropy functional for R2 theory, we also investigate this method in Sec. (2.3.3). 36 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY

2.3.1 Minimal surface condition from the entropy functional

To extremize the functional in Eq. (2.11), we follow the methods of [45, 46, 47]. We denote the surface we are going to extremize w.r.t the action in Eq. (2.11) by Σ. The induced metric on Σ is µ ν hij = ei ej gµν, (2.20) µ µ where gµν is the bulk metric and ei ≡ ∂iX are the basis vectors tangent to the surface µ Σ, X being the bulk coordinates. On the surface Σ, the gir component of the bulk metric µ vanishes. The two normals to the surface are denoted by nr where r = 1, 2 are the transverse directions. The metric tensor in the tangent space spanned by the normal vectors (the metric tensor of the normal bundle of the sub-manifold Σ) is the Kronecker delta:

µ ν δrs = nr ns gµν (2.21) We work in Euclidean signature and set = +1. We use the inverse metric δrs, to raise rµ rs µ indices in the normal directions: n = δ ns . Note that, repeated s indices always imply µ νs µ ν µ ν summation over the transverse directions: ns n ≡ n1 n1 + n2 n2. In this notation, the completeness relation relating gµν, the inverse of the bulk metric, to hij, the inverse of the induced metric, is µν ij µ ν µ νs g = h ei ej + ns n . (2.22) The Gauss and Weingarten equations are µ µ ˆµ ρ ν k µ r µ ∇iej = ∂iej + Γνρei ej − Γijek = −Kijnr µ µ ˆµ ρ ν r µ j µ ∇ins = ∂ins + Γρνei ns − Γsinr = Ksiej . (2.23) Here, ∇ is the Van der Waerden-Bortolotti covariant derivative [45] which acts on a general s···r tensor Ti···j as s···r ˜ s···r s p···r r s···p ∇kTi···j = ∇kTi···j + ΓpkTi···j + ··· + ΓpkTi···j , (2.24) where ∇˜ is the usual covariant derivative associated with the surface Christoffel. This ˆµ Christoffel is related to the bulk Christoffel Γσν as i µ ˆµ σ ν i Γjk = (∂jek + Γσνej ek)eµ . (2.25)

r The Chrisoffel Γis is the Christoffel induced in the normal bundle. It is related to the bulk Christoffel as r µ ˆµ σ ν r Γis = (∂ins + Γσνei ns )nµ . (2.26) This Christoffel can be interpreted geometrically as the freedom to perform rotations of the normal frame. It is, therefore, equivalent to a gauge field Ak, commonly referred to as a twist potential. This field is defined as: 1 A ≡ εrs∂ g , so that Γs = δpsε A , (2.27) k 2 r ks jr rp j CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 37

where εrs is the Levi-Civita symbol. The Gauss identity relating the bulk Riemann with all indices projected in the tangential directions with the surface Riemann is

µ ν ρ σ r r Rµνρσek ei el ej = Rkilj − KklKrij + KikKrjl . (2.28)

The Codazzi-Mainardi relation is

µ ν σ ρ ∇kKrij − ∇iKrkj = Rµνρσek ei ej nr . (2.29)

From Eq. (2.28) we get the Gauss-Codazzi identity

νr µ µr νs ρ σ r s ij R = R − 2Rµνn nr + Rµνρσn n nrns + KrK − KijKs . (2.30)

We now consider an inﬁnitesimal variation of the surface Σ given by Xµ −→ Xµ + δXµ. The change δXµ is µ r µ i µ δX = ξ nr + ξ ei . (2.31) where ξr and ξi are small parameters. For deriving the equation describing the minimal surface we are only concerned with the variation in the normal direction, since the tangential variation leads to a constraint equation. The variation then reduces to

µ r µ δX = ξ nr , (2.32)

µ µ The variation δX in the surface will induce a variation in the basis vectors ei . This can be computed by finding the basis vectors at Xµ + δXµ and parallel transporting them back to Xµ. Taking the difference between the parallel-transported quantity and the original basis vector at the coordinate Xµ, using the identities in Eq. (2.23) and then restricting to normal variation results in µ µ s µ j s δei = ns ∇iξ + ej Ksiξ . (2.33) The details of this calculation are in [45]. As stated in Eq. (2.24), the covariant derivative ∇ acts on ξs as s s s r ∇iξ = ∂iξ + Γirξ . (2.34) The variation in any other tensor quantity can be calculated in a similar way, by parallel transporting the quantity at the new coordinate back to the old coordinate and taking the difference. This gives the variation in the bulk metric as zero. We write down the result for other variations. For details the reader is referred to [45]. The variation of the induced metric is

r δhij = 2ξ Krij , √ √ r δ h = ξ hKr . (2.35) 38 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY

The variation of the extrinsic curvature is

s s s k sµ σ ρ ν r δKij = (−∇i∇jδr + KikKrj + Rµνρσn nr ei ej )ξ , s i s s ki ij sµ σ ρ ν r δK = (−∇ ∇iδr − KikKr + h Rµνρσn nr ei ej )ξ . (2.36)

The covariant derivatives ∇ act all the way to the right. Using these variations we can now compute the change in the action. For this we ﬁrst ν rµ µ ν rρ sσ rewrite the Rµνnr n and Rµνρσnr ns n n terms in the action given in Eq. (2.12) as

νr µ ij ν µ Rµνn nr = R − h Rµνei ej µr νs ρ σ ij ν µ ik jl µ ν ρ σ Rµνρσn n nrns = R − 2h Rµνei ej + h h Rµνρσei ej ekel (2.37)

ij ν µ using the completeness relation in Eq. (2.22). The variation of a term such as h Rµνei ej is given by ij ν µ ij ν µ ij ν µ ij ν µ δ(h Rµνei ej ) = (δh )Rµνei ej + 2h Rµνδ(ei )ej + h δ(Rµν)ei ej (2.38) The ﬁrst two variations can be computed using Eqs. (2.35) and (2.33) respectively. For evaluating the last term we need the variation of the bulk Ricci Tensor which is given by

σ ˆ r δ(Rµν) = nr ∇σRµνξ . (2.39)

The variation in the bulk Ricci scalar and Riemann tensor take a similar form. All these variations are under the integral sign in Eq. (2.12) and we perform a integration by parts where needed, discarding the term that is a total variation. Then using the variations given above we obtain 3: √ √ √ s µ ˆ s δ( hR) = h KsR ξ + n h∇µR ξ , √ √ s √ νr µ νr µ s i ν µ s δ( hRµνn n ) = h KsRµνn n ξ + 2 h ∇ (Rµνn e )ξ − r √ r √ s i σ ij µ ν ˆ s µ ˆ s h n h e e ∇σRµνξ + n h ∇µR ξ , √ √ s i j s √ µr νs ρ σ µr νq ρ σ s i µ ν ρ σ jk s δ( hRµνρσn n n n ) = h KsRµνρσn n n n ξ − 4 h ∇ (Rµνρσn e e e h )ξ + r s √ r √q s j i k i ν µ s ik jl µ ν ρ σ α ˆ s 4 h ∇ (Rµνn e )ξ + hh h e e e e n ∇αRµνρσξ − s i √ i j k l s ν ij µ ρ ˆ s µ ˆ s 2 ns h ei ej ∇νRµρξ + hns ∇µRξ . (2.40)

Similarly the variations for the terms present in the action in Eq. (2.13) are √ √ √ √ s i r s r s ij r δ( hK Ks) =−2 h∇i∇ Krξ + hKrK Ksξ − 2 hK KsijK ξ − √ r s ij µ ν ρ σ r 2 hK Rµνρσh n e n e ξ , √ √ r √i s j √ sij ij r sij r si kj r δ( hKsijK ) =−2 h∇i∇jK ξ + hKrKsijK ξ − 2 hKsijK K ξ − √ r k r sij µ ν ρ σ r 2 hK Rµνρσnr ei nsej ξ . (2.41)

3We thank Joan Camps for valuable discussions CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 39

Adding these variations up with the appropriate factors will give us the equation for the minimal surface for the action in Eq. (2.11) in a general spacetime background. As a check of these equations we now demonstrate that the above results lead to the correct surface equation of motion in the Gauss-Bonnet case. For Gauss-Bonnet the entropy functional for general R2 theory reduces to the Jacobson-Myers functional Z √ 2π 3 2 νr µ µr νs ρ σ s sij SJM = 3 d x h{(1+λL (R−2Rµνn nr +Rµνρσn n nrns +KsK −KsijK )} . (2.42) `P This functional is valid in a general space-time background. Note that the integrand is equal √ to h(1 + λL2R) on using the Gauss-Codazzi identity Eq. (2.30). The surface equation of motion for this theory using this form of the functional is [34],

2 ij K + λL (RK − 2R Kij) = 0 . (2.43)

We now ﬁnd the surface equation of motion by directly varying Eq. (2.42). Using the variation equations Eqs. (2.40–2.41) and simplifying using the identities in Eqs. (2.28–2.29) we get √ √ √ s 2h s jk s h Ks ξ + λL h KsR ξ − 2 hRjkKs ξ √ √ ik jl µ ν ρ σ α ˆ s i µ ν ρ σ kj s + hh h e e e e n ∇αRµνρσξ − 2 h ∇ (Rµνρσn e e e h )ξ √ i j k l s √ s j i k r ij µ ν ρ σ s rij µ ν ρ σ s − 2 hK Rµνρσh ns ei nrej ξ + 2 hK Rµνρσns ei nrej ξ √ µ ν ρ σ il jk si + 2 hRµνρσej ei ekel h Ks ξ . (2.44)

The ﬁrst three terms give precisely the equation of motion for Gauss-Bonnet theory. The rest of the terms add up to zero, as we show in the following. We use the Bianchi identity ˆ on the ∇αRµνρσ factor of the fourth term giving ˆ ˆ ˆ ∇αRµνρσ = −∇σRµναρ − ∇ρRµνσα (2.45) and then rewrite each of these terms as

ik jl µ ν ρ σ α ˆ σ ˆ ik jl µ ν ρ α σ ˆ ik jl µ ν ρ α h h ei ej ekel ns ∇σRµναρ = el ∇σ(h h ei ej ekns Rµναρ) − el ∇σ(h h ei ej ekns )Rµναρ . (2.46) The expression in brackets in the ﬁrst term of the R.H.S is a bulk scalar and therefore this term can be written as

ik jl µ ν ρ α i µ ν ρ σ jk r ik jl µ ν ρ α ∂l(h h ei ej ekns Rµναρ) = −∇ (Rµνρσns ej ei ek h ) + Γslh h ei ej eknr Rµναρ+ m ik jl µ ν ρ α Γjl h h ei emekns Rµναρ , (2.47)

Inserting these expressions in Eq. (2.44) after expanding the second term on the R.H.S of Eq. (2.46) and using the identities in Eq. (2.23) will lead to cancellation of all terms except for the terms in the ﬁrst line of Eq. (2.44). 40 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY

AdS background

We now specialize to the case of AdS background which is the background we will use while ﬁnding the equation of motion using the LM method. In AdS space the Riemann tensor is

Rµνρσ = −C(gµρgνσ − gµσgνρ) , (2.48) where we have deﬁned C = f /L2. Here, L is the length scale associated with the cosmo- ∞ √ ˜ ˜ logical constant and is related to the AdS radius L as L = L f∞. The variable f∞ satisﬁes the following equation for R2 theory

1 1 − f + f 2 (λ + 2λ + 10λ ) = 0 . (2.49) ∞ 3 ∞ 1 2 3 √ For ease of comparison with later results, we also rewrite the variation in hR given in Eq. (2.40) using the Gauss-Codazzi relation Eq. (2.30). The minimal surface equation is then

r 2 r ij r 2 r rij K + L {λ3(RK − 2R Kij + 2∇ K − 2∇i∇jK − r ˜ r ij r r r K K2 + 2KijK2 + K K2 − 2K3 − 18CK )+ 1 2 r 1 r ˜ 1 r ij 11 r λ2( 2 ∇ K − 4 K K2 + 2 KijK2 − 2 CK )+ rij r r r λ1(2∇i∇jK − K K2 + 2K3 − 4CK )} = 0 , (2.50)

sij ij sij ˜ s r si rkj where we have defined K2 = KsijK , K2 = KsK , K2 = KsK and K3 = KsijKk K . Note that these are a set of two equations one for each of the extrinsic curvatures K1, K2. In AdS space we can make a further simplification using Eq. (2.29). The R.H.S of this k i j equation disappears on using Eq. (2.48). We then get the identity ∇ ∇kKr = ∇ ∇ Krij on taking a further covariant derivative of the L.H.S. As explained in Appendix 2.5, in the LM method for a time-independent metric, we can set K1 = K2 = K. We, therefore, also drop the r index and Eq. (2.50) simplifies to

2 ij 3 K + L {λ3(RK − 2R Kij − K + 3KK2 − 2K3 − 18CK)+ 1 2 1 3 1 11 λ2( 2 ∇ K − 4 K + 2 KK2 − 2 CK)+ 2 λ1(2∇ K − KK2 + 2K3 − 4CK)} = 0 . (2.51)

We have also veriﬁed this equation by determining the bulk extremal surfaces for diﬀerent types of boundary entangling regions (sphere, cylinder and slab).

For the Gauss-Bonnet case: λ1 = λ, λ2 = −4λ and λ3 = λ, this equation reduces to the known result in Eq. (2.43). Note that terms cubic in the extrinsic curvature as well as the CK terms are not present in that equation. The Gauss-Bonnet case is special in this sense. CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 41

2 No such simpliﬁcation occurs if we set the value for Weyl theory, λ1 = λ, λ2 = −4λ/3 and

λ3 = λ/6:

λL2 K + (RK − 2RijK + 8∇2K + K3 − 7KK + 10K + 2CK) = 0 . (2.52) 6 ij 2 3 The CK term, in particular, stands out. If we trace the provenance of this term, it comes µ ν ρr σq ij µ ν ρ σ from terms of the form KsRµνρσnr nq n n and Ks Rµνρσnr ei nsej in Eqs. (2.40) and (2.41) — terms that have components normal to the surface. Nevertheless, for AdS background these reduce to a term intrinsic to the surface. In fact, using the Gauss-Codazzi identity, Eq. (2.30), we can rewrite this CK term as ∼ K3 + RK. So far, we have only considered normal variations of the surface. Considering tangential variations leads to a constraint equation. For R2 theory this constraint equation is indistin- guishable from the condition in Eq. (2.29) which is the Codazzi-Mainardi relation.

2.3.2 Minimal surface condition from the Lewkowycz-Maldacena method

We will now derive the surface equation for R2 using the LM method. As already mentioned, the main idea of Ref. [24] is that one can obtain the minimal surface condition by extending the replica trick to the bulk. The bulk will then have a Zn symmetry. Orbifolding by this symmetry will lead to a spacetime in which the fixed points form a codimension-2 surface with a conical deficit. In the n → 1 limit this surface can be identified with the entangling surface. The metric of this surface can be parametrized in gaussian normal coordinates as follows:

2 2ρ(z,z¯) 2ρ(z,z¯) 2 r r s i j ds = e {dzdz¯ + e Ω(¯zdz − zdz¯) } + (gij + Krijx + Qrsijx x )dy dy + 2ρ(z,z¯) r i 2e (Ai + Brix )(¯zdz − zdz¯)dy + ··· . (2.53)

1 2 Here ρ(z, z¯) = − 2 ln(zz¯) and = n−1, while x = z and x =z ¯. This is the most general form of the metric upto terms second order in z(¯z)[30, 31, 33, 48]. The ··· denote higher-order terms. As we will see later, for R2 theory we also need to include third-order terms in the metric expansion. The quantity Kij in this metric is identiﬁed with the extrinsic curvature, while Ai is identiﬁed with the twist potential. Both of these are standard quantities that characterize the embedding of the surface in the bulk. The quantities Ω, Bri and Q in the second-order terms in the metric are not arbitrary, but can be written in terms of Krij, Ai and the components of the curvature tensors. The bulk equation of motion will now contain divergences arising out of the conical singularity of the form z , z2 . However, the matter stress-energy tensor is expected to be 42 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY

ﬁnite. Therefore, we must set all divergences to zero. This condition ﬁxes the location of the entangling surface. The bulk equation of motion for general four-derivative theory is [49]:

1 6 L2 R − g R − g − H = 0 , (2.54) αβ 2 αβ L2 αβ 2 αβ where 1 H = λ g R Rδσµν − 2R R σδµ − 4∇ˆ 2R + 2∇ˆ ∇ˆ R + αβ 1 2 αβ δσµν ασδµ β αβ α β δ δσ 4RαRβδ + 4R Rδαβσ + 1 1 λ ∇ˆ ∇ˆ R + 2RδσR − ∇ˆ 2R + g R Rδσ − g ∇ˆ 2R + 2 α β δαβσ αβ 2 αβ δσ 2 αβ 1 λ − 2RR + 2∇ˆ ∇ˆ R + g R2 − 2g ∇ˆ 2R . (2.55) 3 αβ α β 2 αβ αβ

Gauss-Bonnet theory

Our eventual goal is to find the surface equation of motion for general R2 theory, but it is illuminating to look at the Gauss-Bonnet case first. The Gauss-Bonnet case was addressed in [34, 36, 37] using a metric linear in z(¯z). In this section, we will find the surface equation of motion for this theory using the metric in Eq. (2.53). We first show that the second-order metric in Eq. (2.53) suffices for Gauss-Bonnet theory and inclusion of higher-order terms in this conical metric will not affect the surface equation of motion that we find for this theory from the LM method. The bulk equation of motion for Gauss-Bonnet theory can be obtained from Eq. (2.54) by setting λ1 = λ, λ2 = −4λ and

λ3 = λ giving:

δ δσ σδµ Hαβ = 4RαRβδ − 4R Rδαβσ − 2RRαβ − 2RασδµRβ + 1 δσµν δσ 2 2 gαβ(RδσµνR − 4RδσR + R ) . (2.56)

The surface equation of motion is derived by ﬁnding the divergences in this equation that arise on using the conical metric in the limit z =z ¯ → 0. Terms higher than second- order in the metric will not contribute to the curvature tensors to zeroeth-order in z(¯z), although they might contribute at higher order. This is because the curvature tensors are of dimension 1/Length2 while third-order terms in the metric will be of order 1/Length3. The explicit values of the curvature tensors are listed in Appendix (2.5). These are calculated using a conical metric which is third-order in z(¯z). Note also, that the curvature tensors contain at most divergences of order 1/z. In the above equation of motion all terms are the product of two curvature tensors. Since each curvature tensor can only contribute at most a CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 43

1/z divergence and no third(or higher)-order term occurs at zeroeth order in any curvature tensor, third(and higher)-order terms will be absent in the divergence equations. By the same logic one can see that second-order terms will contribute to the divergence equations. However, in this case, cancellations between terms remove most of the second- order quantities, leaving only the quantities Qzzij and Qz¯zij¯ in the divergence equations. In the z =z ¯ → 0 limit K1 = K2 as explained in Appendix 2.5, so we drop the index r on r K . The divergence in the zz component from Hαβ term in the bulk eom is h i h n oi H = λ(RK − 2KijR ) + e−2ρ(z,z¯)λ − K3 + 3KK − 2K . (2.57) zz z ij z 2 3 Setting this divergence to zero should yield the condition for the extremal surface. There is no divergence in the zz¯ component. The divergence in the zi component is h n H = e−2ρ(z,z¯)λ 2K∇ K − 2K∇ Kj − 2Kj∇ K + 2K ∇ Kkj − zi z i j i i j ij k kj j koi 2Kkj∇iK + 2Kjk∇ Ki . (2.58)

This divergence is equivalent to the constraint equation one gets for the entropy functional (which doesn’t have to be necessarily the Jacobson-Myers functional) using tangential variations of the surface and vanishes similarly by Eq. (2.29). Finally, the divergence in the ij component is

4h n H = e−4ρ(z,z¯)λ 2K KklK + h KK − K K − h K − KK Kk − 4h KQ ij z ik lj ij 2 ij 2 ij 3 ik j ij zz kl k oi + 4hijKklQzz − 8KkiQzzj + 4KijQzz + 4KQzzij + 22 h n oi e−4ρ(z,z¯)λ 2K K − 2K Kk − h K2 + h K . (2.59) z2 ij ik j ij ij 2

In the above equation we have set Qzzij = Qz¯zij¯ . Using the value of the Rzizj component of the Riemann tensor from Appendix 2.5 and setting the background to AdS space, using 1 k Eq. (2.48), we can show that Qzzij = 4 KikKj and as a result the z divergence exactly vanishes. 2 However the z2 divergence will remain. The final step is to take the , z → 0 limit. Depending on the ordering one chooses, there are two ways to do this. One way is to take z → 0 limit first. Physically, this corresponds to looking for a divergence in the bulk equation of motion while there is still a small but non- zero conical deficit parameter . The second way is take → 0 first. The limit is, therefore, an iterated limit – the final result depends on the order in which the limit is taken, so there is an inherent ambiguity in this procedure. In fact, this ambiguity can be made even larger in scope if we take the limit simultaneously in and z. Mathematically, the divergence is a function of the two variables: and z. In this -z plane there are an infinite number of paths 44 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY along which we can take the limit. At least on a mathematical level, there exists no reason why the limit should only be taken along the z = 0 or = 0 path. The path z = 0 is, however, the simplest way to take the limit so as to obtain z → ∞. In −2ρ(z,z¯) this case, all terms containing z are leading divergences while terms containing e /z = (zz¯)/z contribute to sub-leading divergences. Therefore, in this way of taking limits, setting the Hzz divergence to zero will yield two different conditions for the minimal surface. The first condition which, after adding the Einstein term, corresponds to the surface equation of motion is 2 ij K + L λ(RK − 2K Rij) = 0 . (2.60)

This agrees with the surface equation that comes from the Jacobson-Myers functional. How- ever, there will also be an extra constraint [6] of the form

3 − K + 3KK2 − 2K3 = 0 , (2.61) coming from the sub-leading divergence. The Hij divergence will also lead to a similar constraint. The above condition can only be true for very special surfaces and therefore is an over-constraint on the surface. In fact, if these two conditions were to be true simultaneously, the surface equation of motion we would end up getting is:

2 3 c K + αλL (K − 3KK2 + 2K3) = 0. (2.62)

To get this form of the equation, we have used the Gauss identities on AdS space. Here, c = (1 − 2f∞λ) is proportional to the Weyl anomaly and α is a variable that can take any arbitrary numerical value. The surface equation of motion corresponding to the Jacobson- Myers functional can be recovered if α = 1. However, at present nothing within the LM method sets the value of this parameter to one. Note that if α was zero, the minimal surface that we would get is the same as in the Einstein case. It also the minimal surface that would follow if one were minimizing just the Wald part of the entropy functional. In the above paragraph we outlined one way in which the LM method could potentially give rise to the correct surface equation of motion. Let us now explore if we can change the limit-taking procedure itself to get the correct equation. This can be accomplished by choosing a different path in the -z plane to take the limit. Taking the limit along the path = 0 will simply kill off all divergences; this is not surprising since physically this corresponds to turning off the conical deficit in the metric. However, we can pick a path in the , z plane that will kill off the sub-leading divergence but preserve the leading divergence. 2 z 1+v For example, as was shown in [36], taking any path of the form (z) = ( ) , where v is a number greater than one, will keep only the leading divergence. At this point, we can offer no justification of why one should choose this particular way of taking limits. We are merely CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 45 demonstrating that there does exist a way to take limits in the , z plane that leads to the correct surface equation of motion in the Gauss-Bonnet case. This way of taking limits is equivalent to discarding terms suppressed by e−2ρ(z,z¯) and was also used in [30] to show that the LM method leads to the same surface equation of motion in the Lovelock case as can be derived from the entropy functional for Lovelock gravity [27, 30, 50]. This is the way of taking limits that we will use. However, unless one can specify a mechanism or a physical interpretation which reproduces this way of taking limits (which is possible if the metric itself is re-defined), the argument that the LM method reproduces the correct surface equation of motion for Gauss-Bonnet theory remains incomplete. The same ambiguity in taking limits exists for general R2 theory. To remain consistent with the Gauss-Bonnet point, for R2 theory we will continue to take limits as stated in the paragraph above. However, in the general case this is not an ideal solution. As we will see, ∼ K3 terms always occur with the e−2ρ(z,z¯) factor in the divergence equations for R2 theory. This means that if we use the above way of taking limits we will never get such terms at any point in the parameter space. As we saw in Eq. (2.51), the surface equation of motion for R2 theory does contain such terms. However, our goal for general R2 theory is to see to what extent we are able to reproduce the surface equation of motion in Eq. (2.51), while taking the limit in such a manner that the result at the Gauss-Bonnet point agrees with what comes from the Jacobson-Myers functional. It is clear, though, that the question of taking limits in the LM method deserves more study.

The general case

We now work out the divergence equations for the R2 case. For general R2 theory all second- order quantities will enter into the divergences. We can anticipate the eﬀect that terms containing Ω and B will have on the surface equation of motion coming from the LM method. Consider the following components of the bulk Riemann tensor around z =z ¯ = 0:

4ρ(z,z¯) Rpqrs = −3e εˆpqεˆrsΩ , (z=0,z¯=0) 2ρ(z,z¯) Rpqri = 3e εˆpqBri , (2.63) (z=0,z¯=0)

−2ρ(z,z¯) where,ε ˆab is deﬁned asε ˆzz¯ = −εˆzz¯ = e gzz¯. The quantity Ω is therefore equivalent µ ν ρ σ to −1/3Rµνρσnr ns nrns evaluated at z, z¯ = 0. We can determine a numerical value for the r quantities Bi and Ω in the metric by demanding that the bulk Riemann tensor be the AdS solution at zeroeth order. Since for AdS space the Riemann tensor is given by Eq. (2.48), we can write the components of the bulk Riemann tensor on the L.H.S of Eq. (2.63) in terms of the components of the bulk metric. Expanding the metric using Eq. (2.53) and keeping only 46 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY the zeroeth-order terms in z, z¯ we get 1 Ω = − C and B = 0 (2.64) 12 ri

r 4 zz z¯z¯ Therefore, Bi can be set to zero. In writing the divergences, we also ignore Qij and Qij , zz¯ zz¯ the remaining component being Qij = Qij = Qij . For R2 theory the derivative order of the equation of motion is four. That means we should include order z3 terms in the metric, since they can contribute to the divergences. These terms can be parametrized as

2 4ρ(z,z¯) p q r s t r s p i j 2ρ(z,z¯) r s i ds = e ∆pqrstx x x dx dx + Wrspijx x x dy dy + 2e Crsix x (¯zdz − zdz¯)dy . (2.65) This is the most general form of the third-order terms in the metric. Here, we have written the e2ρ(z,z¯) dependence of each term explicitly. As for the second-order quantities, the third- order quantities ∆pqrst, Wrspij and Crsi can be found by calculating the curvature tensors, but 4ρ(z,z¯) µ ν ρ σ to linear order in z(¯z). Then, for example, e ∆pqrst ≡ −1/6∂p(Rµνρσnq nr nsnt ) evaluated at z =z ¯ = 0. Note that the factor of e4ρ(z,z¯) will cancel from both sides on using the AdS background. In fact, this particular term vanishes altogether in this background. On using the metric with the third-order terms listed above to ﬁnd the divergences in the equation of motion we ﬁnd that the Crsi, Wzzzij and Wz¯z¯zij¯ do not contribute. The terms that are relevant are Wzzzij¯ and Wz¯zzij¯ , because as will show below they will lead to unsuppressed CK terms. Without loss of generality, we can set them to be equal and denote this term as

Wij.

4As we saw for the Gauss-Bonnet case, these terms will be present in the divergences, but they will not change our conclusions for R2 theory. CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 47

2 For general R theory, the divergence in the zz component from the Hαβ term in the bulk eom is h H = − 1 (λ + 4λ )∇2K + (2λ + λ + 2λ )∇i∇jK + λ (RK − 2KijR ) + zz z 2 2 3 1 2 3 ij 3 ij i j i 4(−2λ1 + 3λ2 + 14λ3)KijA A − 6(λ2 + 4λ3)KA Ai + i 8(3λ1 + 2λ2 + 5λ3)KΩ − h n oi e−2ρ(z,z¯) (2λ − λ − 6λ )K + 1 (λ + 4λ )K2 + 2(λ + 4λ )Q + z2 1 2 3 2 2 2 3 2 3 h n e−2ρ(z,z¯) − λ K3 + (λ + 7λ )KK − 2(3λ + 2λ + 6λ )K + z 3 2 3 2 1 2 3 3 ij 3 (6λ1 + 5λ2 + 14λ3)KijQ − 2 (λ2 + 4λ3)KQ − oi 4(λ2 + 4λ3)W . (2.66)

The divergences in the other components are 2h n H = e−2ρ(z,z¯) (λ + 1 λ )K3 + (λ − 3 λ − 7λ )KK + zz¯ z 3 4 2 1 2 2 3 2 kl 2(−2λ1 + λ2 + 6λ3)K3 + (2λ1 − 3λ2 − 14λ3)KklQ +

3 oi 2 (λ2 + 4λ3)KQ + 8(λ2 + 4λ3)W , (2.67)

2h n H = e−2ρ(z,z¯) − 1 (2λ + λ )Kk∇ K − (3λ − λ − 6λ )Kkl∇ K − zi z 2 1 2 i k 1 2 3 i kl 1 k l k 4 (3λ2 + 8λ3)K∇iK + (5λ1 + λ2)Ki ∇lKk − λ1K∇kKi + kj k (9λ1 + 2λ2)K ∇kKji − (λ2 + 4λ3)∇iQ − (4λ1 + λ2)∇kQi − 1 2 (10λ1 − 2λ2 − 18λ3)AiK2 − 2 (3λ2 + 12λ3)AiK + jk oi 8(4λ1 + λ2)KijK Ak − 2(λ2 + 4λ3)AiQ , (2.68)

4h n H = e−4ρ(z,z¯) ( 1 λ + 1 λ + 2 λ )h K3 − (7λ + 2λ + 2λ )KK Kk + ij z 3 1 4 2 3 3 ij 1 2 3 ik j kl 2(16λ1 + 4λ2 + λ3)KikK Klj − (λ1 + 3λ2 + 10λ3)hijKK2 − 1 (3λ1 − 2λ3)KijK2 − 3 (λ1 − 18λ2 − 70λ3)hijK3 + k 2(4λ1 + λ2)QijK + 2(λ2 + 4λ3)hijKQ − 8(4λ1 + λ2)KikQj − kl (λ2 + 4λ3)KijQ − 7(λ2 + 4λ3)hijKklQ + 32(4λ1 + λ2)Wij+ oi 32(λ2 + 4λ3)hijW . (2.69)

Whether or not the divergences in the ij, zi and zz¯ components vanish before taking the → 0 limit will depend upon the exact values of the second-order terms. The zi divergence, 48 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY in particular, should be equivalent to the constraint equation coming from the tangential variations and should vanish by the Codazzi relation in Eq. (2.29). As in the Gauss-Bonnet case, the divergence in the ij component is not expected to fully vanish by itself. We therefore take the limit as prescribed in the last section. This reduces the divergences in the zi and ij components to zero. However, because of the presence of the W term there still remains an unsuppressed divergence in the zz¯ component. This divergence can only go to zero if K = 0 or the theory is at the Gauss-Bonnet point. We now examine the divergences in the zz component, to be able to compare it with the surface equation of motion derived using the FPS functional. First looking at the 1/z i j divergence in that component, one can see that it contains the unsuppressed terms KijA A i and KA Ai which are not present in Eq. (2.51). However, these terms can be eliminated in favor of other variables. Consider the Rzizj¯ component of the Riemann tensor expanded around z = 0, z¯ = 0:

1 2ρ(z,z¯) 2ρ(z,z¯) 1 k 1 Rzizj¯ = 2 e Fij − 2e AiAj + 4 KzikKzj¯ − 2 Qzzij¯ . (2.70) (z=0,z¯=0)

ij Using Eq. (2.48) again and multiplying both sides by Kij, we find that the AiAjK term −2ρ(z,z¯) 3 −2ρ(z,z¯) i can be written as ∼ CK + e K + e QK. The AiA K terms can be written in a similar fashion. Since only the CK term is unsuppressed we find CK A A Kij = + ··· and i j 4 3CK A AiK = + ··· , (2.71) i 4 where the dots denote the suppressed terms. Next looking at the e−2ρ(z,z¯)/z divergence we find that the W term will contribute to the surface equation of motion, since this term contains a e2ρ(z,z¯) factor that enhances the divergence to 1/z. This term can be determined by using the following equation

2ρ(z,z¯) ij 2ρ(z,z¯) ∂zRz¯z¯ = −W + 2e K AiAj − 2e ΩK + ··· . (2.72) (z=0,z¯=0) The R.H.S of this equation disappears in the AdS background. Using Eqs. (2.64) and (2.71) we ﬁnd 2e2ρ(z,z¯)CK W = + ··· . (2.73) 3 The Q terms that are also present in this divergence do not contribute since as we show below they are expected to contain only ∼ K2 terms and therefore remain suppressed. Substituting these values in Eq. (2.66), and adding the Einstein term we ﬁnd that the 1/z divergence of the zz component gives rise to the following surface equation of motion:

2 1 2 ij K + L {(2λ1 + 2 λ2)∇ K + λ3(RK − 2K Rij) + λ1C1K + λ2C2K + λ3C3K} = 0 (2.74) CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 49

2 ij where C1 = −4C, C2 = −11C/2 and C3 = −18C. The coeﬃcients of the ∇ K, RK, K Rij and CK terms in the above equation all match with those in Eq. (2.51). Because of the way we are taking limits, the K3 terms that are present in Eq. (2.51) are not present here. Finally we look at the /z2 divergence in the zz component. For this divergence to vanish, we get the condition

1 2 (2λ1 − λ2 − 6λ3)K2 + 2 (λ2 + 4λ3)K + 2(λ2 + 4λ3)Q = 0 . (2.75)

To satisfy this condition at arbitrary points of the parameter space, one has to demand that Q 2 be a function of ∼ K terms, and also λ1, λ2 and λ3. Demanding that Q be independent of λ1,

λ2 and λ3, will pick out a special point in the parameter space (apart from the Gauss-Bonnet point where this condition is trivially satisﬁed). To summarize the results for R2 theory:

1. Apart from the absence of ∼ K3 terms, Eq. (2.74) that we found using the LM method is exactly the surface equation of motion that results from the FPS functional.

2. There are some problematic extra divergences. The zz¯ component of the bulk equation of motion has a divergence that can only disappear at the Gauss-Bonnet point. There is also a second-order 1/z2 divergence in the zz component. This can be taken to ﬁx the value of the term Q; however, it is not possible to do this in a way that is independent of the parameters of R2 theory.

2.3.3 The stress-energy tensor from the brane interpretation

In [24], it was noted that a equation of motion of a cosmic string is the same as the equation for the minimal entangling surface. This is because a cosmic string produces a spacetime with a conical defect with a metric of the form in Eq. (2.53). The equation of motion is given by minimizing its action. For Einstein gravity this is just the Nambu-Goto action and equation of motion of a cosmic string is

K = 0. (2.76)

This condition minimizes the surface area of the string as it sweeps through spacetime. The same thing holds for a cosmic brane. As was done in [30], where it was referred to as the cosmic brane method, this fact can be exploited to construct the entropy functional from the bulk equation of motion. In this section, we will check this construction of [30]. The idea is that the bulk equation of motion in Eq. (2.54) should lead to the cosmic brane as a solution, to linear order in . In 50 2.3. TEST OF THE ENTROPY FUNCTIONAL FOR R2 THEORY particular, this means that L.H.S of Eq. (2.54) should be equal to the stress-energy tensor of the brane. Since the brane is a localized source, the stress-energy tensor will contain delta functions. Once we have found the stress-energy tensor we can identify the associated action δS via Tαβ = . δgαβ Let us see how this works in the Gauss-Bonnet case. In the bulk equation of motion, −2ρ(z,z¯) terms such as ∂z¯∂zρ(z, z¯) correspond to delta functions. We set δ(z, z¯) = e ∂z¯∂zρ(z, z¯). Note that δ(z, z¯) deﬁned this way contains a factor of . The delta divergences in the ij component of the bulk equation of motion to linear order in are then: n Tij = δ(z, z¯) − 4 λ (hijR − 2Rij)+

−2ρ(z,z¯) 2 k o − 2 λ e (hijK2 − hijK + 2KijK − 2KikKj ) . (2.77)

To identify this as the stress-energy tensor coming from the Jacobson-Myers functional (interpreted as a cosmic brane action), the second term should go to zero. This term carries a factor of e−2ρ(z,z¯) as compared to the ﬁrst term and according to our way of taking limits is suppressed. Our result is then in agreement with the claim in [30] that the cosmic-brane method can be used to show that the Jacobson-Myers functional is the right entropy functional for Gauss-Bonnet theory. However, as we will see there are problems for the general four-derivative theory. For R2 theory, the delta divergences in the ij component are CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 51

h Tij = δ(z, z¯) − 4 λ3 (hijR − 2Rij) − 16(6λ1 + 11λ2 + 38λ3)hijΩ +

−2ρ(z,z¯)n 2 e − (λ2 + 2λ3)hijK + 2(λ2 + 3λ3)hijK2 −

2(12λ1 + 4λ2 + 4λ3)Qij − 2(λ2 + 4λ3)Qhij − k 2(4λ1 + λ2 + 2λ3)KijK + 2(14λ1 + 4λ2 + 4λ3)KkjKj ) + o i 16(20λ1 + 11λ2 + 24λ3)AiAj +

−2ρ(z,z¯)n on o e ∂zδ(z, z¯) + ∂z¯δ(z, z¯) − 2(2λ1 + λ2 + 2λ3)Kij + (4λ3 + λ2)hijK −

−2ρ(z,z¯) 4e ∂z∂z¯δ(z, z¯)(λ2 + 4λ3) . (2.78)

Again, barring the term suppressed by e−2ρ(z,z¯), we have checked that the result for this component is of the same form as that produced on calculating the stress-energy tensor from an action equivalent to the FPS functional. The derivative of delta terms like ∂zδ(z, z¯) are typical in the stress-energy tensor of actions containing terms that depend on the extrinsic curvature [47]. However, the zz and zz¯ components of the bulk equation of motion also contain delta divergences that are not suppressed:

2 Tzz = − 4∂z δ(z, z¯)(2λ1 + λ2 + 2λ3) − 2∂zδ(z, z¯)(4λ1 + λ2)K (2.79) and Tzz¯ = − 2 ∂zδ(z, z¯) + ∂z¯δ(z, z¯) 2λ1 + λ2 + 2λ3 K +

4 ∂z∂z¯δ(z, z¯)(2λ1 + λ2 + 2λ3) . (2.80)

Taking the delta divergences in all components into account, the Tµν we have found does not look like the stress-energy tensor for a cosmic brane corresponding to a three-dimensional surface in the ﬁve-dimensional bulk. Note that the extra divergences all vanish for the Gauss- Bonnet theory. The Gauss-Bonnet result therefore stands. However, any attempt to use this method to show that the FPS functional is the correct entropy functional for R2 theory should be able to account for these extra delta divergences.

2.4 Quasi-topological gravity

The lagrangian for quasi-topological gravity [38] contains terms cubic in the Riemann tensor. It can be used to study a class of CFT’s involving three parameters in four dimensions. It has many interesting features including the fact that its linearized equation of motion is two-derivative order. Unitarity for this theory was studied in [51]. 52 2.4. QUASI-TOPOLOGICAL GRAVITY

In Sec. (2.4.1), we ﬁnd the HEE functional for quasi-topological theory using Eq. (2.6) and compute the universal terms is Sec. (2.4.2). In Sec. (2.4.3), we ﬁnd the surface equation of motion for this theory using the LM method.

2.4.1 The entropy functional

The action for quasi-topological theory in ﬁve dimensions is Z 1 5 SQT = − 3 d x L1 + L2 + ν Z5 , (2.81) 2`P where L1 is the Einstein-Hilbert action given in Eq. (3.9) and L2 is the Gauss-Bonnet la- 3 grangian as in Eq. (3.10) with λ1 = λ3 = λ , λ2 = −4λ. The last term is the R lagrangian:

γδ µν αβ β δ η ζ α γ αβγδ Z5 =µ0Rαβ Rγδ Rµν + µ1Rα γ Rβ δ R η ζ + µ2RαβγδR R + αβγ δη αγ βδ β γ α β α 3 µ3RαβγδR ηR + µ4RαβγδR R + µ5Rα Rβ Rγ + µ6Rα Rβ R + µ7R . (2.82) There are two diﬀerent consistent R3 theories. For the ﬁrst theory

3 9 15 18 33 15 µ0 = 0 , µ1 = 1 , µ2 = 8 , µ3 = − 7 , µ4 = 7 , µ5 = 7 , µ6 = − 14 , µ7 = 56 (2.83) 7µL4 and the coupling constant is ν = 4 , while for the second theory 3 60 72 64 54 11 µ0 = 1 , µ1 = 0 , µ2 = 2 , µ3 = − 7 , µ4 = 7 , µ5 = 7 , µ6 = − 14 , µ7 = 14 (2.84) 7µL4 and the coupling constant ν = 8 . The R3 part for the HEE functional is Z √ 2πν 3 SEE, R3 = 3 d x h (LWald, R3 + LAnomaly, R3 ) , (2.85) `P where

zzαβ¯ zαz¯ β zαz β αβρσ LWald,R3 = 6µ0R Rzzαβ¯ + 3µ1 R βRzαz¯ − R βRzαz + µ2 RαβρσR − zz¯ z z¯ α z¯ z α 1 z¯ αβρ z αβ 4RR zz¯ + 2µ3 Rα z¯ R z − Rα z R z¯ + 2 Rαβρ R z¯ + µ4(2R αzβR + z 2 zz zα αβ z 2 (R z) − R Rzz) + 3µ5R Rzα + µ6 RαβR + 2RR z + 3µ7R . (2.86) The symbols z andz ¯ in the above expression label the two orthogonal directions while the indices α, β, ... are the usual bulk indices. The expression for the anomaly part is

ij 3 3 2 ikjl LAnomaly, R3 =µ0(12 K2 Qij − 6 K4) − µ1( 2 K4 − 2 K2 + 3 KijKklR )− 2 2 µ2(6 K2 − 2 K2 K − 8 K2 Q + 4 K2R)− 1 2 ij ij ij µ3(2 K4 + 2 K2 − K2 Q − 2 K2 Qij − 2 K Qij K + 2 K2 Rij)− 2 ij ij µ4(2 K3 K − K2 K − 2K Qij K + 2 K Rij K)− 3 2 3 2 3 2 1 4 2 2 µ5( 4 K2 K − 2 K Q) − µ6( 2 K2 K − 2 K − 2 K Q + K R) . (2.87) CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 53

ij lk where K4 = K KjlK Kki . In calculating the anomaly part from Eq. (2.6), we have used the value of Bi = 0 that we found in Sec. (2.3.2). This is the reason that while terms involving Bi are supposed to contribute to Eq. (2.6), the above equation does not contain any terms containing Bi. The full HEE functional has contributions from the Einstein and R2 part also which are given in Eqs.(2.12) and (2.13).

2.4.2 Universal terms

In this section, we will demonstrate that our HEE functional for the quasi-topological gravity produces the correct universal terms. For the general structure and calculation of the universal term of the entanglement entropy in four dimensions, see [52]. These central charges can be easily calculated using the technique of [53]. We follow the procedure given in [34] for R2 theory. Here we sketch the main steps of this calculation. We will minimize Eq. (2.85) for a bulk surface with a spherical and cylindrical boundary. We will carry out this procedure for the ﬁve-dimensional bulk AdS metric as shown in the section (2.3). Following the analysis of section (2.3) we set ρ = f(z), τ = 0 in the metric in Eq. (2.14) and minimize the entanglement entropy functional (whose R3 part is given in Eq. (2.85)) on this codimension 2 surface to ﬁnd the Euler-Lagrange equation for f(z). Using the solution for f(z) we evaluate the entropy functional to get the EE.

p 2 2 For the spherical boundary, we get f(z) = f0 − z which gives the EE as f S = −4a ln( 0 ) . (2.88) EE δ

Here, δ is the UV cut-oﬀ that comes from the lower limit of the z integral and f0 is the radius of the entangling surface. The value of a is

2 3 π L 2 a = (1 − 6f∞λ + 9f µ) . (2.89) 3/2 3 ∞ f∞ `P For this case, the entire contribution comes from the Wald entropy as the extrinsic curvatures are identically zero. z2 For the cylindrical boundary, we ﬁnd f(z) = f0 − + ... leading to 4f0 cH f S = − ln( 0 ) . (2.90) EE 2R δ The value of c corresponding to the theory in Eq. (2.83) is

2 3 π L n 2 2 o c = 1 − 2f∞λ + 9f µ + f µ(42µ1 − 336µ2 − 56µ3) , (2.91) 3/2 3 ∞ ∞ f∞ `P 54 2.4. QUASI-TOPOLOGICAL GRAVITY while that corresponding to the theory in Eq. (2.84) is

2 3 π L n 2 2 o c = 1 − 2f∞λ + 9f µ − f µ(168µ2 + 28µ3) . (2.92) 3/2 3 ∞ ∞ f∞ `P

2 The 1 + 9f∞µ part is the usual Wald entropy contribution, while the remaining part comes from the anomaly part. After putting in the values of µ’s given in Eqs.(2.83) and (2.84) we obtain

2 3 π L 2 c = (1 − 2f∞λ − 3f µ) (2.93) 3/2 3 ∞ f∞ `P for both theories. These results for the universal terms agree with those calculated in [34] for the two quasi- topological theories. Note from Eqs. (2.91) and (2.92) that only a few terms from LAnomaly, R3 have contributed to the universal term. Terms of the form ∼ K4 do not contribute to this calculation at all. Since Q ∼ K2, terms of the form ∼ K2Q also do not contribute.

2.4.3 Minimal surface condition

We now ﬁnd the surface equation of motion for quasi-topological gravity using the LM method. For ease of calculation, we set all second-order quantities and cross-components in the metric in Eq. (2.53) to zero. The bulk equation of motion for this theory is [49]:

1 6 L2 R − g R − g − H − νF = 0 , (2.94) αβ 2 αβ L2 αβ 2 αβ αβ

where Fαβ is deﬁned in [38, 49]. The z divergence in the zz component of the equation of motion coming from the Fαβ term is h F 1 = ( 3 µ − µ − µ − 3 µ − 4µ − 12µ )Rij∇2K − ( 1 µ + µ + 6µ )RijK R + zz z 2 1 2 3 2 5 6 7 ij 2 2 6 7 ij 1 ij 1 1 2 i (µ2 + 6 µ6 + 3µ7)RijR K + 2 (µ6 + 2 µ4)K∇ R + (µ4 + µ3 + 4µ2)∇i∇ K − 3 l k ij 1 kl i j (3µ1 − 8µ − 2 − 3µ3 − µ4 + 2 µ5)∇ Rlijk∇ K − 2 (µ4 + 3µ1)K ∇ ∇ Rkijl − 3 5 1 3 i j ( 4 µ1 − 2 µ2 − µ3 − 2 µ4 − 4 µ5 − 2µ6 − 6µ7)R∇ ∇ Kij + 1 ij i 4 (µ4 + 2µ3 + 8µ2)K ∇i∇jR . (2.95)

While we haven’t computed the surface equation of motion that one gets on minimizing the functional in Eq. (2.85), this is not very hard to do using the methods of Sec. (2.3.1) and Mathematica5. The main point is, however, that the surface equation of motion that one

5We have used the Xact package for a number of calculations in this chapter CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 55 will get from the entropy functional will contain K4 terms that are absent in the above divergence. Other divergences are also present in the zz component: h n F 2 = e−2ρ(z,z¯) 1 (3µ + 19µ + 2µ + 14 µ )R KikKjl − ( 7 µ + 18µ )RK + zz z2 2 1 2 3 3 6 ijkl 2 6 7 2 3 2 3 2 (2µ2 + 2 µ6 + 6µ7)RK + (µ4 − 2 µ5)K∇ K − 4 3 4 ij 3 i j ( 3 µ2 − µ4 + 2 µ5 + 3 µ6)K ∇i∇jK − (µ4 − 2 µ5)K∇ ∇ Kij + 2 3 2 ij k (3µ1 − 3 µ2 − µ3 + µ4 − 2 µ5 − 3 µ6)∇kK ∇ Kij − 2 3 2 i k j (3µ1 + 3 µ2 + µ3 + 3µ4 − 2 µ5 + 3 µ6)∇ Kij∇ Kk − 3 ij jk i oi (8µ2 − µ4 + 2 µ5)RijK K + (3µ1 + 8µ2 − µ4 − 2µ6)RijK Kk + h n oi e−4ρ(z,z¯) (3µ − 2µ − 2µ )K + (µ − 3µ − 2µ )KK . (2.96) z3 1 2 3 3 4 5 6 2

As for R2 theory, these divergences can be used to determine second and higher-order terms in the metric. At linear-order in the metric, divergences in all other components of the equation of motion go to zero if we take the limit as mentioned in Sec. (2.3.2).

2.5 Discussion

We found the surface equation of motion for general R2 theory and quasi-topological gravity using the generalized gravitational entropy method of [24]. We found that these do not match exactly with what can be derived by extremizing the HEE functional for these theories – the HEE functional being calculated using the formula proposed in [30, 31]. Let us summarize our findings regarding R2 theory. First, the leading-order terms on both sides do match. In fact, barring ∼ K3 terms, the surface equation of motion that follows from the LM method is precisely the surface equation of motion that follows from the FPS functional. The main problem with the LM method is that there are divergences in components other than the zz component, for a general higher-derivative theory. In the Gauss-Bonnet case, there are ways we can take the limit to set these divergences to zero. However, the effect of taking the limit in this way is to remove all ∼ K3 divergences from all components of the equation of motion. This means that we do not get any ∼ K3 terms in the surface equation of motion using the LM method. No matter what the HEE functional for R2 theory is, it is unlikely that no ∼ K3 terms will occur in its surface equation of motion at any point in its parameter space. Even after taking the limit as prescribed, for general R2 theory, there remain extra divergences in the bulk equation of motion. It is impossible to set these 56 2.5. DISCUSSION divergences to zero at all points of the parameter space, although this can be done for specific points like the Gauss-Bonnet point. As we discussed, the absence of ∼ K3 terms is the R2 equation of motion is an artifact of the way limits have to be taken in the LM method for the Gauss-Bonnet case. The limit can also be taken in such a way so as to preserve ∼ K3 terms. It is worth recapitulating the results this way of taking the limit gives for Gauss-Bonnet theory. As we showed, using the second-order conical metric, the bulk equation of motion for Gauss-Bonnet theory, before taking the limit, has divergences only in the zz and ij components. There is no divergence in the zz¯ component, while the divergence in the zi component turns out to be a constraint equation that vanishes by itself on using the Codazzi-Mainardi relation on AdS space. This same constraint equation results from the Jacobson-Myers functional, as well, on taking tangential variations of the surface. It is not clear what the relevance of the divergence in the ij component is in the LM method. Were we to ignore this divergence, the surface equation of motion that would result from the zz component for Gauss-Bonnet theory, after taking the limit, is c K = 0, where c is proportional to the Weyl anomaly. This equation is clearly not what comes from the Jacobson-Myers functional. However, the resulting minimal surface is what one obtains on extremizing just the Wald entropy part of the functional. It would be interesting to check whether the zz component of R2 theory also leads to the same result. One of the pending issues with the LM method is to fix the ambiguity present in the limit- taking procedure. However, fixing this by itself does not seem enough to simultaneously cure the two problems present for R2 theory: the absence of ∼ K3 terms and the presence of extra constraints; although, it can remove one of these problems from the list. The ambiguity in the limit-taking procedure is not unique to the LM method. Similar, though not exactly the same, issues occur in studies of co-dimension two branes in the context of brane-world gravity [54]. It is possible that a further modification to the LM method will fix these problems; on the contrary, it may be that one cannot get rid of it in any way. The problem of extra divergences is related to the derivative order of the bulk equation of motion and seems to spring from the pathology of the general R2 theory itself. In this sense, it is not surprising that we encounter it for general higher-derivative theories. Higher-derivative theories are known to suffer from problems regarding unitarity [55, 56, 57]. These problems seem to be manifested in the LM method in the inability to remove all divergences, that occur on using the conical metric, from the bulk equation of motion. What does our analysis say about the validity of the formula proposed in [31, 30] as the entanglement entropy functional? For general R2 theory as we demonstrated the leading- order terms match on both sides, which stops short of being a validation of the proposal for this theory. This test, at present, is similar in scope in refining conjectured entropy functionals for higher-derivative theories as the test whether the entropy functional leads to CHAPTER 2. HOLOGRAPHIC ENTANGLEMENT ENTROPY FUNCTIONALS: A DERIVATION 57 the correct universal terms. Also we showed, for quasi-topological theory the universal terms are not sensitive to terms of the form ∼ K4 in the entropy functional (similar statement applies for other higher-derivative theories) and one can change these terms and still have the universal terms come out to be correct. The LM method, therefore, in its current form has limitations that make it ineffective in testing proposed entropy functionals for generic higher-derivative gravity theories. The fact that the LM method only works for specific theories may indicate one of two things. One possibility is that entropy functionals only exist for specific theories such as Lovelock theories, for which the result of the surface equation of motion from the existing entropy functional and the LM method coincide. The other possibility, as mentioned before, is that the LM method needs some modification. In this context, it is also desirable that alternate methods to test entropy functionals be developed. Our argument can be extended for other Lovelock theories in general dimensions and can be shown that extremization condition can be derived consistently. But for for general theories it fails. In short LM method in its current form only picks out some particular theories of gravity. After our work [44], it was shown in the [58], that one can possibly modify the LM method by breaking replica symmetry inside the bulk spacetime, but keeping the symmetry preserved at the boundary. This generates new terms in the conical metric and enhances the

Zn symmetry group inside the bulk. Using this proposal one can show that, it is possible to cancel the oﬀending K3 terms for the Gauss-Bonnet theory without taking the limits as shown in this chapter. This methods works only when one treats the Gauss-Bonnet coupling perturbatively. But still it remains to be investigated, whether it is possible to cure the problem for general four-derivative theories.

Appendix

A: Conical Metric

Near the conical singularity the bulk metric can also be written as

2 −2 2 2 −2 p j i j ds = ρ(x, y) (dx + dy ) + ρ(x, y) apj dx dx + gij dx dx . (2.97)

The two-dimensional part is written in cartesian coordinates x and y and ρ(x, y) = px2 + y2. We have written the metric upto terms ﬁrst order in x(y). The co-dimension two surface (Σ) is located at x = 0 and y = 0. The metric gij can be written down order by order in x(y) after expanding around the surface Σ as

gij = hij + x ∂xgij Σ + y ∂ygij Σ + ··· . (2.98) 58 2.5. DISCUSSION

The surface tensor hij is independent of x and y. The variable apj ∼ O(x). The extrinsic curvatures for the co-dimension two surface (Σ) are deﬁned as

β β ˆδ α Ksij = ej ∇insβ Σ = ej (∂insβ − Γαβei nsδ) Σ . (2.99)

β Expanding the Christoﬀel in terms of the metric and using the fact that the ﬁrst term ej ∂insβ vanishes it follows that 1 1 Kxij = ∂x gij , Kyij = ∂ygij . (2.100) 2 Σ 2 Σ

We now make the simplifying assumption that the metric gij is independent of the co-ordinate y. Under this assumption, the extrinsic curvature Kyij vanishes as ∂ygij vanishes. The complex coordinates z andz ¯ used in the metric in Eq. (2.14) are related to x and y as z = x + iy, z¯ = x − iy . (2.101)

In these coordinates the metric is

2 2ρ(z,z¯) i j 2ρ(z,z¯) i ds = e (dzdz¯) + gijdx dx + 2e Ai(¯zdz − zdz¯)dy , (2.102) where

gij = hij + z Kzij +z ¯Kzij¯ + ··· . (2.103)

The extrinsic curvatures in this coordinate system are related to Kxij and Kyij as K + i K K − i K K = xij yij , K = xij yij . (2.104) zij 2 zij¯ 2

Since Kyij = 0 we have

Kzij = Kzij¯ . (2.105) Similar considerations apply to the second-order quantities Q.

B. Curvature Tensors

In this appendix, we list components of the curvature tensors for the metric in Eq. (2.53), that do not appear in the main text. We retain only terms uptil zeroeth-order in z, z¯. The components of the Christoﬀels are Γz = − , Γz¯ = − , Γz = −e−2ρ(z,z¯) Kz¯ , Γz¯ = −e−2ρ(z,z¯) Kz , zz z z¯z¯ z¯ ij ij ij ij 1 1 1 Γi = Ki , Γi = Ki , Γi = gil(∂ g + ∂ g − ∂ g ) , zj 2 zj zj¯ 2 zj¯ jk 2 j lk k lj l jk z z¯ Γ zi = −2Ai , Γzi¯ = 2Ai . (2.106) REFERENCES 59

The components of the Riemann tensor are

2ρ(z,z¯) k k Rpqij = 2e εˆpqFij + (KpjkKqi − KpikKqj) , 1 k Rzizj = 4 KzjkKzi − Qzzij − 2z Kzij , 1 2ρ(z,z¯) 2ρ(z,z¯) 1 k 1 Rzizj¯ = 2 e Fij − 2e AiAj + 4 KzjkKzi¯ − 2 Qzzij¯ , 1 Rpijk = 2 (∇kKpij − ∇jKpik) , 1 −2ρ(z,z¯) Rikjl = Rikjl + 2 e (KzilKzjk¯ + Kzil¯ Kzjk − KzijKzkl¯ − Kzij¯ Kzkl) , (2.107) where Fij ≡ ∂iAj − ∂jAi. The components of the Ricci tensor are

1 j Rzi = 2 (∇ Kzji − ∇iKz) , 1 ij 1 Rzz = 4 KzijKz − 2 Qzz − 2z Kz , 1 ij 1 2ρ(z,z¯) i Rzz¯ = 4 Kzij¯ Kz − 2 Qzz¯ − 2e (AiA − 3Ω) , −2ρ(z,z¯) k k 1 1 Rij = e Kzj¯ Kzik + KzjKzik¯ − 2 Kzij¯ Kz − 2 KzijKz¯ − 2Qzzij¯ + Rij − 8AiAj . (2.108)

As in the main text, ∇ used in the above equations is the Van der Waerden-Bortolotti covariant derivative [45] deﬁned in Eq. (2.24). The Ricci scalar is

i −2ρ(z,z¯) ij R = R + 24Ω − 16AiA − e KzKz¯ − 3Kzij¯ Kz + 4Qzz¯ . (2.109)

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3.1 Introduction

In the Chapter.(2) we have shown that the identification of the entanglement entropy with the generalized gravitational entropy has opened up the avenue for systematically generalizing holographic entanglement entropy (EE) for more general bulk theories of gravity other than Einstein gravity. This understanding is crucial in order to understand systematics of how finite coupling effects in the field theory modify entanglement entropy. In this chapter we will only focus on those particular theories which arise from “classical” and local higher derivative corrections to the bulk theory1. Now our main objective will be to compute the universal term for EE for various higher derivative gravity theories using generalized entropy method. Rest of this chapter is mainly based on the work done with Prof. Aninda Sinha and Dr. Menika Sharma [2].

3.2 Generalized entropy and Feﬀerman-Graham expansion

Let us start by recapitulating generalized entropy method one more time. We will see following [3], the generalized gravitational entropy is deﬁned as,

S = −n∂n(ln[Z(n)] − n ln[Z(1)])n=1 , (3.1) where ln[Z(1)] is identified with the Euclidean gravitational action for which the period of the Euclidean time is 2π and the boundary condition for other fields collectively denoted as φ present in the action is φ(0) = φ(2π) . ln[Z(n)] is identified with the Euclidean gravitational action In for which the period of the Euclidean time is 2πn and the boundary condition for φ is still φ(0) = φ(2π) . This is the usual replica trick. Translating this fact for the holographic

1there are other type of corrections which arise from “quantum” or loop corrections to the eﬀective action which would include non-local eﬀects. Interested readers are referred to [1].

69 70 3.2. GENERALIZED ENTROPY AND FEFFERMAN-GRAHAM EXPANSION

case we can define In for a regularized geometry on a cone whose opening angle is 2π/n. We can analytically continue this for non integer n and then can compute the entropy. Also while evaluating ln[Z(n)] we can perform the time τ integral from 0 to 2π and multiply it by n so that ln[Z(n)] = n ln[Z]2π . The entropy calculated using this method is equal to the area of some codimension 2 surface where the time circle shrinks to zero which can be shown to be the minimal surface in Einstein gravity [3]. In this section we will show that this procedure also gives the correct entanglement entropy for higher curvature gravity theories. To compute the EE we have to start with some specific boundary geometry for the nth solution. Then we can construct our bulk spacetime using the Fefferman-Graham expansion. We will consider the following two 4-dimensional metrics following [4],

2 2 2 2 n 1−n 2 2 2 ds = f(r, b)dr + r dτ + (f0 + r d cos(τ)) dφ + dz cylinder (3.2) 2 2 2 2 n 1−n 2 2 2 2 dssphere = f(r, b)dr + r dτ + (f0 + r d cos(τ)) (dθ + sin θdφ )

r2+b2n2 where, f(r, b) = r2+b2 . For b → 0 and n → 1 limit these two metrics reduce to the cylinder and the sphere. This metric is known as “squashed cone” metric. The key point in Eq. (3.2) as compared to earlier regularizations e.g., [3] is the introduction of a regulator in the extrinsic curvature terms. This is needed since otherwise the Ricci scalar would go like (n − 1)/r and would be singular. Another important point is that b is a regulator which at this stage does not have an restriction except that f(0, b) = n2. In AdS/CFT we do not expect an arbitrary parameter to appear in the metric. b is here a dimensionful quantity having the dimension α>0 of r . So b must be proportional to f0(n − 1) such that it goes to zero as n → 1. We can take the metrics in Eq. (3.2) as boundary metrics and construct the bulk spacetime using the Feﬀerman-Graham expansion. Notice that our starting point is a smooth metric. At the end of the calculation, when we remove the regulators and compute EE, we will separately check what the contribution from the singularities is going to be. In the best case scenario, although the boundary metric will be singular once the regulator is removed, the bulk metric will at most be mildly singular, namely the on-shell bulk action will not be singular, following the terminology used in [3]. As in [3] we could have done a conformal transformation to pull out 2 2 dr2 a factor of r such that the r, τ part of the metric looks like dτ + r2 which would make the time-circle non-shrinking. We can use this form of the metric with a suitable regularization and do the calculation after verifying that there are no singularities in the bulk. Since this is a conformal transformation of a smooth metric, the results for the universal part of the EE will remain unchanged. One can write the bulk metric as,

(0) (2) dρ2 (g + ρg + .....) ds2 = L˜2 + ij ij dxidxj . (3.3) 4ρ2 ρ CHAPTER 3. ENTANGLEMENT ENTROPY FROM GENERALIZED ENTROPY 71

(2) To evaluate the log term we will need the gij coeﬃcient and here we will use Eq. (3.2) as (0) gij . We will consider here a 5 dimensional bulk lagrangian. In this case,

L˜2 1 g(2) = − (R(0) − g(0)R(0)) , ij 2 ij 6 ij

(0) (0) (0) (2) where Rij and R are constructed using gij . Note that in all subsequent calculations gij (2) will play an important role. The structure of gij is independent of the form of the higher derivative terms present in the action. Only terms proportional to n − 1 in the on-shell bulk action contributes to the SEE. The calculation is similar in spirit to the way that Weyl anomaly is extracted in AdS/CFT, e.g., [5] except that the n − 1 dependence comes from the neighbourhood of r = 0 in the bulk action. In the next section we proceed to give details of this.

Regularization procedure

To illustrate the regularization procedure in some detail, we start with some simple examples 2 (2) involving curvature polynomials . We calculate gij and evaluate the following integral , Z 5 √ µν I1 = d x g RµνR . (3.4)

Following3 [4], in the integrand, we put r = bx then expand around b = 0 and pick out the 0 O(b ) term. The r integral is between 0 < r < r0. This makes the upper limit of the x integral to be r0/b which goes to infinity. We will be interested in the log term so we extract 1 first the coefficient of ρ term which has the following form, Z Z ∞ dρ 2 2 2n−3 3 I1 = b dτ d y dx (n − 1) ζ(x, n)(bx) + O((n − 1) ) . (3.5) ρ 0 We have here shown only the leading term. Note that at this stage the integrand is proportional to (n − 1)2 whereas we need get something proportional to (n − 1). The integral over x will give a factor of 1/(n − 1). We will now expand ζ(x, n) around n = 1 and then carry out the integral over x. After expanding around n = 1 this leads to

2 I1 = (n − 1)ζ1 + O(n − 1) + ··· . (3.6)

Note that the rn factor in the cylindrical and the spherical parts in Eq. (3.2) were crucial in reaching this point. ζ1 is just a quantity independent of the regularization parameters b, d, 2We thank Sasha Patrushev for discussions on this topic. 3Alternatively we could have done the expansion around x = 0 ﬁrst, since it was assumed in [4] that the metric is valid between 0 < r < b f0. Then we could have integrated x in the neighbourhood of x = 0. The results are identical. 72 3.2. GENERALIZED ENTROPY AND FEFFERMAN-GRAHAM EXPANSION

, 0. The same procedure is applied for other curvature polynomial integrals. For example, Z √ 5 µνρσ 2 I2 = g d xRµνρσR = (n − 1)ζ2 + O(n − 1) + ··· , Z (3.7) √ 5 2 2 I3 = g d xR = O(n − 1) + ··· .

3.2.1 Four derivative theory

Let us now consider the general R2 theory lagrangian as shown below.

L = L1 + L2 , (3.8) where 12 L = R + (3.9) 1 L2 is the usual Einstein-Hilbert lagrangian with the cosmological constant appropriate for ﬁve- dimensional AdS space and

L2 L = λ R Rαβγδ + λ R Rαβ + λ R2 (3.10) 2 2 1 αβγδ 2 αβ 3 is the R2 lagrangian. Also we will henceforth consider only a 5 dimensional bulk spacetime unless mentioned otherwise. The boundary of this spacetime is at ρ = 0 . We then evaluate the total action 1 and extract the ρ term and carry out the τ integral. We put r = b x and expand Eq. (3.8) around b = 0 . Then we pick out the O(b0) term.

Z 2n 1 dρ 2 2 (bx) 3 S = − 3 dx d y (n − 1) a1 3 + O((n − 1) ) , (3.11) 2`P ρ x where A(x) a1 = . (3.12) 2 5/2 2 4 18 b f∞ f0 (1 + x ) A(x) is a function of x . For the cylinder we get,

3 2 8 6 4 2 8 6 4 2 A(x) = πL f∞ λ1 4x + 16x + 43x + 36x + 9 − 2 20x + 80x + 161x + 108x + 27 8 6 4 2 8 6 4 2 (λ2 + 5λ3) + 6f∞ 5x + 20x + 38x + 24x + 6 − 3 8x + 32x + 59x + 36x + 9 . (3.13)

We then carry out the x integral. Z ∞ 1 dρ 2 A1(x, n) S = − d y , (3.14) 3 2 5/2 2 2 2`P ρ 36 b f∞ (n − 1) f0 x 0 CHAPTER 3. ENTANGLEMENT ENTROPY FROM GENERALIZED ENTROPY 73 where 3 2 2n 4 2 2 A1(x, n) = πL (n − 1) (bx) (n − 1)x 2F1 2, n + 1; n + 2; −x f∞(5λ1 − 14(λ2 + 5λ3))

4 2 2 + 6f∞ − 3 + 2(n − 1)x 2F1 3, n + 1; n + 2; −x f∞(5λ1 − 14(λ2 + 5λ3)) 2 2 4 2 4 + 6f∞ − 3 + 2F1 4, n + 1; n + 2; −x 4f∞λ1x (1 − n) − 40f∞λ2x (1 − n) 2 4 4 4 2 − 200f∞λ3x (1 − n) + 30f∞x (1 − n) − 24x (1 − n) − 9f∞λ1(1 + n) 2 2 + 54f∞λ2(1 + n) + 270f∞λ3(1 + n) − 36f∞(1 + n) + 27(1 + n) . (3.15)

For the cylinder after doing the expansion around n = 1 and the remaining integrals (note 2 that ρ = z in the coordinates used in [6] and so ln δρ = 2 ln δ),

cH f S = − ln( 0 ) . (3.16) EE 2R δ

1 2 Here we have used 1 = f∞ − 3 f∞(λ1 +2λ2 +10λ3) and c is given in Eq. (2.19). For the sphere we proceed similarly. In this case, expanding Eq. (3.11) around b = 0 we get ,

Z 2n 1 dρ 2 2 (bx) 3 S = · · · − 3 dx d y (n − 1) a1 3 + O((n − 1) ) , (3.17) 2`P ρ x where A(x) a1 = . (3.18) 2 5/2 4 2 4 72 b f∞ f0 (1 + x ) A(x) is a function of x . For the sphere we get, 3 4 10 2 4 10 4 8 2 4 8 A(x) = −πL sin(θ) 300 b λ3x f∞ − 45 b x f∞ + 600 b λ3x f∞ − 90 b x f∞

4 6 2 4 6 4 10 4 8 4 6 2 2 10 2 + 300 b λ3x f∞ − 45 b x f∞ + 36 b x + 72 b x + 36 b x − 680 b λ3R x f∞ 2 2 10 2 2 8 2 2 2 8 2 2 6 2 2 2 6 + 84 b R x f∞ − 1920 b λ3R x f∞ + 216 b R x f∞ − 120 b λ3R x f∞ + 36 b R x f∞ 2 2 4 2 2 2 4 2 2 10 2 2 8 2 2 6 2 2 4 + 1120 b λ3R x f∞ − 96 b R x f∞ − 60 b R x − 144 b R x − 36 b R x + 48 b R x 2 4 2 2 6 2 2 6 4 2 4 4 8 6 4 + 2λ1f∞ − 3b x + 1 x + 2b R 7x + 24x − 3x − 20 x + 4R x − 73x + 242x 2 2 4 2 2 6 2 2 6 4 2 4 + 361x + 54 + 4λ2f∞ 15b x + 1 x − 2b R 17x + 48x + 3x − 28 x 4 8 6 4 2 4 2 4 8 2 4 8 + 4R 13x − 13x + 230x + 301x + 54 + 4320λ3R f∞ + 1040λ3R x f∞ − 192R x f∞ 4 6 2 4 6 4 4 2 4 4 4 2 2 − 1040λ3R x f∞ − 168R x f∞ + 18400λ3R x f∞ − 2760R x f∞ + 24080λ3R x f∞ 4 2 4 4 8 4 6 4 4 4 2 4 − 3144R x f∞ − 576R f∞ + 168R x + 264R x + 2208R x + 2328R x + 432R ) . (3.19) 74 3.2. GENERALIZED ENTROPY AND FEFFERMAN-GRAHAM EXPANSION

After doing the x integral,

Z ∞ 1 dρ 2 A1(x, n) S = − d y , (3.20) 2`3 ρ 2 5/2 4 2 P 144 b f∞ n (n + 1) f0 x 0 where A1(x, n) is a function of x and n . 3 2n 4 2 2 A1(x, n) = πL (n − 1) sin(θ)(bx) − 8(n + 1)R f∞(λ1(145x (n − 1) + 54n) + 2(λ2 + 5λ3)

2 2 2 2 2 (85x (n − 1) + 54n)) − 3f∞(n(35x + 24) − 35x ) + n 75x + 54 − 75x 2 4 4 2 + 2F1 4, n + 1; n + 2; −x (−72(n − 1)nR x (λ1 + 2 (λ2 + 5λ3)) f∞ − 3f∞ + 3 ) 2 2 4 2 2 2 + 2F1 3, n + 1; n + 2; −x (8(n − 1)nR x (f∞(λ1(15b + 328R ) − 2 (λ2 + 5λ3) 2 2 2 2 2 2 2 21b − 232R ) + 6 3b − 46R f∞ − 9b + 192R )) + 2F1 1, n + 1; n + 2; −x 4 4 2 2 4 2 2 4 2 2 4 ((n − 1)nx (−36b + 60b R + f∞(45b − 84b R + 2f∞(λ1(3b − 14b R + 580R ) 4 2 2 4 4 4 2 − 2(λ2 + 5λ3)(15b − 34b R − 340R )) − 840R ) + 600R )) + 2F1 2, n + 1; n + 2; −x 4 2 4 2 2 4 4 2 2 4 (−3(n − 1)nx (2f∞(λ1(b + 2b R − 264R ) − 2(λ2 + 5λ3)(5b − 2b R + 168R )) 4 2 2 4 4 2 2 4 + 3(5b − 4b R + 136R )f∞ − 12(b − b R + 24R ))) . (3.21)

For the sphere after doing the expansion around n = 1 and the remaining integrals ,

f S = −4 a ln( 0 ) , (3.22) EE δ

1 2 where we have used 1 = f∞ − 3 f∞(λ1 + 2λ2 + 10λ3) and a is given in Eq. (2.19). Thus we get the expected universal terms using the regularization proposed in [4].

3.2.2 New Massive Gravity

As an example for a calculation of generalized gravitational entropy in other dimensions, we consider the New Massive Gravity action in three dimensions [7] and use the notation in [8] Z 1 3 √ 2 2 µν 3 2 S = − d x g R + 2 + 4λL (RµνR − R ) . 2`P L 8

2 Here 1 − f∞ + f∞λ = 0. The entropy functional for this is not intrinsic as compared to the three dimensional Einstein gravity and is given by Z 2π √ 2 µ ν 1 s 3 SEE = dx gxx 1 + 4λL ([Rµνns ns − K Ks] − R) . (3.23) `P 2 4 CHAPTER 3. ENTANGLEMENT ENTROPY FROM GENERALIZED ENTROPY 75

The integral is over the one dimensional entangling region. s denotes the two transverse direction. We calculate the generalized gravitational entropy following the same procedure as used above. The two dimensional squashed cone metric is given by

ds2 = f(r, b)dr2 + r2dτ 2 . f0 in this case also corresponds to the radius of the entangling surface. In 3 dimensions [9, 10] L˜2 g(2) = − R(0)g(0) + t (3.24) ij 2 ij ij Only divergence and trace of tij are known.

(0) ij (0) i gij t = R , ∇ tij = 0 . 2b2 (n2 − 1) R(0) = − . (3.25) (b2n2 + r2)2 2 Using 1 − f∞ + f∞λ = 0 and we get, 1 Z dρ Z 2π Z r=f0 L(rb2(n2 − 1)(1 + 2f λ) S = ··· + dτ dr √ ∞ + ··· . (3.26) 1/2 2 2 2 2 2 3/2 2`P ρ 0 r=0 f∞ b + r (b n + r )

Note that tij does not enter in the calculation of the universal term. After doing the integrals we get  s  Z dρ πL (1 + 2f λ) 1 b2 + f 2 S = ··· + √ ∞ − 0 + ··· . (3.27)  2 2 2  ρ `P f∞ n b n + f0 Then expanding around b = 0 and n = 1 we get the correct universal term c f S = ln( 0 ) , (3.28) EE 3 δ

c 2πL(1+2f∞λ) where, 3 = 1/2 . f∞ `P 3.2.3 Quasi-Topological Gravity

The six-derivative action for quasi-topological gravity is given below [11], 1 Z √ 12 L2λ L47µ 5 (3.29) S = − 3 d x g R + 2 + GB + Z5 2`P L 2 4 where,

µνρσ µν 2 GB = RµνρσR − 4RµνR + R and 3 9 15 Z = R ν σR α βR µ ρ + R RµνρσR − R Rµνρ Rσα + R RµρRνσ 5 µ ρ ν σ α β 8 µνρσ 7 µνρσ α 7 µνρσ (3.30) 18 33 15 + R RσαRµ − R RαβR + R3 . 7 µσ α 14 αβ 56 76 3.3. COMMENT ABOUT SINGULARITIES IN THE METRIC

Following exactly the same procedure we can derive the holographic entanglement entropy for this six derivative gravity theory and obtain the correct universal terms.. For the sphere we get,

2 3 4π L 2 f0 SEE = − (1 − 6f∞λ + 9f µ) ln( ) . (3.31) 3/2 3 ∞ δ f∞ `P

For the cylinder 2 3 π L H 2 f0 SEE = − (1 − 2f∞λ − 3f µ) ln( ) . (3.32) 3/2 3 ∞ δ 2f∞ `P R

3.2.4 α03 IIB supergravity

The action for this follows from [12] Z 1 5 √ 12 6 S = − 3 d x g R + 2 + L γκ5 (3.33) 2`P L where, 1 κ = C CρβµσCαδγ Cν − C Cαβ Cµρ Cνσδγ . 5 αβµν ρ δγσ 4 αβµν ρσ δγ 1 03 6 Cαβµν is the Weyl tensor in 5 dimensions. In the context of IIB string theory, γ = 8 ζ(3)α /L . For this theory we ﬁnd that the universal parts of EE do not get corrected compared to the Einstein case. This is expected since from the perspective of the AdS/CFT correspondence, the C4 correction correspond to 1/λ corrections and the anomalies are not expected to receive such corrections. Recently the eﬀect of the C4 correction on Renyi entropy was analysed in [13].

3.3 Comment about singularities in the metric

(2) There are singularities in the ﬁve dimensional metric coming entirely from gij . We expand the metric around r = 0 . Upto the leading order the metric is shown below.

For the sphere (diagonal components are gρρ, grr, gττ , gθθ, gφφ ),

 L2  2 0 0 0 0 4f∞ρ 2  0 (n−1) cos(τ)L + 1 0 0 0   f0 rf∞ ρ  2  0 0 r2 − L (n−1) r cos(τ) 0 0  . (3.34)  ρ f0f∞   2   f0  0 0 0 ρ 0  2 2  f0 sin (θ) 0 0 0 0 ρ REFERENCES 77

For the cylinder,

 L2  2 0 0 0 0 4f∞ρ 2  0 (n−1) cos(τ)L + 1 0 0 0   2f0 rf∞ ρ  2  0 0 r2 − L (n−1) r cos(τ) 0 0  . (3.35)  ρ 2f0f∞   2   0 0 0 f0 0   ρ  1 0 0 0 0 ρ

The grr component is singular in r. The other components are non singular. However it is easy to see that the determinant does not have a singularity at r = 0. The singularity in the metric gives rise to singularities in the components of the Riemann tensor. We have explicitly checked that these singularities do not enter in the higher derivative actions considered in this paper. Hence these are mild singularities in the sense used in [3]. Note that in order to (2) calculate the universal part of EE in four dimensions only gij is important.

3.4 Discussion

The newly proposed regularization in [4] yields the expected universal terms in the EE in higher derivative gravity theories dual to four dimensional CFTs. We considered the Feﬀerman-Graham metric with the regularized metrics in [4] as the boundary metric. Then we computed the generalized gravitational entropy as proposed in [3]. The universal log terms worked out to be as expected. We showed that upto the order we are interested in, the singularities in the metric are mild. As pointed out in [3] we could also have done a conformal transformation of the boundary metric with conical singularity such that it is non-singular and then done the calculation. We expect the results to be identical. One can possibly use other regularization scheme and compute EE using generalized entropy to obtain correct results, but we will demonstrate the importance of this particular regularization in the next chapter when we will try to connect EE with the Wald entropy formula.

References

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[5] S. ’i. Nojiri and S. D. Odintsov, “On the conformal anomaly from higher derivative gravity in AdS / CFT correspondence,” Int. J. Mod. Phys. A 15, 413 (2000) [hep- th/9903033]. A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, “Holo- graphic GB gravity in arbitrary dimensions,” JHEP 1003, 111 (2010) [arXiv:0911.4257 [hep-th]].

[6] L. -Y. Hung, R. C. Myers and M. Smolkin, “On Holographic Entanglement Entropy and Higher Curvature Gravity,” JHEP 1104 (2011) 025 [arXiv:1101.5813 [hep-th]].

[7] E. A. Bergshoeﬀ, O. Hohm and P. K. Townsend, “Massive Gravity in Three Dimen- sions,” Phys. Rev. Lett. 102, 201301 (2009) [arXiv:0901.1766 [hep-th]].

[8] A. Sinha, “On the new massive gravity and AdS/CFT,” JHEP 1006, 061 (2010) [arXiv:1003.0683 [hep-th]].

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[12] D. J. Gross and E. Witten, “Superstring Modiﬁcations of Einstein’s Equations,” Nucl. Phys. B 277, 1 (1986). M. B. Green and C. Stahn, “D3-branes on the Coulomb branch and instantons,” JHEP 0309, 052 (2003) [hep-th/0308061]. M. F. Paulos, “Higher derivative terms including the Ramond-Ramond ﬁve-form,” REFERENCES 79

JHEP 0810, 047 (2008) [arXiv:0804.0763 [hep-th]]. S. S. Gubser, I. R. Klebanov and A. A. Tseytlin, “Coupling constant dependence in the thermodynamics of N=4 supersymmetric Yang-Mills theory,” Nucl. Phys. B 534, 202 (1998) [hep-th/9805156]. A. Buchel, J. T. Liu and A. O. Starinets, “Coupling constant dependence of the shear viscosity in N=4 supersymmetric Yang-Mills theory,” Nucl. Phys. B 707, 56 (2005) [hep-th/0406264]. R. C. Myers, M. F. Paulos and A. Sinha, “Quantum corrections to eta/s,” Phys. Rev. D 79, 041901 (2009) [arXiv:0806.2156 [hep-th]]. A. Buchel, R. C. Myers, M. F. Paulos and A. Sinha, “Universal holographic hydrodynamics at ﬁnite coupling,” Phys. Lett. B 669 (2008) 364 [arXiv:0808.1837 [hep-th]]. W. H. Baron and M. Schvellinger, “Quantum corrections to dynamical holographic thermalization: entanglement entropy and other non-local observables,” arXiv:1305.2237 [hep-th].

[13] D. A. Galante and R. C. Myers, “Holographic Renyi entropies at ﬁnite coupling,” JHEP 1308 (2013) 063 [arXiv:1305.7191 [hep-th]]. 80 REFERENCES Connection between entanglement 4 entropy and Wald entropy

4.1 Introduction

As discussed in the Chapter.(2), there is a striking similarity between black hole entropy and entanglement entropy (EE) in Einstein gravity. In the black hole case, there exists a simple generalization of the area law for calculating the entropy of a black hole in any general higher-derivative gravity theory, known as the Wald entropy [1,2,3]. It is by now well known, in an arbitrary theory of gravity, taking the Wald entropy functional in AdS space will give rise to the wrong universal terms in EE [4,5]. A prescription is given for evaluating EE in [6,7], suitably modifying the Wald entropy functional for general surfaces based on the generalized entropy principle. In the Chapter.(2) of this thesis we have shown that, for any arbitrary theory of gravity it is not possible to derive those conjectured entropy functionals [8]. On the other hand, based on the Noether charge prescription one can derive the Wald entropy functionals, there by increasing its geometrical signiﬁcance. In this chapter our main objective is to explore the possibility of connecting EE with the Wald entropy as that we will give us a concrete holographic derivation of EE based on Noether charge method, enabling us to avoid the conﬂicts as pointed out in Chapter.(2). In this chapter we will consider the background constructed out of the squashed cone metric in Eq. (3.3) and evaluate the Wald entropy for various theories of gravity in this background on a particular surface. This will produce the correct universal terms in EE. After that we will comment on the connection with Iyer-Wald prescription for computing entropy for black holes with dynamical horizon, opening up a possibility of connecting EE rigorously with the Noether charge. In the Chapter.(3) we have shown that if we evaluate generalized entropy [9] in the squashed cone background we will get the correct universal terms for EE. In this chapter it will be clear why that particular background is important, as that will play a crucial role in our quest to connect EE with the Wald entropy.

81 82 4.2. WALD ENTROPY

4.2 Wald Entropy

Before proceeding further, let us mention the key steps in Wald’s derivation of the entropy functionals for black holes for general theories of gravity [2]. Let us consider a diffeomorphism invariant lagrangian L. We denote all the dynamical fields collectively by φ. In general φ carries indices depending on the nature of the fields present. Varying L with respect to φ we get, Z Z Z d d d µ d xδL = d xE. δφ + d x∇ Φµ (4.1)

First piece gives the equation of motion and the second piece is a surface term. Now one can construct a Noether current out of this surface term as follows

µ µ µ J = Φ (φ, Lζ φ) − ζ L, (4.2) where ζµ denotes the killing vectors associated with the diﬀeomorphism invariance. Using the equation of of motion one can write down,

J = dQ (4.3) where Q is the Noether charge. At this stage several ambiguities enter in the calculation. For example, one can add a closed form to Q to deﬁne a new charge Q˜ → Q + dΨ such that,

J = dQ˜ (4.4) still holds. Also one can add a total derivative term to L which will not aﬀect the equation of motion but it will change the expression for Q such that Eq. (4.3) still holds. For more detailed discussion of these ambiguities interested readers are referred to the original reference [2]. But it was shown by Iyer and Wald in the [2] that all these ambiguities vanish on a bifurcation surface and one can write down a ﬁrst law like relation for the entropy. Z κ δ Q = δS (4.5) 2π wald where κ is the surface gravity related to the temperature of the black hole and the charge is integrated over the horizon of the black hole. From this we one can read of the Wald entropy solely in terms of geometrical quantities. Z √ d−2 ∂L Swald = d x h ˆαβˆγδ . (4.6) ∂Rαβγδ This expression is evaluated on the black hole horizon, which is a codimension-2 surface. 1 2 2 1 Here ˆαβ = nαnβ − nαnβ is the binormal corresponding to the two transverse directions 1, 2 . CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALD ENTROPY 83

One crucial point is that, Wald formula is applicable only on a bifurcation surface. For a static black hole, the horizon is a bifurcation surface for which the extrinsic curvatures vanish. For surfaces with non zero extrinsic curvature one cannot apply this formula. Now let us see what it gives for our case. We will evaluate Eq. (4.6) on the background constructed in Eq. (3.3) on a codimension-2 surface r = 0 , τ = 0. Rest of this chapter is mainly based on the work done with Prof. Aninda Sinha and Dr. Menika Sharma [10].

4.3 Four derivative theory

The Wald entropy is calculated from Eq. (3.8). We have to ﬁrst evaluate the following,

∂L 1 αγ βδ αδ βγ 2 αβγδ 1 βδ αγ βγ αδ αδ βγ αγ βδ = (g g − g g ) + L λ1R + λ2 g R − g R − g R + g R ∂Rαβγδ 2 4 1 + λ R gαγgβδ − gαδgβγ 2 3 (4.7) and after some simpliﬁcations we get,

Z √ 2 2π 3 L ν rµ µ ν rρ sσ Swald = 3 d x h 1 + (2λ3R + λ2Rµνnr n + 2λ1Rµνρσnr ns n n ) . (4.8) `P 2 In this section we will show that starting with the boundary metrics in Eq. (3.2) we can construct a bulk spacetime on which Swald will produce the expected universal parts for the entanglement entropy for both cylindrical and spherical region. Note that (4.8) diﬀers from (2.11) by the O(K2) terms.

4.3.1 Cylinder

As we will show, a particular form of the regularization b = α(n − 1)1/2 appearing in the Eq. (3.2), where α is some number which we will determine later (it will turn out to be surface dependent but theory independent), is needed to get the correct universal term. Recall that the only restriction on b was that f(r, b) has to be n2 in the r = 0 limit. However, in holographic calculations we expect that the bulk metrics will only depend on the AdS radius, the radius of the entangling region and n. As such we can expect that the only way that b → 0 would arise in holographic calculations is such that b is some positive power of (n − 1). Now we will evaluate Eq. (4.8) using Eq. (3.3) using the cylinder metric to be its boundary. 1 Then we extract the coeﬃcient of the ρ term. We set τ = 0 . There is no integral over r 84 4.3. FOUR DERIVATIVE THEORY in the Wald entropy as the entangling surface is located at r = 0, τ = 0 . We put r = b x . After that we expand around x = 0 and then expand around n = 1 . We retain only the n independent part as other terms vanish in n → 1 limit. Below we quote some intermediate steps after expanding in ρ, r and n respectively. It is important to take the limits in r, n in that particular in order to get the correct result [11]. After doing the ρ expansion we pick 1 out the ρ term of (4.8) which is shown below.

2π Z A(x, n) Swald = ··· + 3 dρdφdz + O(ρ) + ··· , (4.9) `P ρ where

L3 (n2 − 1) d−n ((4λ + 20λ − 2λ ) f − 1) (2f dn − d (n2 + n + x2 − 2) (bx)n) A(x, n) = 2 3 1 ∞ 0 . 2 3/2 2 2 2 24b f∞ (n + x )

Then expanding A(x, n) around x = 0 we get,

L3 (n2 − 1) f ((4λ + 20λ − 2λ ) f − 1) A(x, n) = 0 2 3 1 ∞ + ··· . (4.10) 2 4 3/2 12 b n f∞

If 2f √ b = √ 0 n2 − 1β(n) , 3 where β(1) = 1 we get upon further expanding A(x, n) around n = 1

3 L (1 + 2 (λ1 − 2 (λ2 + 5λ3)) f∞) A(x, n) = − 3/2 + O(n − 1) + ··· . (4.11) 16f0f∞

Notice that the choice for b was independent of the theory, i.e., in this case of λi’s. Finally we get, 2 3 π L H(1 + 2f∞(λ1 − 2λ2 − 10λ3)) f0 Swald = − ln( ) . (4.12) 3/2 3 δ 2f∞ `P f0 This is precisely what is expected.

4.3.2 Sphere

1 We proceed similarly for the sphere case. First we expand in ρ and pick out the ρ term.

2π Z A(x, n) Swald = ··· + 3 dρdθdφ + O(ρ) + ··· . (4.13) `P ρ CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALD ENTROPY 85

Here

L3d−2n sin(θ) A(x, n) = 4λ f (b2x2d2n(n2 + x2)2 − d2 (n4 − n3x2 + 3n2x2 + nx2 12b2f 3/2x2(n2 + x2)2 1 ∞ + x4 − x2)(bx)2n + d (n2 − 1)R x2(n2 + n + x2 − 2)(b dx)n − (n2 − 1)R2x2d2n) 2 2 2n 2 2 2 2 4 2 3 2 − (2(λ1 + 2(λ2 + 5λ3))f∞ − 1)(−2b x d (n + x ) + d (n (3x + 2) + n x + 3n2x4 − nx2 − x4 + x2)(bx)2n + d (n2 − 1)Rx2(n2 + n + x2 − 2)(b d x)n − (n2 − 1)R2x2 d2n) (4.14)

Then expanding A(x, n) around x = 0 we get 1,

L3 sin(θ)2b2n4 (4 (λ + λ + 5λ ) f − 1) + (n2 − 1) f 2 ((−2λ + 4λ + 20λ ) f − 1) A(x, n) = 1 2 3 ∞ 0 1 2 3 ∞ . 2 3/2 4 12b f∞ n (4.15) Only the x independent term is shown. If (for consistency checks see below)

√ 2 b = f0 n − 1β(n) (4.16) where β(1) = 1, expanding around n = 1 we get,

3 L sin(θ) (1 − 2 (λ1 + 2 (λ2 + 5λ3)) f∞) A(x, n) = − 3/2 + O(n − 1) + ··· . (4.17) 4f∞

As in the cylinder case, notice that the choice for b is theory independent. Finally we get,

2 3 4π L (1 − 2f∞(λ1 + 2λ2 + 10λ3)) f0 Swald = − ln( ) (4.18) 3/2 3 δ f∞ `P

We have ﬁxed b for both the cylinder and the sphere case. In all the subsequent calculations of Wald entropy we will use these same values for b.

1Remember that at this stage n = 1 + . Thus we will drop x2n compared to x2. 86 4.4. QUASI-TOPOLOGICAL GRAVITY

4.4 Quasi-Topological gravity

The Wald entropy is calculated for (3.29) using (4.6) . For this case,

∂L 1 αγ βδ αδ βγ 2 αβγδ 1 βδ αγ βγ αδ αδ βγ αγ βδ = (g g − g g ) + L λ1R + λ2 g R − g R − g R + g R ∂Rαβγδ 2 4 1 7µL4 µ + λ R gαγgβδ − gαδgβγ + (3µ (RαργσRβ δ − RαρδσRβ γ )) + 2 [(gαγgβδ − gαδgβγ) 2 3 4 1 ρ σ ρ σ 2 µ R Rµνρσ + 4 RRαβγδ] + 3 [gβδRαρσµRγ − gβγRαρσµRδ − gαδRβρσµRγ + gαγRβρσµRδ µνρσ 4 ρσµ ρσµ ρσµ ρσµ µ − 2RγρRαβδ + 2RδρRαβγ + 2RβρRα γδ − 2RαρRβ γδ] + 4 (Rρσ[gβδRα γ − gβγRα δ ρ ρ ρ ρ 2 ρ σ ρ σ 3µ − gαδRβ γ + gαγRβ δ ] + [RαγRβδ − RαδRβγ]) + 5 [gβδRασRγ − gβγRασRδ ρ σ ρ σ 4 σ σ µ − gαδRβσRγ + gαγRβσRδ ] + 6 R gβδRαγ − gβγRαδ + gαγRβδ − gαδRβγ σ σ 2 3 + (gαγgβδ − gαδgβγ)R Rµν + µ (R2[gαγgβδ − gαδgβγ]) . µν 2 7 (4.19)

Now the coeﬃcients are,

3 9 15 18 33 15 µ = 1 , µ = , µ = − , µ = , µ = , µ = − , µ = , 1 2 8 3 7 4 7 5 7 6 14 7 56

2 and λ2 = −4λ1, λ3 = λ1 = λ. Proceeding similarly as mentioned for the R theory we get the expected universal terms. For the cylinder 2,

2 3 π L H 2 f0 Swald = − (1 − 2f∞λ − 3f µ) ln( ) . (4.20) 3/2 3 ∞ δ 2f∞ `P R

For the sphere,

2 3 4π L 2 f0 Swald = − (1 − 6f∞λ + 9f µ) ln( ) . (4.21) 3/2 3 ∞ δ f∞ `P

Again note that the choice for α did not depend on the theory.

2The c and a coeﬃcients for an arbitrary higher derivative theory can be easily calculated using the short-cut mentioned in the appendix of [12]. CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALD ENTROPY 87

4.5 α03 IIB supergravity

The Wald entropy is calculated for (3.33) using (4.6) . For this case,

∂L 1 αγ βδ αδ βγ 6 1 βγ αµδν ρση βδ αµγν ρση = (g g − g g ) + L γ (g C CνρσηCµ − g C CνρσηCµ ∂Rαβγδ 2 3 1 + gαδCβµγνC C ρση − gαγCβµδνC C ρση) + (gαγgβδ − gαδgβγ)(C µ ρCσηνζ C νρση µ νρση µ 6 σ ν ηρζµ 1 1 − C ρσCµνηζ C ) + (gβδCαρζσC Cγ µν − gαδCβρζσC Cγ µν 2 µν ηρζσ 6 ρσµν ζ ρσµν ζ 1 − gβγCαρζσC Cδ µν + gαγCβρζσC Cδ µν) + (gβδCαρζσCγµ νC ρσµν ζ ρσµν ζ 6 ρ ζσµν αδ βρζσ γµ ν βγ αρζσ δµ ν αγ βρζσ δµ ν − g C C ρ Cζσµν − g C C ρ Cζσµν + g C C ρ Cζσµν) αρ σ βµδη γ βρ σ αµδη γ αρ σ βµγη δ βρ σ αµγη δ + (C µ C C ρησ − C µ C C ρησ − C µ C C ρησ + C µ C C ρησ) 1 2 − (Cγδσζ Cβ Cα µρ + Cαβσζ Cδ Cγ µρ) + (gαδCβρζνC Cγµ σ 2 ζµρ σ ζµρ σ 3 ρσνµ ζ βδ αρζν γµ σ βγ αρζν δµ σ αγ βρζν δµ σ − g C CρσνµC ζ + g C CρσνµC ζ − g C CρσνµC ζ ) . (4.22)

Proceeding similarly as mentioned for the R2 theory we get the expected universal terms. For the cylinder, 2 3 π L H f0 Swald = − 3 ln( ) . (4.23) 2`P R δ For the sphere, 2 3 4π L f0 Swald = − 3 ln( ) . (4.24) `P δ As expected, for this case the universal terms are independent of the higher derivative correction.

4.6 Connection with Ryu-Takayanagi

The Ryu-Takayanagi calculation involves the minimization of an entropy functional3. For both the sphere and the cylinder, one can check that minimizing the Wald area functional in the Feﬀerman-Graham background for squashed cones leads to the correct universal terms provided we choose b as mentioned above. Recall that the Wald entropy functional in AdS spacetime was not the correct one [4,5]. However, our background is not AdS and it turns out that the Wald entropy functional leads to the correct universal terms. We show this for

3We thank Rob Myers for discussions on this section. 88 4.6. CONNECTION WITH RYU-TAKAYANAGI

α the cylinder, the sphere case working similarly. Putting r = R(ρ) = r0 + r1ρ around ρ = 0 leads to r0 = 0 and the equation

n αn+1 2 n 2α cnr1 ρ − 4r1Rc α(α − 2)ρ = 0 , where we have shown the leading terms which would contribute around n = 1. If we set n = 1 we recover the result α = 1, r1 = −1/(4f0) for a cylinder–this is expected. The n = 1 boundary geometry is just flat space with the dual bulk being AdS. Hence we expect to recover the RT result. However if n = 1 + , then it is easy to see that either r1 = 0 or α = 2 or r1 = −1/(4f0) and α = 1 + . As in the RT case, only the linear term in R(ρ) would have affected the universal term–since α 6= 1 if n = 1 + we find that there is no linear term. For n 6= 1 the minimal surface is at r = 0 = τ. This is the reason why the Wald entropy on the r = 0 = τ surface and the RT entropy functional approach give the same result for the universal terms in the squashed cone background. We now point out a direct comparison between the calculation done in AdS spacetime and that in the squashed cone background for the sphere in what follows. The Ryu-Takayanagi prescription was implemented in the following way for a spherical entangling surface. Consider the AdS5 metric with the boundary written in spherical coordinates L˜2 ds2 = (dz2 + dt2 + drˆ2 +r ˆ2dθ2 +r ˆ2 sin2 θdφ2) . (4.25) z2 2 Now putr ˆ = f(z) = f0 + f2z + ··· and t = 0 and minimize the relevant entropy functional. Implicitly our analysis says that this surface and the r = 0 = τ surface in the coordinate system we have been using are related. Since in both cases the extrinsic curvatures vanish we can attempt to make a direct comparison. In order to do this we make a coordinate transformation: dz dρ p1 + f 0(z)2 = . (4.26) z 2ρ 2 2 2 2 2 2 Around ρ = 0 we will find z = ρ − 2f2 ρ + ··· and f(z) /z = f0 /ρ + 2f0f2(1 + f0f2) + ··· . Now around ρ = 0, the metric on the r = 0 = τ surface takes the form dρ2 ds2 = L˜2[ + K(ρ)(dθ2 + sin2 θdφ2)] , (4.27) 4ρ2 where f 2 L˜2 K(ρ) = 0 − (2b2n4 + (n2 − 1)f 2) . ρ 6b2n4 0 This also shows that for n 6= 1 minimal surface is at r = 0 = τ . Now choosing b as in Eq. (4.16), expanding upto O((n − 1)0) and comparing with the RT calculation we find f2 = −1/(2f0). This is exactly what we would have got if we minimized the RT area functional (or the relevant higher derivative entropy functional) in AdS space. This also serves as a consistency check for the choice of b. CHAPTER 4. CONNECTION BETWEEN ENTANGLEMENT ENTROPY AND WALD ENTROPY 89

4.7 Comments on the connection with the Iyer-Wald prescription

Why does the Wald entropy functional lead to the correct result in our case? Wald’s formula in Eq. (4.6) is valid for a surface which is a local bifurcation surface on which the Killing field vanishes. For a bifurcation surface, the extrinsic curvatures vanish. SEE mentioned in (2.11) differs from Swald only by the extrinsic curvature terms. The Noether charge method of [1] needs a bifurcation surface to remove various ambiguities [2,3]. According to the prescription of Iyer and Wald [2], in order to compute the entropy for horizons which are not bifurcate, e.g., dynamical horizons, the curvature terms in ∂L are replaced by their boost invariant ∂Rabcd counterparts [2]. To do this we have to construct a boost invariant metric from our original 1 2 metric. Let gab be our starting d dimensional metric with the two normals na, nb . The boost 1 2 invariant part of gab will only have terms with the same number of n , n . We then consider a d − 2 dimensional surface and find a neighbourhood of it O such that for any points x belonging to this neighbourhood, we can find a point P which lies on a unit affine distance on a geodesic with a tangent vector va on the d − 2 dimensional plane perpendicular to this surface under consideration. Now we assign a coordinate system U, V, x1, ...xd−2 for the point a 1 2 x where U, V are the components of v along na and na. A change of normals under the a a b −1 a −1 boosts n1 → αn1, n2 → α n2 will change the coordinates as follows U → αU, V → α V.

Now we Taylor expand gab around Uand V ,

(0) gab = gab + U∂g + V ∂g + UV ∂∂g + ...... (4.28) We have shown the expansion schematically. Under boosts, the terms linear in U, V do not remain invariant. The prescription in [2] is to drop these terms. The UV term is invariant a ∂ a ∂ a under the boost. One important point to note is that , ψ = U( ∂U ) −V ( ∂V ) is a Killing field of the metric. This means that Lie derivative of gab with respect to ψ is zero. Effectively, we have constructed a new spacetime in which the original dynamical horizon becomes a bifurcate Killing horizon. The evidence for the existence of this bifurcation surface would be that extrinsic curvatures for this surface in the bulk background vanishes. Our entangling surface is a codimension- 2 surface. Now we calculate the extrinsic curvatures for this surface in the bulk Fefferman- Graham metric. There will be two of them—one along the direction of the normal (τ)n for τ = 0 and the other one along the normal (r)n for r = 0. We start with the 5 dimensional metrics given in Eq. (3.3). The non-zero components of the normals are 1 1 (τ)n = √ , (r)n = √ . τ gττ r grr

(τ) (r) With these we calculate the two extrinsic curvatures Kµν and Kµν . Then we put r = b x 90 4.8. UNIVERSALITY IN RENYI ENTROPY and τ = 0 as before. As the entangling surface is located at r = 0, τ = 0 we further do an (τ) (r) expansion around x followed by an expansion in n. Now Kab = 0 whereas Kab = A(x, n, ρ) is some function of x , n and ρ . First we expand it around x = 0 and then we do an expansion (r) around n = 1 . We find that Kab = 0 . Thus effectively the Fefferman-Graham construction is the same as the Iyer-Wald prescription, provided we take the limits in the manner prescribed in [11]. The replacement of i j n i j rKijdx dx by r Kijdx dx plays a key role in this construction. Recall that this was needed to keep the boundary Ricci scalar finite. Also another important point to notice that for the squashed cone metric there is no time like killing vector as the metric components are dependent on τ . The Wald-Iyer prescription calls for calculating the Wald functional in the context of black hole entropy where there exists a time like killing vector. But in the metric (3.2) the cos(τ) factor which breaks the time translational symmetry is accompanied by a factor of rn . In our calculation we have taken the r → 0 limit first and then the n → 1 limit. Thus the cos(τ) multiplied by rn is suppressed in this way of taking limits. For this reason we have an approximate time-translational symmetry in our new space time. Upto this point the discussion is independent of the choice of b. Now when one wants to evaluate the Wald entropy functional with this squashed cone metric one needs to specify b as mentioned in the previous sections for the sphere and the cylinder to obtain the correct universal terms. As there is no integral over r in the Wald entropy functional, the final result obtained will be b dependent as we have found and hence we have to choose b accordingly.

4.8 Universality in Renyi entropy

Before closing out this chapter let us mention an interesting application of this generalized entropy. In [13, 14, 15, 16] it was shown that for spherical entangling surfaces in four dimensions the Renyi entropy has a universal feature. Namely

∂nSn|n=1 ∝ cT .

In four dimensions cT ∝ c, the Weyl anomaly. If we use Eq. (4.15) and identify it as the 4 expression for Sn with the choice for b given below it , then we indeed find that this is true! This also works for the six and eight derivative examples. Thus this approach enables us to check some information away from n = 1. Further, as a bonus, we can predict what happens in the case of a cylindrical entangling surface where holographic results for the Renyi entropy are not available. If we use Eq. (4.10) or its analog for the six and eight derivative examples, we find that ∂nSn|n=1 ∝ cT still holds. It will be interesting to explicitly verify this in field theory.

4 In order to get the proportionality constant to work out, we will need to adjust ∂nβ(n)|n=1 in b. REFERENCES 91

4.9 Discussion

In this chapter we computed the Wald entropy on the r = 0 = τ co-dimension 2 surface in the Fefferman-Graham metric and found that it gives the correct universal terms for both spherical and cylindrical surfaces. In order to get the expected results, we needed to choose a surface dependent but theory independent regularization parameter. Recall that in bulk AdS space, from the entropy functional way of computing EE in general theories of gravity, one needed to use the entropy functional proposed in [6,7] , which differed from the Wald entropy functional by extrinsic curvature terms as shown in the Chapter.(2). These extrinsic curvature terms are important to get the correct universal piece for any entangling surface with extrinsic curvature. Whether EE can be thought of as a Noether charge needs further investigation. Our findings in this chapter seems to suggest that this may indeed be true. The Fefferman-Graham metric is the analog of the Iyer-Wald metric used to compute the entropy for dynamical horizons in [2]. Our conjecture then is that the Wald entropy (after appropriately fixing the regularization) evaluated on the r = 0 = τ co-dimension two surface in the Fefferman-Graham metric is going to capture the expected universal terms for any entangling surface. Recently it was pointed out in the [17], that the ambiguities in the Nother charge method can be fixed by demanding that the resulting entropy functional satisfies a generalized second law [18] when evaluated on any arbitrary surface. Also it has been shown that, those ambiguities correspond to the extrinsic curvature dependent terms of the proposed entropy functionals as shown in Eq. (2.6) upto O(K2). This opens up the possibility of deriving these holographic entropy functionals [6,7] directly from the Noether charge method and that will also solidify our conjecture of connecting EE with the Noether charge.

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[18] A. C. Wall, “Ten Proofs of the Generalized Second Law,” JHEP 0906 (2009) 021 [arXiv:0901.3865 [gr-qc]]. S. Sarkar and A. C. Wall, “Generalized second law at linear order for actions that are functions of Lovelock densities,” Phys. Rev. D 88 (2013) 044017 [arXiv:1306.1623 [gr-qc]]. 94 REFERENCES Constraining gravity using 5 entanglement entropy

5.1 Introduction

One can constrain gravity in several interesting ways by using quantum entanglement. In this chapter we focus on one of them and in the next chapter we discuss another one. From Chapter.(2) and Chapter.(3) it is evident that the extremal surface plays a crucial role in evaluating entanglement entropy (EE), in fact the entropy functionals have to be evaluated on the extremal surface to get the correct universal terms. In this chapter we will demonstrate that in the context of higher derivative gravity theories, one can constrain the coupling of the higher derivative terms by demanding the smoothness of the extremal surface. More speciﬁcally, we will derive constraints on the Gauss-Bonnet (GB) coupling by demanding that the entangling surface for sphere, cylinder and the slab close oﬀ smoothly in the bulk. The slab case was considered before in the Ref. [1]. The GB action is given in the Eq. (2.8) with

λ1 = λ3 = λ and λ2 = −4 λ and the corresponding entropy functional is the Jacobson-Myers (JM) functional as shown in the Eq. (2.42). At the onset note that treating the truncated GB gravity on its own leads to problems with entanglement entropy as was pointed out in the Ref. [1]. In particular if we consider an entangling surface that topologically looks like

M2 × R, then the R term in the JM entropy functional becomes topological. Adding more handles to the entangling surface will allow us to lower the entanglement entropy arbitrarily if λ > 0. Since this particular sign of λ happens to arise in many consistent examples in string theory (see for eg.[2]), this hints at a problem in interpreting GB gravity on its own as a model for theories describing c 6= a–of course, there is no reason to suspect any inconsistencies if this is just the ﬁrst perturbative correction in an inﬁnite set of higher derivative corrections. We will not have anything to add to this observation. We will simply focus on what constraints arise on the GB coupling by demanding smoothness and compare the result with the causality/positive energy constraints [3,4,5] as shown in Eq.(5.1). 7 9 − ≤ λ ≤ . (5.1) 36 100 To elaborate a bit more, we note at this point that the GB coupling λ is bounded. Following the Refs. [3,4] for the calculation of the three point correlation function of stress tensor one

95 96 5.2. SMOOTHNESS OF ENTANGLING SURFACE

needs to compute a energy ﬂux which comes form the insertion of ijTij , where ij and Tij are the polarization tensor and stress tensor respectively. Demanding the positivity of this energy ﬂux in the holographic set up we get the following three constraints and from those we obtain bounds on λ . These coincide with the bounds arising from micro-causality [6]. 9 Tensor channel : 1 − 10f λ ≥ 0 ⇒ λ ≤ ∞ 100 3 1 Vector channel : 1 + 2f λ ≥ 0 ⇒ − ≤ λ ≤ (5.2) ∞ 4 4 7 1 Scalar channel : 1 + 6f λ ≥ 0 ⇒ − ≤ λ ≤ ∞ 36 4 From this we get 7 9 − ≤ λ ≤ . 36 100 2 2 This is the same as the condition 1 − 4f∞λ − 60f∞λ ≥ 0. Rest of this chapter is based on the work done with Dr. Shamik Banerjee, prof, Aninda Sinha, Apratim Kaviraj and Kallol sen [7]. 5.2 Smoothness of entangling surface

The general strategy we will adopt is the following. The entangling surface equation follows from Eq.(2.43), coming from minimizing JM functional. We will consider 5 dimensional AdS spacetime. L˜2(dz2 + dt2 + dΣ2) ds2 = (5.3) z2 and choose a constant time slice t = 0. When we consider a spherical entangling , we parametrize dΣ in terms of spherical polar coordinates. dΣ2 = dr2 + r2(dθ2 + sin(θ)2dφ2) (5.4) and the extremal surface is given by r = f(z). We will consider a cylindrical surface also. For that we parametrize dΣ in terms of cylindrical polar coordinates,

dΣ = du2 + dr2 + r2dθ (5.5) where u is the coordinate along the length of the cylinder and the extremal surface is characterized again by r = f(z). For the slab, we simply write dΣ in cartesian coordinate and the corresponding extremal surafce is given by x = f(z), where x is one of the coordinates of Σ.

Now let us assume that the surface f(z) closes oﬀ smoothly at z = zh inside the bulk spacetime. Around this point, let us assume ∞ X α+i f(z) = ci(zh − z) . (5.6) i=0 CHAPTER 5. CONSTRAINING GRAVITY USING ENTANGLEMENT ENTROPY 97

0 We need to determine α and ci’s. At z = zh, f (z) → +∞ since the tangent to the surface will be perpendicular at that point. This means that 0 < α < 1 and c0 > 0. Using these two conditions, we will ﬁnd that λ will be bounded. Cylinder

Consider the cylinder case ﬁrst. In cylindrical coordinates, assume the required hypersurface to have the form r = f(z). From Eq. (6.72), we get the following equation,

00 0 0 2 0 2 zf (z) 6f∞λzf (z) + f(z) (4f∞λ + 1)f (z) − 2f∞λ + 1 − f (z) + 1 (5.7) 0 0 0 2 f (z) z(4f∞λ + 1)f (z) + 3f(z) f (z) − 2f∞λ + 1 − 2f∞λz + z = 0 . We take the trial solution Eq. (5.6) and determine an appropriate α. We obtain α = 1/2, 3/2. We will drop the second solution since this will lead to a conical tip. Expanding the eom in powers of (zh − z) and setting the leading order term to 0, we get 4 roots of c0. We take the two positive ones, r2q z (1 + 4f λ ± p1 − 10f λ + 16f 2 λ2) . (5.8) 3 h ∞ ∞ ∞ √ With f∞ = (1− 1 − 4λ)/2λ, this puts some constraints on λ. Since the bottom sign vanishes in the λ → 0 limit, we will ignore this solution. For the other case, we have 7 λ ≤ . (5.9) 64 The quantities inside the square root have to be positive to make the root real. If we look 2 2 carefully we will ﬁnd that 1 − 10f∞λ + 16f∞λ has to be positive. This is almost same as 2 2 that of the tensor channel constraint except for the extra additional factor of 16f∞λ . That 9 is why we get a bigger bound instead of λ< 100 .

Sphere

The eom reads,

00 0 2 0 2 2 0 2 zf (z) 12f∞λzf(z)f (z) + f(z) (4f∞λ + 1)f (z) − 2f∞λ + 1 + 6f∞λz − f (z) + 1 2 0 0 2 2 0 0 2 6f∞λz f (z) + 2zf(z) (4f∞λ + 1)f (z) − 2f∞λ + 1 + 3f(z) f (z) f (z) − 2f∞λ + 1 = 0 . (5.10)

We get only α = 1/2 as a solution to the indicial equation. We get six roots of c0 from the leading order of eom. Three of them are positive: √ q √ p 2zh , 4f∞zhλ ± 2 2zh f∞λ(−1 + 2f∞λ) . (5.11)

The positivity of the ﬁrst root cannot give us any constraint on λ. The other two roots go to zero as λ goes to zero so we will ignore them. 98 5.3. DISCUSSION

Slab

The eom reads,

0 2 0 0 3 0 2 00 − 3(1 − 2f∞λ + f (z) )(f (z) + f (z) ) + z(1 − 2f∞λ + (1 + 4f∞λ)f (z) )f (z) = 0 (5.12)

We get α = 1/2, 1 which give non-zero c0. Arguing as before we will only consider α = 1/2.

Here we get the following positive solution for c0: r2 c = pz + 4f z λ . (5.13) 0 3 h ∞ h Demanding this to be positive, we get 5 1 − ≤ λ ≤ . (5.14) 16 4 This agrees with [1]. Thus together with the constraints from the cylinder we have 5 7 − ≤ λ ≤ . (5.15) 16 64 We can recast this inequality as one for a/c where a, c are the Euler and Weyl anomaly coefficients respectively for a 4d CFT. This gives us 1 a 5 ≤ ≤ . (5.16) 3 c 3 Quite curiously, the lower bound 1/3 is precisely what appears in non-supersymmetric theories in the Refs. [3,4,5], in particular for a free boson. The upper bound of 5 /3 corresponds to a free theory with one boson and two vector fields. For a non-supersymmetric theory, the bound on a/c worked out1 in the Refs. [3,4] was 1 /3 ≤ a/c ≤ 31/18. Just to point out in words, the 1/3 came from the cylinder calculation while the 5/3 came from the slab. The causality constraints on the other hand translates into 1/2 ≤ a/c ≤ 3/2. We compare the different bounds on λ in Fig. (5.1). As is clear, the causality constraints are the tightest.

5.3 Discussion

We have considered different entangling surfaces and demanded that these close off smoothly in the bulk. In Gauss-Bonnet gravity, this led to the coupling being constrained. The spherical entangling surface did not lead to any constraints on the coupling while the cylindrical and slab entangling surfaces did. It will also be interesting to find if there are other entangling surfaces which lead to a tighter bound and if the bounds are stronger than the causality constraints. Moreover one can generalize this for arbitrary R2 theory and try get the constraints on the coupling. 1Note 31/18 ≈ 1.72 while 5/3 ≈ 1.67. REFERENCES 99

Figure 5.1: Comparison between the various constraints on the GB coupling. The length of the line represents the range of allowed λ.

References

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6.1 Introduction

In this chapter we will discuss one more way of constraining gravity based on quantum entanglement. Certain entanglement measures such as relative entropy [1], which roughly speaking tells us how distinguishable two states are, need to be positive in a unitary theory. The positivity of this quantity was studied in holographic field theories with two derivative gravity duals in [2]–related work include [3]. In the context of quantum field theories with holographic dual gravity descriptions, one can ask what this inequality translates into. Let us begin by discussing relative entropy. Relative entropy between two states ρ and σ is defined as S(ρ|σ) = tr (ρ log ρ) − tr (ρ log σ) . (6.1) As reviewed in appendix A, in quantum mechanics, this quantity is positive for a unitary theory. In [2], relative entropy was discussed in the holographic context. The state σ was chosen to be the reduced density matrix for a spherical entangling surface. In this case, σ ≡ e−H /tr e−H with H being the modular hamiltonian. It can be easily shown that (see Refs. [2,4]) S(ρ|σ) = ∆H − ∆S, (6.2) where ∆H = hHi1 − hHi0 and ∆S = S(ρ) − S(σ) with S(ρ) = −tr ρ log ρ being the von Neumann entropy for ρ and is the entanglement entropy for a reduced density matrix ρ. Then the positivity of S(ρ|σ) would require,

∆H ≥ ∆S. (6.3)

Now we can calculate the modular hamiltonian for the sphere [5], from the formula,

Z 2 2 d−1 R − r H = 2π d x T00 . (6.4) r

Here Tµν is the d-dimensional ﬁeld theory stress tensor and 00 is the time-time component.

We know how to compute Tµν in holography. The Ryu-Takayanagi prescription (and known generalizations [6]) gives us a way to compute ∆S. Thus we can check if and how the inequality ∆H ≥ ∆S is satisﬁed. In [2], many examples were considered and in each case

101 102 6.1. INTRODUCTION it was shown that this inequality is respected in Einstein gravity. If we consider a small excitation around the vacuum state then to linear order in the perturbation ∆H = ∆S. This can be shown to be equivalent with the linearized Einstein equations [7,8]. This equality has been recently shown to hold for a general higher derivative theory of gravity in [9]. It is thus very interesting to ask what constraints we get at the nonlinear order. We will address this question for the special case of a constant stress tensor for the case where the holographic entanglement entropy is given by the Ryu-Takayanagi prescription—in other words, we will ask if even at non-linear level we get Einstein gravity. We find that the constraints arising from relative entropy give us a larger class of models than just Einstein gravity. However, we show that there exists matter stress tensor for which the bulk null energy condition is violated everywhere except at the Einstein point. This in turn implies that relative entropy can continue to be positive although the bulk null energy condition is violated. In fact we can ask the question the other way round: are there examples where the relative entropy is negative but the bulk null energy condition still holds? We will give an example where this happens. Thus the connection between energy conditions and the positivity of relative entropy, which in some sense is reminiscent of the connection between energy conditions and the laws of thermodynamics, appears to be less direct than what one would have expected. In order to get some intuition about what feature of gravity ensures the positivity of relative entropy, we extend the calculations in [2] to higher derivative theories. In particular we focus on Gauss-Bonnet gravity in 5 bulk dimensions [10, 11, 12] since in this context there is a derivation [13, 14] of the corresponding entropy functional [15, 16, 17]. We find that in all examples that we consider, the positivity of the two point function of the stress tensor guarantees that the relative entropy is positive. In particular we show this for a constant field theory stress tensor as well as for a disturbance that is far from the entangling surface. At this point we should emphasize that, the inequality for relative entropy can only be explicitly checked when the modular hamiltonian is known. Unfortunately, currently this is not known for cases when the entangling region is a cylinder or a slab. This chapter is organized as follows. In section (6.2), we consider constraints arising from the positivity of relative entropy in a holographic set up where the entanglement entropy is given by the Ryu-Takayanagi entropy functional. These constraints arise at a quadratic order in a perturbation with a constant field theory stress tensor. In section (6.3), we turn to the study of relative entropy in Gauss-Bonnet holography. In section (6.4), we investigate the relative entropy for an anisotropic plasma which breaks conformal invariance. We find that the relative entropy in this case is negative and we suggest some possible explanations for this. We conclude in section (6.5). The appendix contains further calculations relevant for the rest of the chapter. We will use capital latin letters to indicate bulk indices and greek letters to indicate boundary indices. Lower case latin letters will indicate an index pertaining CHAPTER 6. RELATIVE ENTROPY 103 to the co-dimension 2 entangling surface. 1

6.2 Relative entropy considerations

In this section we will use the results in [2] to derive certain constraints at nonlinear order that arise due to the positivity of relative entropy. In Feﬀerman-Graham coordinates, the bulk metric can be written as L2 ds2 = dz2 + g dxµdxν . (6.5) z2 µν

For Einstein gravity, the bulk equations of motion allow us to systematically solve for gµν as an expansion around the boundary z = 0 (see eg.[19]). The idea here is to see what mileage we get if we do not know what the bulk theory is but we demand that the relative entropy calculated using the Ryu-Takayanagi entropy functional is positive. We want to calculate the quadratic correction to the entanglement entropy for the following form of boundary metric, L2 g = η + azdT + a2z2d(n T T α + n η T T αβ) + ··· , (6.6) µν z2 µν µν 1 µα ν 2 µν αβ

`d−1 where a = 2 P . This form is consistent with Lorentz invariance for a constant T . We will d Lˆd−1 µν treat Tµν as a small perturbation to the vacuum. At linearized order, it has been shown in Refs. [7,8,9] Einstein equations arise from the condition ∆ H = ∆S. We wish to investigate what happens at the next order. We will keep n1 and n2 arbitrary and derive constraints on them arising from the inequality ∆H ≥ ∆S. Our analysis follows [2] very closely, the only change being that we will not specify n1 and n2 to be the Einstein values. Since at linear order (the argument will be reviewed in the next section) we have the equality ∆H = ∆S d and since T00 from the holographic calculation is just given by the coeﬃcient of the z term in the metric, the inequality implies ∆S ≤ 0 at quadratic order. Thus our task is to calculate (2) ∆ S, the quadratic correction to ∆S, as a function of n1, n2. The analysis below is valid for d > 2. We start with the Ryu-Takayanagi prescription for calculating entanglement entropy in holography, 2π Z √ S = dd−1x h . (6.7) `d−1 P √ From Taylor expansion one can show that the quadratic correction to h is, √ 1√ 1√ 1√ δ(2) h = h(hijδh )2 + h δhijδh + h hijδ(2)h . (6.8) 8 ij 4 ij 4 ij 1 The paper by Erdmenger et al [18] deals with a related idea of looking for pathological surfaces in certain higher derivative theories of gravity and is of some relevance in this context. Interested readers are referred to that. 104 6.2. RELATIVE ENTROPY CONSIDERATIONS

The induced metric is, L2 h = g + ∂ z∂ z . (6.9) ij ij z2 i j √ 2 2 This is evaluated at the extremal surface z = z0 + z1 = R − r + z1. Hence, at 0-th order, the metric and its inverse are, 2 2 i j L xixj ij z0 ij x x hij = 2 ηij + 2 and h = 2 η − 2 . (6.10) z0 z0 L R In ∆(2)S, we get 3 kinds of second order contributions. To be systematic, we write, Z √ d−1 (2) d x δ h = A(2,0) + A(2,1) + A(2,2) , (6.11)

2 2 where schematically, these are the (δg) , z1δg and z1 contributions respectively. To calculate the ﬁrst term, we can set z1 = 0. Then δ(2)h δh = aL2zd−2T and ij = a2L2z2d−2(n T T α + n η T T αβ) . (6.12) ij ij 2 0 1 iα j 2 ij αβ This gives, Z n r2 n n r2 A = Ld−1a2 dd−1x Rzd T T i0 1 + (d − 1)n − n + (T )2 2 (d − 1) − 2 (2,0) 0 i0 2 2 2 R2 00 2 2R2 n n n r2 1 n 1 n +T T ij 1 + 2 (d − 1) − 2 − − 1 xixjT T 0 + xixjT T k − 1 ij 2 2 2R2 4 2R2 i0 j ik j 2R2 2R2 1 + T 2 − T 2 − 2TT , 8 x x

i j Tij i where Tx = x x R2 and T = Ti . The last two terms in (6.11) are same as they appear in [?] . Quoting the result, Z 2 2 i j k d−1 d−1 R z0 i Tij 2 i j i j z0x x x ∂kz1 A(2,1) = L a d x T z1 − 2 x ∂iz1 + 2 2z0x ∂ z1 − z1x x − 2 , 2z0 R R R (6.13) Z 2 2 2 2 i 2 i 2 d−1 d−1 R d(d − 1)z1 z0(∂z1) z0(x ∂iz1) (d − 1)x ∂iz1 A(2,2) = L d x d 2 + 2 − 4 + 2 . (6.14) z0 2z0 2R 2R 2R We can ﬁnd z1 by minimizing A(2,1) + A(2,2), which gives, aR2zd−1 z = − 0 (T + T ) . (6.15) 1 2(d + 1) x Plugging this and summing we get from Eq. (6.11), Z √ Z i j k d−1 (2) d−1 2 d−1 2 2 2 i0 x x TikTj d x δ h = L a d x c1T + c2Tx + c3Tij + c4Ti0T + c5 2 R (6.16) i j 0 x x Ti0T + c j + c TT , 6 R2 7 x CHAPTER 6. RELATIVE ENTROPY 105

2 where unlike [2] , the coeﬃcients c1 ··· c7 are dependent on n1 and n2,

(R2 − r2)(d−4)/2 c = −4(1 + d)2n (r2 − R2)2(r2 − (d − 1)R2) (6.17) 1 8(1 + d)2R 2 +R2(2(d2 + 2d − 1)r4 + (1 − 5d2)r2R2 + (2d2 − d − 1)R4) , (6.18) 1 (−4+d) (−r2 + R2) 2 ((1 − 5d2) r2R3 + (−3 + d(3 + 4d))R5) c = , (6.19) 2 8(1 + d)2 (−r2 + R2)d/2 (−2n r2 + (−1 + 2n + 2(−1 + d)n )R2) c = 2 1 2 , (6.20) 3 4R (−r2 + R2)d/2 (n R2 − 2n (r2 − (−1 + d)R2)) c = 1 2 , (6.21) 4 2R (d2 − (1 + d)2n ) R (−r2 + R2)d/2 c = 1 , (6.22) 5 2(1 + d)2 n c = − 1 R −r2 + R2d/2 , (6.23) 6 2 1 (−4+d) (−1 + d)R3 (−r2 + R2) 2 ((1 − 3d)r2 + (1 + 2d)R2) c = . (6.24) 7 4(1 + d)2

Now we integrate the expression (6.16) over the (d − 2)-sphere on the boundary. We use the trick, Z Z d−1 i j k l d−1 2n d x f(r)x x x x ··· n pairs = N(δijδkl ··· + permutations) d x f(r)r , (6.25) where N is some normalization constant. For n = 1, N = 1/(d − 1); and for n = 2, N = 1/((d − 1)2 + 2(d − 1)). The ﬁnal result comes out in the form 3, Z √ d−1 2 d−1 2 2 2 d x h = a L Ωd−2 C1T + C2Tij + C3Ti0 , (6.26) with √ 2−3−dd (1 + 4 (d2 − 1) n ) πR2dΓ[d + 1] C = 2 , (6.27) 1 2 3 (d − 1) Γ 2 + d √ 2−3−dd πR2dΓ[1 + d] C = −1 − 2d + 4(d + 1)n + 4 d2 − 1 n , (6.28) 2 2 3 1 2 (d − 1) Γ 2 + d −1−d √ 2d 2 d(n1 + 2(d − 1)n2) πR Γ[1 + d] C3 = − 3 . (6.29) (d − 1)Γ 2 + d

2 There appears to be an overall sign missing for c6 in [2]. 3 The expression for C3 in [2] after substituting for n1, n2 is oﬀ by a factor of d/(d+2) although the overall sign is correct. This appears to be related to the opposite sign used for c6. We have cross-checked our results on mathematica for various cases and the notebook may be made available on request. 106 6.2. RELATIVE ENTROPY CONSIDERATIONS

Now we must demand that ∆(2)S ≤ 0. We can write ∆(2)S = V T MV with V being a

(d−1)(d+2)/2 dimensional vector with the independent components of Tµν as its components. Then demanding that the eigenvalues of M are ≤ 0 will ensure ∆(2)S ≤ 0. This leads to

n1 + 2(d − 1)n2 ≥ 0 , (6.30) 2 2d + 1 − 4(d + 1)n1 − 4(d − 1)n2 ≥ 0 , (6.31) 2 d + 2 − 4(d + 1)n1 − 4d(d − 1)n2 ≥ 0 . (6.32)

Figure 6.1: (colour online) For d > 2 we get the allowed n1, n2 region to be the blue triangle above for a generic stress tensor. The region above the blue solid line and below the blue dashed and dotted lines are allowed from the relative entropy positivity. For d → ∞ the region 1 1 collapses to a line 0 ≤ n1 ≤ 1 indicated in green. The Einstein value (n1, n2) = ( 2 , − 8(d−1) ) is shown by the black dot. The region below the solid red line and above the dashed and dotted red lines are allowed by the null energy condition. By turning on a generic component of the stress tensor only the Einstein value is picked out. By switching oﬀ certain components of the stress tensor, various bands bounded by the solid, dashed and dotted lines are picked out.

We get the region indicated in Fig. (6.1) allowed by this set of inequalities. One interesting CHAPTER 6. RELATIVE ENTROPY 107

observation is that when d → ∞, then the allowed region becomes the interval 0 ≤ n1 ≤ 1 with n2 = 0 coinciding with the Einstein result. The area of the triangle is given by

d2 Area = . (6.33) d 8(d + 1)2(d − 2)

Notice that the (extrapolated) Aread=2 is inﬁnity. This makes sense since in d = 2 we expect 2 2 constraints on only 2 eigenvalues (since T and Tij are no longer independent) which will give us an unbounded region. Further Aread→∞ → 0 which leads to a line interval for d → ∞ as shown in Fig. (6.1). At this stage, we have a wider class of theories that are allowed by the inequality than the Einstein theory. The other theories need extra matter in addition to Einstein gravity to support them. As such we could ask if the matter needed satisﬁes the null energy condition.

As an example consider turning on a constant T01 in d = 4. Then we ﬁnd 1 12 R − g (R + ) = T bulk , (6.34) AB 2 AB L2 AB

bulk with TAB working to be h3 T bulk = 16z6T 2 (δn + 4δn )δz δz + (δn + 6δn )δ0 δ0 − (δn + 6δn )δ1 δ1 AB 01 2 1 2 A B 1 2 A B 1 2 A B X i i i (6.35) − 2(δn1 + 3δn2) δAδB . i=2,3 m Here δn1 = n1 −1/2 and δn2 = n2 +1/24. Using this we ﬁnd that the null energy condition bulk A B TAB ζ ζ ≥ 0 leads to

bulk bulk T00 + T22 = −δn1 ≥ 0 , (6.36) 5 T bulk + T bulk = δn + 12δn ≥ 0 , (6.37) 00 zz 2 1 2

bulk bulk with T00 + T11 = 0. These simplify to n1 ≤ 1/2 and n2 ≥ −1/24. Thus the region in ﬁg.1 that respects the null energy condition is smaller than that allowed by the positivity of relative entropy. For a general constant stress tensor in general d we proceed as follows. We note that for a metric of the form in Eq. (6.5), with gµν a function of z only, we have [20]

0 κ Rµν = Rµν − (z∂zKµν + KKµν − 2KµκKν ) , (6.38)

Rµz = 0 , (6.39) 2 µν µν z Rzz = −g z∂zKµν + K Kµν , (6.40) 0 µν 2 µν R = R − (2zg ∂zKµν + K − 3K Kµν) , (6.41) 108 6.2. RELATIVE ENTROPY CONSIDERATIONS

1 0 where Kµν = 2 z∂zgµν. Here denotes a quantity computed with gµν. Using these it is straightforward (but tedious) to compute (setting L = 1 for convenience, deﬁning Sµν = α αβ n1TµαTν + n2ηµνTαβT and aborbing the factors of a into Tµν; the raising and lowering of µ indices on Tµν,Sµν are done with ηµν. Also we have used Tµ = 0.)

µν 2 µν µν d µα ν µν 2d g = z [η − T z + (T Tα − S )z ] , (6.42) 1 K = − (2η − (d − 2)zdT − 2(d − 1)z2dS ) , (6.43) µν 2z2 µν µν µν 1 Kν = − [2δν − dzdT ν + dz2d(T T αν − 2Sν)] , (6.44) µ 2 µ µ µα µ 1 K = − [2d + dz2d(T T αβ − 2Sα)] , (6.45) 2 αβ α z2 Kµν = − [2ηµν − (d + 2)zdT µν + 2(d + 1)z2d(T µT να − Sµν)] , (6.46) 2 α 1 zgµν∂ K = [4d + z2d(4d(d − 2)Sα − d(d − 4)T T αβ)] , (6.47) z µν 2 α αβ z2d K Kµν = d + [d(d + 4)T T αβ − 8dSα] , (6.48) µν 4 αβ α 1 K Kκ = [4η − 4(d − 1)zdT + z2d(d2T κT − 4(2d − 1)S )] . (6.49) µκ ν 4z2 µν µν µ κν µν Using these we ﬁnd

bulk 2d−2 αβ Tzz = −d(d − 1)z TαβT (δn1 + dδn2) , (6.50) bulk 2 2d−2 κ αβ Tµν = d z −δn1TµκTν + ηµνTαβT (δn1 + (d − 1)δn2) . (6.51)

Here δn1 = n1 −1/2 and δn2 = n2 +1/(8(d−1)) i.e., the deviations from the Einstein values. Now we are in a position to ask if the matter supporting this bulk stress tensor satisﬁes the bulk bulk null energy conditions or not. First we note that T00 + T11 ≥ 0 immediately leads to

2 2 2 − d (−T00 + Tij)δn1 ≥ 0 . (6.52)

2 2 This leads to a definite sign for δn1 if and only if (−T00 + Tij) has a definite sign. But in general, there is no reason for this combination to have a definite sign. So we are led to suspect that for a generic stress tensor, δn1 = 0. To confirm this suspicion let us look at bulk bulk Tzz + T00 .

bulk bulk Tzz + T00 = h 2 2 −d (d − 1)T00(δn1 + 2dδn2) + Tij[(2d − 1)δn1 + 2d(d − 1)δn2]

2 i +T0i[(2 − 3d)δn1 − 4d(d − 1)δn2] . (6.53)

As in the relative entropy analysis, we write the RHS as V T MV where V is a (d−1)(d+2)/2 dimensional vector whose non-zero independent components are the T00,Tij,T0i’s. Then we CHAPTER 6. RELATIVE ENTROPY 109 demand that the eigenvalues of M are positive for the null energy condition to hold for a generic constant traceless stress tensor Tµν. This yields

(3d − 2)δn1 + 4d(d − 1)δn2 ≥ 0 , (6.54)

(2d − 1)δn1 + 2d(d − 1)δn2 ≤ 0 , (6.55)

δn1 + 2(d − 1)δn2 ≤ 0 . (6.56)

Only for δn1 = δn2 = 0 are these inequalities satisﬁed for d > 2. Thus the null energy condition picks out the Einstein value if we ask if for a generic constant stress tensor the 2 O(T ) terms are supported by matter. Of course as we saw for d = 4 we can turn on T0i and set everything else to zero, there would be a region in the n1, n2 parameter space where the null energy condition and the positivity of the relative entropy would hold (this corresponds to the region between the red and blue solid lines in Fig. (6.1). For the generic case, only the Einstein value is picked out. To emphasis, that the Einstein value was picked out for the generic case, relied only on the null energy condition analysis and did not rely on the positivity of the relative entropy. To summarize, we found that there exists a larger class of theories in the (n1, n2) parameter space than just the Einstein theory. However, except at the Einstein point, we found that there always exists some matter stress tensor which violates the bulk null energy condition. 6.3 Relative entropy in Gauss-Bonnet holography

In this section we will calculate relative entropy for excited states in Gauss-Bonnet gravity. For deﬁniteness, we will consider d = 4 or 5-dimensional bulk. We will follow the conventions in [11]. The total action is given by

I = Ibulk + IGH + Ict , (6.57) where Z √ 12 λ I = d5x g R + + L2(R RABCD − 4R RAB + R2) . (6.58) bulk L2 2 ABCD AB The generalized Gibbons-Hawking term is given by [21] Z 1 4 √ 2 µν 1 3 IGH = − 3 d x γ K − λL (2GµνK + (K − 3KK2 + 2K3) . (6.59) `P 3 µν α β γ Here Gµν = Rµν −1/2γµνR made from the boundary γµν, K2 = KµνK and K3 = Kβ Kγ Kα. α Kµν is the extrinsic curvature and K = Kα . The counterterm action Ict is needed for the cancellation of the power law divergences in Itot. For our case this works out to be [22, 23] ˜ (L and f∞ are deﬁned below) 1 Z √ 3 L˜ I = d4x γc + c Rˆ , (6.60) ct 3 1 ˜ 2 `P L 4 110 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY

ˆ 2 where R is the four dimensional Ricci scalar and c1 = 1 − 3 f∞λ and c2 = 1 + 2f∞λ . The equations of motion are given by 1 6 λL2 R − g R − g − H = 0, (6.61) AB 2 AB L2 AB 2 AB where

1 2 CD CDEF C HAB = gAB(R − 4RCDR + RCDEF R ) − 2RRAB + 4RA RCB 2 (6.62) CDE CD − 2RACDERB − 4R RCABD .

AdS5 given by L˜2 ds2 = dz2 − dt2 + dx2 + dx2 + dx2 (6.63) z2 1 2 3 √ ˜ 2 where L = L/ f∞ with 1 − f∞ + λf∞ = 0. The dual CFT is characterized by the central charges c, a appearing in the trace anomaly[11, 24]: π2L˜3 π2L˜3 c = 3 (1 − 2λf∞) , a = 3 (1 − 6λf∞) . (6.64) `P `P The CFT stress tensor two point function is given by 40c hT (x)T (0)i = I (x) , (6.65) µν ρσ π2(x2)4 µν,ρσ where I is a function of x and the positivity of the two point function leads to c > 0. We will need the formula for the holographic stress tensor (see [25]) ˜ 1 2 1 3 L 1 Tµν = 3 [Kµν − gµνK + λL (qµν − gµνq)] − c1γµν + c2[Rµν(γ) − γµνR(γ)] , (6.66) `p 3 L˜ 2 2 where µν q = h qµν α αβ αβ 2 αβ qµν = 2KKµαKν − 2KµαK Kβν + Kµν(KαβK − K ) + 2KRµν + RKµν − 2K Rαµνβ − α 4R(µKν)α . The terms proportional to c1, c2 come from Ict. We also note that the GB coupling λ is bounded. Following [26] for the calculation of the three point correlation function of stress tensor one needs to compute a energy ﬂux which comes form the insertion of ijTij , where ij and Tij are the polarization tensor and stress tensor respectively. Demanding the positivity of this energy ﬂux in the holographic set up we get the following three constraints and from those we obtain bounds on λ . These coincide with the bounds arising from micro-causality [27]. 9 Tensor channel : 1 − 10f λ ≥ 0 ⇒ λ ≤ ∞ 100 3 1 Vector channel : 1 + 2f λ ≥ 0 ⇒ − ≤ λ ≤ (6.67) ∞ 4 4 7 1 Scalar channel : 1 + 6f λ ≥ 0 ⇒ − ≤ λ ≤ ∞ 36 4 CHAPTER 6. RELATIVE ENTROPY 111

From this we get 7 9 − ≤ λ ≤ . 36 100 2 2 This is the same as the condition 1 − 4f∞λ − 60f∞λ ≥ 0.

6.3.1 Linear order calculations

We are interested in considering the excited state to be a perturbative excitation of the ground state. At linear order in the perturbation ∆H = ∆S. Let us review the argument of

[2] why. Let ρ0 be a reference state. Now let ρ(α) be a continuous family of states dependent on a parameter α that runs over all possible values. We choose the parametrization such that ρ(α = 0) = ρ0. Now relative entropy vanishes for two states that are equal. So we must have S(ρ(0)|ρ0) = 0 and also S(ρ(α → ±)|ρ0) → 0+ > 0 where is a small positive valued number denoting a small perturbation from the reference state ρ0. This means at α = 0 we must have, d(S(ρ(α)|ρ0))/dα = 0. Or equivalently at the linear order of the perturbation ,

∆H = ∆S, (6.68)

4 which follows from Eq. (6.2). We can demonstrate this with a simple example . Let ρ0 to be the vacuum of the CFT4 whose holographic dual is the empty AdS5 (our linearized results are a sub-case of the more general case worked out in [9]),

L˜2 ds2 = dz2 − dt2 + dx2 + dx2 + dx2 (6.69) z2 1 2 3

We choose ρ1 to be the dual of a metric which is being perturbed around the empty AdS. Following [2], we take the perturbation to be of the form,

3 `P 2 X 2n (n) δgµν = z z T . (6.70) ˜3 µν 2L n

(n) To keep track of the perturbation we keep the components of Tµν proportional to a small number . We compute the entanglement entropy from the Jacobson-Myers functional, Z √ Z √ 2π 3 2 4π 2 2 S = 3 d x h(1 + λL R) + 3 λL d x h K . (6.71) `P `P

Here, hab is the induced metric on the minimal surface and R and K are respectively the intrinsic ricci scalar and extrinsic curvature evaluated on that surface. To simplify notation, we will set L = 1 . The minimal surface equation is given by

2 ij K + λL (RK − 2RijK ) = 0 , (6.72)

4The change in entanglement entropy for excited states in GB holography has been considered in [28]. 112 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY which was derived in [13, 29] following [30]. For the spherical entangling surface in the unperturbed metric the following continues to be an exact solution √ 2 2 z = z0 = R − r . (6.73)

In the perturbed case, it changes to

z = z0 + z1 . (6.74)

However note that we obtained z0 by extremization. Hence z1 can only contribute to a quadratic order in and not at linear order. Thus at linear order we can set z1 = 0. Using z = z0 to compute (6.71) and then extracting the terms proportional to gives us ∆S.

Now we can calculate the modular hamiltonian, from the formula in Eq. (6.4), where T00 is obtained in holography using Eq. (6.66). Since T00 = 0, for empty AdS, this directly gives ∆H. Now we will demonstrate the equality in Eq. (6.68) by considering a special case (we have checked that this holds in the other examples considered below as well). (n) (0) Using Gauss-Bonnet eom, we can determine Tµν in terms of the lowest mode Tµν . It (0) (0) turns out they are all derivatives of Tµν . To keep it simple we take Tµν to be a constant. Also note that to satisfy GB eom, we must have traceless and divergenceless conditions on (0) Tµν , (0)µ (0)µ T µ = 0 and ∂µT ν = 0 . (6.75) Consider an isotropic perturbation E E E T (0) = E, , , (6.76) µν 3 3 3

Note that this satisﬁes the conditions in Eq. (6.75). However the holographic dual tensor Tµν (0) is not same as T µν. We compute it from Eq. (6.66) as, E E E T = (1 − 2f λ) E, , , (6.77) µν ∞ 3 3 3 Now using (6.4) one gets5, 8π2L˜3ER4 ∆H = 3 (1 − 2f∞λ) . 15`P √ As discussed before, we can compute ∆S from (6.71) with z = R2 − r2, and then take out the order coeﬃcients. We obtain, √ 2 2 2 E (R (3 + 30f∞λ) − r (1 + 58f∞λ)) h(1 + λL R) = − 3/2 (6.78) 6f∞ R

5 There is a typo in Eq. (6.29) in [9] for hTµν i. There is a factor of 2 missing in front of the term proportional to a1 in that expression. Taking this into account our expression agrees with their both for GB and for the general R2 theory discussed in appendix C. CHAPTER 6. RELATIVE ENTROPY 113 from which we calculate, 8π2L˜3ER4 ∆S = 3 (1 − 2f∞λ) . (6.79) 15`P This demonstrates ∆H = ∆S for an isotropic perturbation.

6.3.2 Quadratic corrections

Now we turn to the more interesting case of quadratic corrections which lead to inequalities. We take the following form for the boundary metric,

2 d 2d α αβ z gµν = ηµν + z Tµν + z (n1TµαTν + n2 ηµνTαβT ) + ··· (6.80) where compared to Eq. (6.6) we have absorbed a factor of a into the stress tensor. We need to

ﬁx the numbers n1 and n2. By plugging into the GB equations of motion given by Eq. (6.61), we ﬁnd that

− 3(n1 + 4n2) + f∞(1 + 6n1λ + 24n2λ) = 0 (6.81)

2 3 2 n1(9 − 17f∞ + 25f∞λ) − 4(4f∞λ − 3n2(1 − 9f∞ + 17f∞λ)) = 0 (6.82)

2 Solving the two equations and using the relation 1 − f∞ + f∞λ = 0 we get

1 1 + 2f∞λ 1 1 + 6f∞λ n1 = and n2 = − . (6.83) 2 1 − 2f∞λ 24 1 − 2f∞λ These results match with the λ = 0 case given in [2]6.

6.3.3 Constant Tµν

The next step is to calculate the second order change in ∆S. For a general but constant stress tensor we can guess the following form of the second order correction of entropy from Lorentz invariance, (2) 2 ij 0i ∆ S = C1T + C2TijT + C3T0iT (6.84) where T denotes the trace of the spatial part of the stress tensor Tµν. The latin indices run from 1 to 3, and denote the spatial part of a tensor. They are raised with ηij. Our task is to identify the constants Ci’s for a non-zero λ. The only condition on the stress tensor is that it

6 Notice a curious fact. If we demanded that n1 ≥ 0 and n2 ≤ 0, or in other words even in GB gravity they have the same sign as in Einstein gravity then with c > 0, we would get

1 + 2f∞λ ≥ 0 , 1 + 6f∞λ ≥ 0 .

But these are nothing but the scalar and vector channel constraints in Eq. (6.67)! These leads us to wonder if entanglement entropy knows about the causality constraints. 114 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY is symmetric and traceless. To do the perturbative analysis we assume that the components of the stress tensor are proportional to a perturbative parameter . Also we have absorbed a parameter a in the stress tensor. The background metric will be changed in the quadratic √ 2 2 order as given in (6.6). Now, assume that the minimal surface z0 = R − r is modiﬁed as

z = z0 + z1 . (6.85) z1 contributes at the quadratic order in the JM functional (6.71). So it is sufficient to consider only the first order fluctuation to the entangling surface. Next we expand the entropy functional upto quadratic order and then extract the terms proportional to 2 which gives the quadratic correction to the entropy. We vary it with respect to z1. This gives us (2) the equation of motion for z1. We find the solution and put it back to ∆ S. Since it was shown that at linear order, ∆S = ∆H, at second order we must have ∆(2)S > 0. We get the following equation for z1 ,

xixj (1 − 2f λ) ∂2(z z ) − ∂ ∂ (z z ) − (R2 − r2)2 (T + 3T ) = 0 , (6.86) ∞ 0 1 R2 i j 0 1 x with the solution, R2z3 z = − 0 (T + T ) . (6.87) 1 10 x Notice that the equation is the same as what appears in the Einstein case upto the overall factor of (1 − 2λf∞). The Gibbons Hawking term doesn’t contribute to the action when we put in the solution. Alternatively, we could have taken the action and integrated all terms 00 involving z1 (x)’s by part and cast it in the conventional form. The surface term resulting from this will cancel with the appropriate Gibbons-Hawking term. We have checked both approaches and have got the same result. Integrating the resulting action over the volume of the entangling region, we obtain the second order correction to the entropy,

3 ˜3 (2) 8π L (1 − 2f∞λ) 2 2 2 ∆ S = − 3 C1T + C2Tij + C3Ti0 , (6.88) `P

C1,C2,C3 are same as the Einstein values obtained in section 2.

Note that this is just a factor of (1−2f∞λ) times what is obtained in the Einstein gravity (the Einstein result was manifestly negative). This can be cross-checked easily on a computer by suitably turning on various components of the stress tensor and identifying various tensor structures. Now from the discussions in the previous sections, it is clear that this quantity has to be negative. The only constraint to ensure ∆(2)S < 0 is

1 − 2f∞λ > 0 . (6.89) CHAPTER 6. RELATIVE ENTROPY 115

This is equivalent to saying the central charge c > 0 which also is the condition needed for the positivity of the two point function of the ﬁeld theory stress tensor. The condition λ < 1/4 ensures that this holds. If this inequality on λ did not hold, the corresponding vacuum would have ghosts [11].

6.3.4 Shockwave background

Up to this point we have only considered constant stress-tensor. It is interesting to ask if we get non-trivial constraints for Tµν not constant. To explore a nontrivial case of non-constant

Tµν, consider the following 5 dimensional metric

L˜2 ds2 = (dz2 + dx dxµ + f(t + x )W (z, x , x )(dt + dx )2) (6.90) z2 µ 3 1 2 3 where µ = 1, 2.

The above metric solves the GB equation exactly, given that W (z, x1, x2) satisﬁes the following diﬀerential equation, 3 ∂2W + ∂2 W + ∂2 W = − ∂ W, (6.91) z x1 x2 z z with no constraint on f(t + x3). If f = δ(t + x3) then this is the shockwave metric considered for example in [26] to derive constraints on higher derivative gravity theories. We will set f = 1 and in a slight abuse of terminology continue to refer the metric as a shockwave.

W (z, x1, x2) is taken as

L˜2z4 W (z, x1, x2) = 2 0 2 0 2 3 . (6.92) (z + (x1 − x1) + (x2 − x2) )

0 0 Here (x1, x2) represent the point where the disturbance is peaked. Since in our calculations 0 0 we perturb the background metric, we should choose x1 and x2 to be outside the entangling region. With this in mind we proceed with the second order calculation. Next we consider 0 a shockwave disturbance localized just outside the entangling surface. We will set x2 = 0 in (6.92). We start with the following metric which is obtained by expanding W around z = 0 and retaining the ﬁrst two terms in the expansion,

˜2 4 ˜2 3 6 ˜2 4 2 L 2 µ z L 3z L 2 ds = 2 (dz + dxµdx + ( 2 0 2 3 − 2 0 2 4 )(dt + dx3) ) (6.93) z (x1 + (x2 − x2) ) (x1 + (x2 − x2) ) The factors have been inserted to keep track of the order of the expansion and matches with 3 the power appear in the denominator. If we write the entangling surface as z = z0 + z1 6 then the quadratic terms in z1 will involve which is at a higher order than the second order term in the metric above. Thus we expect to see an inequality ∆H > ∆S with the 116 6.3. RELATIVE ENTROPY IN GAUSS-BONNET HOLOGRAPHY

above metric setting z1 = 0. We thus evaluate the entropy functional considering only the unperturbed entangling surface and expand it upto 4 and pick out the 4 term which gives the ﬁrst leading order change in the relative entropy. The integrand is shown below, Z 5 (2) 2π 3L 2 2 2 2 2 2 2 2 ∆ S = dx3dx1dx2 (x3 + x2 + x1 − R )(40f∞(x3 + x2 + x1 − R ) `3 5/2 2 0 2 6 P 2Rf∞ (x3 + (x2 − x2) ) 2 2 0 2 4 2 2 0 2 2 0 2 (4R (x3 + (x2 − x2) ) − 4(x3 + x3(x1 + 2x2(x2 − x2)) + (x1 + x2)(x2 − x2) ))λ + 16f∞ 2 2 2 2 2 0 2 2 2 2 2 0 (R − x3 − x2 − x1)(x3 + (x2 − x2) )(2R − 13x3 − 2x1 − 13x2 + 12x2x2)λ 2 0 2 2 2 2 2 2 − (x3 + (x2 − x2) ) (60f∞(x3 + x − 2 )λ − R (1 + 18f∞λ) + x1(1 + 18f∞λ))) . (6.94) √ √ 2 2 2 2 Then we perform the integration over x3 which goes from − R − r to R − r and x1 = r cos(θ) , x2 = r sin(θ) . Now after some algebraic manipulation we can write the integrand as,

5 (2) 2πL 2 ∆ S = (f1 + f2 sin(θ) + f3 sin(θ) ) (6.95) 3 5/2 2 02 0 6 2`P f∞ R(r + x2 − 2rx2 sin(θ)) where f1, f2, f3 are some function of r and λ . Integral over θ goes from 0 to 2π and integral over r goes from 0 to R. We ﬁrst perform the θ integral. To perform the θ integral we have used the following integral identity: Z 2π dθ 2π = √ , 2 2 0 a + b sin(θ) a − b Finally we get,

Z R 5 5 3 2 4 (2) 2π L 30 (8a + 40a b + 15ab ) π ∆ S = dr − f1(− ) 3 5/2 2 2 11/2 `P 0 240f∞ R (a − b ) 4 2 2 4 5 3 2 4 90b (8a + 12a b + b ) π 30 (4a + 41a b + 18ab ) π + f2( ) − f3( ) , (a2 − b2)11/2 (a2 − b2)11/2 (6.96)

2 2 02 0 where, a = r + x2 and b = −2rx2 . Next we perform the r integration. The leading contribution in ∆(2)S comes form the lower limit of the r integral which is shown below.

2 5 (2) π L 0 ∆ S = (1 − 2f∞λ)f(x2) , (6.97) 3 5/2 2 96`P f∞ R 0 where, f(x2) is a negative valued function given by

p 02 02 04 06 02 04 06 08 −1 0 x2 − 1 (−136 + 72x2 − 56x2 + 15x2 ) − 3 (32 − 16x2 + 36x2 − 22x2 + 5x2 ) Csc (x2) f(x0 ) = 2 02 9/2 (x2 − 1) (6.98) CHAPTER 6. RELATIVE ENTROPY 117

and plotted in Fig. (6.2). To satisfy, ∆S ≤ ∆H we will get, 1 − 2f∞λ ≥ 0 or in other words c > 0. Note that in order for us to be able to expand in small z, the perturbation needs to be located far away from the entangling surface. This is because in the denominator in W 2 2 0 2 we had z + x1 + (x2 − x2) . When we plug in z = z0, the maximum value for z is R and 0 this happens when x1 = x2 = 0. Thus we will need R x2 for the expansion to be valid. It will be interesting to see what happens as we move the perturbation closer and closer to the entangling surface. However this appears to be a very hard problem.

0 Figure 6.2: Negative of the function f(x2) is plotted which is a positive valued function

6.3.5 Correction from additional operators

In this section we consider perturbed states in which certain additional operators acquire nontrivial vacuum expectation value. Our analysis will follow [2]. The holographic dual of these operators will involve additional massive fields in the bulk. We will show that even for such cases in Gauss-Bonnet gravity, the relation ∆H > ∆S will hold. Again we are in AdS5 with the bulk action given by, Z √ 12 λL2 1 1 I = d5x −G R + + R2 − 4R2 + R2 − (∂φ)2 − m2φ2 , (6.99) L2 2 AB ABCD 2 2 where we have added a massive scalar field which acts as a bulk dual of a scalar operator of dimension ∆. When m2 = ∆(4 − ∆), the field φ behaves asymptotically as, φ = γOz∆ . (6.100) 118 6.4. RELATIVE ENTROPY FOR AN ANISOTROPIC PLASMA

Now we can work out the stress tensor corresponding to this from the formula,

1 1 T = ∂ φ∂ φ − g ((∂φ)2 + m2φ2) . (6.101) AB 2 A B 4 AB This will result in the following change to the boundary metric boundary metric,

2 d X 2n (n) 2∆ X 2n (n) z δgµν = az z Tµν + z z σµν (6.102) n n where we must have, 2 (0) γ 2 σµν = − ηµνO . (6.103) 12(1 − 2f∞λ)

(n) in order to satisfy Gauss-Bonnet eom. The higher modes, namely σµν (n > 0) are composed (0) of derivatives of σµν . As in [2], we consider O to be slowly varying and, hence, neglect the higher modes. It is not necessary to find any correction to the entangling surface. There are two different perturbations, both in their first orders, and using the z0 minimal surface to compute ∆S will suffice. The correction to entropy will have two parts,

∆S = ∆T S + ∆OS. (6.104)

The ﬁrst part, ∆T S comes from the holographic boundary stress tensor Tµν, and its the same as what we calculated before for the linear order. The second part comes from the scalar ﬁeld and is obtained by calculating the area functional with the metric of Eq. (6.102).

3/2 2∆ 2 π R γ (−2 + 3∆)Γ[−1 + ∆]Ωd−2 2 ∆OS = − 1 O . (6.105) 48 a Γ 2 + ∆

Note that the result is independent of λ. Since the result is negative it seems the metric already knows of the positivity of relative entropy even for Gauss-Bonnet provided the unitarity bounds are respected.

6.4 Relative entropy for an anisotropic plasma

We now want to turn our attention to a holographic anisotropic plasma–there is going to be a surprise in store. We consider the holographic dual of the deformed N = 4 SYM where the deformation is generated by anisotropy along one spatial direction viz.

1 Z S = S + θ(z) Tr F ∧ F, (6.106) N =4 8π2 CHAPTER 6. RELATIVE ENTROPY 119

θ is the ﬁeld generating anisotropy along the z direction. The holographic dual is the Einstein- dilaton-axion system given by Z Z 1 √ 12 1 2 1 2φ 2 1 √ Sbulk = 3 −g(R + 2 − (∂φ) − e (∂χ) ) + 3 −γ2K, (6.107) 2`P M L 2 2 2`P ∂M where φ is the dilaton and at the level of the solution is taken to be a function of the AdS radius only and χ is the axion dual to the gauge theory θ-term, responsible for inducing anisotropy, which is taken to be χ = ρx3. This model was proposed and studied in detail in [31]. The low anisotropy regime corresponding to ρ/T 1 in this model is unstable [31]. The metric equations are given by (L = 1) 1 R − Rg − 6g = T , (6.108) MN 2 MN MN MN where the bulk matter stress tensor is given as 1 1 1 1 T = ∂ φ∂ φ − (∂φ)2g + e2φ∂ χ∂ χ − e2φ(∂χ)2g . (6.109) MN 2 M N 4 MN 2 M N 4 MN The metric, φ and χ equations can be written as 1 1 R + 4g − ∂ φ∂ φ − e2φ∂ χ∂ χ = 0, MN MN 2 M N 2 M N ∇2φ − e2φ(∂χ)2 = 0, (6.110) ∇2χ = 0 .

The metric in the FG coordinates is given by dz2 1 ds2 = + γ (z, xi)dxµdxν, (6.111) z2 z2 µν where ρ2 γ = −1 + z2 + ..., tt 24 ρ2 γ = γ = 1 − z2 + ..., (6.112) x1x1 x2x2 24 5ρ2 γ = 1 + z2 + ..., x3x3 24 If we introduce a temperature, the modification to the metric will start at O(z4). Further, the scalar field introduces a new scale which breaks scale invariance explicitly and the trace of the boundary stress tensor is now non-zero. It needs to be checked if the null energy condition is satisfied by the bulk stress tensor TMN given by Eq. (6.109). Contracting the above with the null vectors ξµ we have 1 T ξM ξN = [(∂ φ)2 + e2φ(∂ χ)2], (6.113) MN 2 ξ ξ 120 6.4. RELATIVE ENTROPY FOR AN ANISOTROPIC PLASMA

M M N 2 where ∂ξ(φ, χ) = ξ ∂M (φ, χ) and ξ ξ gMN = ξ = 0. Since the bulk scalar axion follows the profile χ = ρx3 then M x3 ξ ∂M χ = ρξ , (6.114) whereas the dilaton field φ depends on the radial coordinate. The NEC for the bulk stress µ ν tensor becomes by contracting with the null vectors Tµνξ ξ as ρ2 ρ2 2φ x3 2 2φ Tx3x3 = e (ξ ) = e ≥ 0, 2 2 (6.115) 1 T = (∂ φ)2 ≥ 0 . uu 2 ξ Thus we have explicitly verified that the bulk stress tensor satisfies the null energy condition. We now want to verify the calculation for the relative entropy in this low anisotropy regime. As mentioned before, the low anisotropy phase is thermodynamically unstable. We can thus try to see what happens to the relative entropy in such a phase. Also note that we are considering Einstein gravity for which the entropy functional is the Ryu-Takayanagi one.

Further in the low anisotropy regime, we are interested in, since we are expanding γµν upto O(z2) (assuming a small entangling surface Rρ 1) and the stress tensor appears at O(z4), we have ∆H = 0. Here the state σ is the vacuum state which corresponds to ρ = 0 and is conformally invariant. Thus the modular hamiltonian will be the same as in Eq. (4). Thus we only need to compute the change in the entanglement entropy. Furthermore, at leading order in ρ we expect to see an inequality and as such we do not need to evaluate the change in the entangling surface.

p 2 2 2 3 Putting in the solution for the entangling surface f(x1, x2, x3) = R − x1 − x2 − x3 we have √ 1 2 2 2 2 2 2 2 2 2 2 4 h = 2 2 2 3 2 [48R + (R − x1 − x2 − x3)(3R − 5x3 + x1 + x2)ρ ] + O(ρ ) . 48(R − x1 − x2 − x3) R (6.116) The entanglement entropy then becomes Z 2π 1 2 2 2 2 2 2 2 2 2 2 S = 3 dx1dx2dx3 2 2 2 3 2 [48R +(R −x1−x2−x3)(3R −5x3+x1+x2)ρ ] . `p 48(R − x1 − x2 − x3) R (6.117)

In spherical polar coordinates x3 = r cos θ, x1 = r sin θ sin φ, x2 = r sin θ cos φ where (θ, φ) are spherical polar coordinates we have 2 Z 2 2 2 2πρ (3R − 2r − 3r cos 2θ) 2 ∆1S = 3 3 2 r sin θdθdφdr . (6.118) `p 48(R − r R) Carrying out the (θ, φ, r) integrals we ﬁnd (on reinstating L factors) π2ρ2R2L3 5 ∆1S = 3 (− − log[ ]) . (6.119) 6`p 3 2R CHAPTER 6. RELATIVE ENTROPY 121

Here is a cutoﬀ and r = R − (since → 0 corresponds to z → 0 it is related to the UV cutoﬀ). The log-divergence is due to the breaking of conformal invariance by the excited state. However, notice that in the limit of → 0, the result leads to ∆1S > 0 and hence the positivity of relative entropy is violated. Since the positivity of relative entropy in quantum mechanics depends on unitarity (reviewed in appendix A), this leads to the following possible interpretations:

1. There are additional contributions which we are missing and they are required for the positivity of the relative entropy to hold in this case. One could speculate that there are additional saddle points of the bulk gravity theory which contribute to the entanglement entropy. It will be interesting to ﬁnd out those saddle points and see if they “unitarize” the problem7.

2. Holographic relative entropy positivity needs further conditions than just bulk unitarity. It could be that the derivation of the positivity does not go through in any straightforward manner to quantum ﬁeld theory.

3. In the low anisotropy regime, may be there is a loss of bulk unitarity that is not immediately apparent.

All possibilities need further investigation. Let us first briefly comment on the third possibility. Expanding the linearized equations near the boundary and upto linear order in ρ we have 2 2 ρ ( + )h = 0, ( + )h + L χ = 0, (6.120) L2 ij L2 Mx3 2 Mx3 1 x3 φ1 − 2ρ∂ χ1 = 0, χ1 = 0, (6.121) where hMN , φ1 and χ1 are metric, φ and χ fluctuations respectively and i, j take values apart from x3. ∇A is evaluated using the AdS5 metric. Here LMN ≡ δMx3 ∂N + δNx3 ∂M is a linear operator. The coupling between the metric and χ fluctuation is of the form Hh + Lχ = 0, Hχ = 0. But this form is similar to what arises in the context of logarithmic conformal field theories which are non-unitary [33]. Thus one should check if there are log modes in the fluctuations. We can do this following [34]. According to the arguments in [34] 2 log modes arise if the form of the equations is ( + a) hµν = 0. Let us check what the form 8 4 of the equations are when we decouple them. Using ∇Aχ1 = ∇Aχ1 − L2 ∇Aχ1, we find 7This is very similar to the resolution of information loss paradox in case of eternal AdS Black Holes as formulated by Maldacena [32]. The exponentially small correlation as required by the unitarity arises form the periodically identified Euclidean AdS, although this is not the dominant contribution to the canonical ensemble. 8Useful identities can be found for eg. in the appendices of [35] 122 6.5. DISCUSSION

that the decoupled equation for hMx3 takes the form 2 4 ( + )( + )h = 0 , L2 L2 Mx3 while for φ we get 4 ( + )φ = 0 . L2 2 Neither of the four derivative linear operator is of the form ( + a) and hence following the arguments in [34] there are no log modes so the dual field theory is not a log CFT. Naively it may appear that the propagator for say the φ field will look like 1/(p2(p2 + m2)) = 1/m2(1/p2 − 1/(p2 + m2)), and hence the theory is non-unitary. However, this is not true since in addition to the decoupled form of the equations above, the relations in Eq. (6.121) still have to hold–any loss of unitarity would have shown up in the asymptotic fall offs in the field. Thus it appears that the other two possibilities become plausible. In closing this section, we note that in [2] it was argued that the relative entropy should increase as the radius of the entangling surface increases. In our case since ∂RS(ρ1|ρ0) = π2ρ2RL3 −∂R∆1S ≈ 3 (log[ ]) < 0 and hence this monotonicity would also appear violated. 3`p 2R

6.5 Discussion

In this chapter we used holographic entanglement to constrain gravity in interesting ways. First, we started with the Ryu-Takayanagi entropy functional (which holds for Einstein gravity) and considered what constraints arise at nonlinear order on the metric by demanding that relative entropy is positive. At linearlized level, it is now known that for the spherical entangling surface ∆H = ∆S leads to linearized equations for any higher derivative theory of gravity [9]. We considered a constant field theory stress tensor. At the next order, we found interesting constraints on the terms allowed by the positivity of relative entropy. These were more general than what arises from Einstein gravity. We analysed energy conditions for matter that could support these additional theories. We showed that the additional theories could be supported by matter that violates the null energy condition. In other words, holographic relative entropy can be positive although the bulk null energy condition is violated. It is an important open problem to understand if this feature persists for a more general stress tensor. We also gave an example of a model which corresponds to an anisotropic plasma, where for small anisotropy, the relative entropy is negative. This occurred even though the bulk stress tensor satisfied the null energy condition. We gave some possible explanations for this. We will leave further investigations of similar models as an open problem. Second, we analysed the inequality in Gauss-Bonnet gravity for a given class of small perturbations around the vacuum state. We found that for all our examples, the positivity CHAPTER 6. RELATIVE ENTROPY 123 of the stress tensor two point function ascertained that this inequality was respected. On the bulk side this corresponds to metric fluctuations having positive energy. The simplicity of the final result does cry out for a simpler explanation for our findings. Although the intermediate integrals involved appeared very complicated, the final result was simply proportional to the Weyl anomaly c. It would be nice to find a simple explanation for this finding. Some preliminary studies of the general four derivative theory has been made in appendix B. Another interesting open problem is to consider a disturbance close to the entangling surface. We were able to consider a disturbance that was localized far from the entangling surface and show that the relative entropy is positive. Whether the constraints change as one moves the disturbance closer to the entangling surface is an open problem. In the previous chapter we considered different entangling surfaces and demanded that these close off smoothly in the bulk. In Gauss-Bonnet gravity, this led to the coupling being constrained. Now, suppose we knew how to extend the relative entropy results for the spherical entangling surface to other surfaces. Then the smoothness criteria above seems to constrain the coupling of the higher derivative interaction. This suggests that implicitly the relative entropy inequality knows about this. Since the positivity of relative entropy seems to rely only on the unitarity of the field theory, this raises the question if there is any conflict with unitarity if one is outside the allowed region for the coupling. It will be interesting to investigate this question since apriori there does not appear to be any such conflict in the dual gravity.

Appendix

A: Positivity of Relative entropy

Here we review the proof in quantum mechanics leading to the positivity of relative entropy. This can be found in Nielsen and Chuang’s book listed in [1]. We deﬁne relative entropy as,

ln(tx + (1 − t)y) ≥ t ln(x) + (1 − t) ln(y) for 0 ≤ t ≤ 1 . (6.126)

It is easy to generalize this to,

ln (x1t1 + x2t2 + ... + xmtm) ≥ t1 ln(x1) + t2 ln(x2) + ... + tm ln(xm) (6.127)

m X where ti = 1 and 0 ≤ ti ≤ 1 ∀i ∈ [1, m] . i=1

The equality follows if for some p, we have tp = 1. Using this, and (6.125) we can write, X X − Pij pi ln qj ≥ −pi ln ri where ri = Pijqj . (6.128) j j Hence we get, X pi X ri S(ρ|σ) ≥ p ln = − p ln . (6.129) i r i p i i i i Now note that, ln x ≥ x − 1. This gives X ri X ri S(ρ|σ) ≥ − p ln ≥ − p 1 − , i p i p i i i i X = (pi − ri) = 0 . (6.130) i Hence, S(ρ|σ) ≥ 0 and the equality follows when ρ = σ. To repeat, the only assumption that went in the proof was the unitarity of the quantum theory. So, whenever we have a unitary theory we can expect relative entropy to be positive. CHAPTER 6. RELATIVE ENTROPY 125

B: Relative entropy for R2 theory in shockwave background

In this section we want to sketch the calculation for the relative entropy in shockwave background for a general R2 theory9 where the disturbance is located very far away from the entangling surface. The action for this theory is shown below,

Z √ 12 L2 I = d5x G R + + λ R2 + λ R RAB + λ R RABCD . (6.131) L2 2 3 2 AB 1 ABCD

1 2 In this case, f∞ satisﬁes 1 − f∞ + 3 f∞(λ1 + 2λ2 + 10λ3) = 0. We start of with the shockwave metric as given in eq.(6.90) . We have explicitly checked that this is still a solution for the R2 theory. Next we quote the area functional for this theory [6, 14, 36],

Z √ 2 2π 3 L A B 1 i A B C D i ab SEE = 3 d x h 1+ (2λ3R+λ2(RABni ni − K Ki)+2λ1(RABCDni nj ni nj −KabKi )) . `P 2 2 (6.132)

Here i denotes the two transverse directions to the co-dimension 2 surface z = f(x1, x2, x3) and t = 0 and Ki’s are the two extrinsic curvatures along these two directions pulled back to the surface and a, b are three dimensional indices. Then we proceed in the same way as √ 2 2 0 before. We set z = z0 = R − r . Also as before we set x1 = 0 and without loss of any 0 3 generality and we will expand the integrand around x2 = ∞ . First we expand upto O( ) which is the linearized term and hence should yield ∆H = ∆S. The expression for ∆(1)S is

2 5 4 (1) 16π L R ∆ S = (1 + 2f∞(λ1 − 2(λ2 + 5λ3)) . (6.133) 5/2 3 06 15f∞ `P x2

The λi dependence has packaged into being proportional to c for the general theory [?]. Using the results of [9] (eq.(6.29) in that paper with the typo mentioned in footnote 4 taken into account), we ﬁnd that ∆H = ∆S at this order as expected. Then we expand (6.132) upto 4 order and pick out the 4 term which gives us the ∆(2)S. Note that for a general R2 theory the surface term is not known. So we can only do this calculation for the disturbance located very far away from the entangling surface such that we do not have to consider the perturbation to the entangling surface as this will contribute to some order higher than 4. Further since the extrinsic curvatures are each proportional to 3 and hence the O(K2) terms would be proportional to O(6), they will not contribute. The result before carrying out the

9The corresponding entropy functional will be useful in studying relative entropy in non-unitary log CFTs–for recent applications for entanglement entropy in these theories, see [37]. 126 REFERENCES integrations is shown below,

Z 5 (2) 2π 3L 2 2 2 2 2 ∆ S = dx3dx1dx2 ((x + x + x − R )[R (2f∞(23λ1 + λ2 − 10λ3) + 1) `3 5/2 08 1 2 3 P 2f∞ Rx2 2 2 2 − 60f∞λ1 x1 + x2 − x3(2f∞(23λ1 + λ2 − 10λ3) + 1)]) . (6.134) √ √ 2 2 2 2 Then we perform the integration over x3 which goes from − R − r to R − r and x1 = r cos(θ) , x2 = r sin(θ) . Now after some algebraic manipulation we can write the integrand as, 48π2L5R6(1 + 2f (13λ + λ − 10λ )) ∆(2)S = − ∞ 1 2 3 . (6.135) 5/2 3 08 35f∞ `P x2 Note that this is not proportional to c for this theory. Since for generic values of the couplings

λi, the bulk theory is non-unitary this may not be surprising. This may be indicative of the fact that rather than depending only on the two point function of the stress tensor, the higher point functions also contribute as in the second reference in [26]. The bulk theory will make sense as an effective theory where the couplings are small. In this circumstance, we can use field redefinitions to make the theory equivalent to Gauss-Bonnet with λ ∝ λ1. For the Gauss-Bonnet value λ1 = λ3 = λ , λ3 = −4λ it reduces to, 48π2L5R6(1 − 2f λ) ∆(2)S = − ∞ , (6.136) 5/2 3 08 35f∞ `P x2 which is proportional to c for the GB theory.

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7.1 Introduction

In this chapter we will briefly discuss an application of holographic entanglement entropy (EE). We have observed throughout this thesis that, this quantity is sensitive to UV physics and the leading divergence obeys the area law. This indicates that the EE is not a well defined observable in the continuum limit. In 2d, Casini and Huerta [1] devised a method to extract the universal contribution to the entanglement entropy. Liu and Mezei [2,3] generalized this prescription to higher dimensions. The resulting quantity, known as the renormalized entanglement entropy (REE), is UV finite and local on the scale of the entangling region. The evolution of REE with respect to the size of the entangling region can be used as a probe to realize the RG flow. Moreover, for vacuum states the REE for a spherical entangling region provides a c-function parametrizing the RG flow 1. We want to compute the REE for states that break Lorentz invariance due to the presence of a finite charge density. While holographic RG flows for vacuum states correspond to domain walls (i.e. solutions interpolating between same dimensional AdS)[12, 13, 14], the flow for charged states can be described in terms of black holes/branes. Specifically, we want to study the REE for BPS black solutions in N = 2, 4d FI gauged supergravity. Starting with the work of Cacciatori-Klemm [15] these models have been studied extensively over the last few years [16, 17, 18], culminating with the full construction of static BPS solutions for all symmetric models by [19, 20]. These are solutions that interpolate between AdS4 and AdS2 × Σk, where Σk is the surface of constant sectional curvature with k = −1, 0, 1. Since these objects interpolate between different AdS spaces they are interesting from the holographic perspective. Morever, for the STU model these solutions have a M-theory realization through an embedding into the de Wit-Nicolai N = 8 theory [21]. This chapter is organized as follows. In section (7.2) we outline the computation for the REE for the black brane solutions. In section (7.3), we summarize the BPS black objects in

1 For more applications of REE in the context of holographic RG ﬂows, interested readers are referred to some of these references [4,5,6,7,8,9, 10, 11] and the references there in.

131 132 7.2. RENORMALIZED ENTANGLEMENT ENTROPY

AdS4, followed by the computation for the REE in section (7.4). Finally, in Appendix (7.5), we discuss the symplectic invariant in N = 2. This chapter is based on the work [22] done with Dr. Shajid Haque and Dr. Alvaro veliz-Osorio.

7.2 Renormalized Entanglement Entropy

In this section we outline a general procedure to obtain the universal contributions to the entanglement entropy for quantum systems that can be described holographically by a metric of the form 2 2 2 −2 2 2 2 ds = −a dt + a dr + b dΩk. (7.1)

2 In the above expression, dΩk is the line element of a surface of constant sectional curvature k. Clearly we must demand AdS asymptotics. Hence, as r → ∞ the metric takes the form r r a → b → . (7.2) l4 l4 We wish to compute the entanglement entropy for a subsystem A consisting of a disk Σ(R) of radius R. From the Ryu-Takayanagi prescription [28], we know that this quantity corresponds to the area of an extremal surface attached to ∂Σ(R) going into the bulk (see Fig. 7.1). For the metric (7.1) this problem corresponds to the Plateau problem for the functional

Z R 2π 2p −2ψ 2 ψ S(R) = 2 dρ ρ b 1 + e r˙ e ≡ ab. (7.3) lp 0

2π In the subsequent calculations we will absorb the factor 2 into S(R). From the above lp functional, it can be showed that the proﬁle r(ρ) of the minimal surfaces can be found from the ODE r˙ b0 r¨ − ψ0 r˙2 + − 2 e2ψ 1 + e−2ψ r˙2 = 0, (7.4) ρ b 0 where ˙ = ∂ρ and = ∂r. This equation is supplemented with the boundary conditions

r(0) = r0 > 0r ˙(0) = 0. (7.5)

Solutions of equations (7.4) and (7.5) correspond to extremal surfaces attached to the boundary of a disk at inﬁnity and whose tip is at r = r0. Moreover, the depth of the tip can be related to the size R of the entangling disk at the boundary, R corresponds to the value of ρ for which r(ρ) → ∞. Therefore, each surface can be labeled either by the size of the entangling region R or by the depth it reaches in the bulk r0 (see Fig. 7.1). Once we have found the proﬁler ˜(ρ) of the minimal surface, we are instructed to plug it into the functional (7.3) in order to obtain the holographic entanglement entropy. However, CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENT ENTROPY 133

R3 > R2 > R1

Boundary Ρ

r Horizon 1 r2 r3

Figure 7.1: Minimal surfaces in AAdS one must be careful since the resulting quantity is divergent. We should, therefore, regularize this integral first. We introduce a UV cut-off in the following way–let 1 and restrict the values of r such that l 4 > . (7.6) b(r) Usingr ˜(ρ), we can translate this bulk cut-off into a boundary cut-off, i.e., we must consider only ρ < R, where l 4 = . (7.7) b(˜r(R)) Then we can compute the finite quantity

Z R S(R, ) = dρ A(ρ, R), (7.8) 0 where A(ρ, R) stands for the integrand of (7.3) evaluated on the solutionr ˜(ρ). In order to systematically obtain the universal contribution to (7.8), which we call hereafter renormalized entanglement entropy, we use the operator introduced in [1,2]. In four bulk space-time dimensions, the renormalized entanglement entropy is given by d S(R) ≡ R − 1 S(R, ). (7.9) dR This quantity can be alternatively written as

Z R ∂R ∂A S(R) = A(R,R) R + R − A . (7.10) ∂R 0 ∂R 134 7.3. BPS BLACK OBJECTS IN ADS4

As an illustration let us brieﬂy discuss the application of this procedure for pure AdS4. For this geometry equation (7.4) yields l2 r˜(ρ) = 4 . (7.11) pR2 − ρ2

2 −1 This corresponds to an extremal surface reaching into the bulk until r = l4R . Using equation (7.7) we ﬁnd r 2 R = R2 − . (7.12) 2 For this simple case we can compute equation (7.10) explicitly. First we ﬁnd Rρ A(R, ρ) = , (7.13) (R2 − ρ2)3/2 and then Z R R dρ A(R, ρ) = − 1, (7.14) p 2 2 0 R − R which diverges as we take → 0. However, the additional terms in (7.10) contribute as follows: 2 ∂R R A(R,R) = 2 2 3/2 ∂R (R − R ) Z R 2 ∂A R dρ = − 2 2 3/2 . (7.15) 0 ∂R (R − R )

So we ﬁnd that for AdS4 S(R) = 1. (7.16) This result is consistent with the interpretation of the REE as a c-function probing the holographic renormalization group ﬂow [14, 29, 30, 31, 32].

7.3 BPS black objects in AdS4

We wish to apply the techniques presented in the previous section to an interesting class of 1 solutions of the form (7.1), namely 4 -BPS black objects in N = 2, FI gauged supergravity. These correspond to zero temperature solutions supported by scalars and abelian gauge ﬁelds.

They are parametrized by two vectors of 2nv + 2 real quantities ! ! pI gI Γ = and G = , (7.17) qI gI

I I where p and qI are the magnetic and electric charges of the gauge ﬁelds while g and gI are the parameters of the Fayet-Iliopoulos potential. In (7.17) the index I = 1, . . . , nv + 1, where CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENT ENTROPY 135

nv is the number of vector multiplets considered. These parameters are not independent. In fact, they are subject to the symplectic constraint

hG, Γi = k, (7.18) where just as in (7.1), k labels the horizon topology, e.g., k = −1, 0, 1 indicates spherical, flat and hyperbolic respectively. In the following discussion we focus only on the properties of the spacetime metric and leave the behavior of the scalars and gauge fields aside. In terms of the quantities (7.17), the warp factors are given by [19, 20] 1 eψ = a b = I (G)1/4 r + hG, Bi r b = I (H)1/4, (7.19) 4 2 4 where I4(V ) is defined in equation (7.33) in the Appendix, and H is a symplectic vector of linear functions H = A r + B. (7.20) Here, A and B are constant symplectic vectors. The former can be obtained directly from the FI parameters 1 A = I (G)−3/4dI (G), (7.21) 2 4 4 while the latter is given by a combination of the charges and FI parameters, dictated by the algebraic equation 1 dI (B,B,G) = Γ, (7.22) 4 4 where dI4 is defined in equation (7.35) in the Appendix. Moreover, physical consistency requires to choose solutions of (7.22) that fulfill the constraints

hG, Bi > 0 I4(B) > 0. (7.23) Therefore the construction of a BPS solution is reduced to a purely algebraic problem.

The warp factors (7.19) correspond to a metric that interpolates between AdS4 at infinity 2 (UV) and an AdS2 × Σk near horizon (IR) geometry as r → 0. The AdS radii corresponding to these UV and IR geometries are given by 1 l = I (G)−1/4 l = I (B)1/4hG, Bi−1. (7.24) 4 4 2 2 4 Moreover, the entropy (density for k=0) is proportional to 1 b2 −−−→IR σ2 = pI (B). (7.25) 0 4 4 At this point we want to remind the reader that these solutions are also accompanied by flowing scalars. Due to the attractor mechanism [16, 17] the scalars flow from constant to constant. These scalars can be thought of as coupling constants and giving rise to the notion of an attractive RG flow [23]. 136 7.4. REE FOR BPS BLACK BRANES

7.4 REE for BPS black branes

In this section we compute the renormalized entanglement entropy as discussed in section (7.2) for solutions of the kind presented in section 7.3. In the following computation we restrict ourselves to black objects with flat horizons (k = 0), i.e., black branes. Furthermore, we will consider solutions of the STU-model. This model captures the essential features of extremal black holes in N ≥ 2, d = 4 theories [26]. In the STU-model, the structure constants for the prepotential (see Appendix 7.5) are given by cijk = |ijk|. By plugging the warp factors (7.19) into equation (7.4), we obtain an explicit, albeit complicated, differential equation for r in terms of ρ. Now we need to set the parameters that will support the solution i.e. charges/FI parameters. To each such charge/FI configuration we can associate three symplectic invariant combinations, which correspond to the AdS length scales and the entropy density. As we will see in the following our results depend only on these three quantities. In order for the solution to be regular these quantities must not vanish and we choose the charges/FI parameters accordingly. Hereafter we will consider solutions supported by non-vanishing charges/FI parameters

i 0 (q0, p ; g , gi) i = 1, . . . , nv + 1 (7.26) or 0 i (p , qi; g0, g ) i = 1, . . . , nv + 1. (7.27) In the following discussion we will display the results for the first configuration (7.26). For the second configuration we have verified explicitly that we get completely analogous results. It is straightforward to extend the following discussion to other configurations as well. Moreover, the reader must keep in mind that the results that follow are invariant under symplectic transformations of the kind discussed in [25, 26]. Given the intricacy of the resulting ODE describing the extremal surfaces’ profile, we proceed to solve it numerically. Moreover, in order to realize the program outlined in section

7.2 we are compelled to produce a large sample of such minimal surfaces ri(ρ) with i being an index for the sample (see Fig. 7.2). Now by introducing a cut-off it is possible to compute numerically (7.8), creating thus a list of regularized areas Si() corresponding to each of the extremal surfaces ri(ρ). Hence we are left with a list of points (Ri,Si()), which can be interpolated to find S(R, ). Finally, from this function we construct the desired renormalized entanglement entropy S(R) for a given set of charges/FI parameters. We must point out that by construction S(R) is a cut-off independent quantity. This behavior is exhibited by our numerical computations as we tune to ever smaller values. The resulting REE is depicted in Fig. 7.3 for a particular example. However the observed behavior is generic regardless of the values chosen for the parameters (7.26). First of all, as CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENT ENTROPY 137

Asymptopia

R R3 r 4 R2 R1 0.5

0.4

r(Ρ) 0.3

0.2

r4 : 0.1

r1 Ρ 0.0 0.0 0.2 0.4 0.6 0.8 Horizon

1 2 3 0 Figure 7.2: Some extremal surfaces for a p = 2, p = p = 1 and −g = g1 = g2 = g3 = 1 black brane

S(R) → 1 as R → 0 in agreement with (7.16), the REE then decreases monotonically until it reaches a minimum S∗ when the entangling disk has a radius R∗. After reaching that critical value, S(R) starts to increase and approaches the value σ0 as we get closer to the horizon.

S(R)

0.9

0.8 Σ0

0.7 UV IR

0.6

0.5

Smin R 0.2 0.4 0.6 0.8 Rmin

Figure 7.3: Renormalized entanglement entropy for a p1 = 2, p2 = 1, p3 = 1/2 and −g0 = g1 = g2 = g3 = 1 black brane 138 7.5. DISCUSSION

We wish to explore how the values of S∗ and R∗ depend upon the charges/FI parameters Γ and G. First of all, it is clear that these parameters must enter only through symplectic invariant combinations. The warp factors (7.19) can be speciﬁed in terms of the invariant quantities {I4(G), hG, Bi,I4(A),I4(B)}. From equation (7.24, 7.25) we can identify them as

{l2, l4, σ0,I4(A)}. But a closer inspection shows that I4(A) can be expressed in terms of l4.

Hence, there are three independent invariants upon which S∗ depends, namely the AdS radii and the entropy density. In order to ﬁnd a pattern, we start by identifying a subclass of parameters for which one of the symplectic invariants is held ﬁxed. One such family is given by 1 2 3 g1 = g2 = g3 p = p = p . (7.28) The crucial point here is that for black branes with these kind of parameters the near- 1 horizon AdS2 radius l2 is independent of the value of p . Let’s see how this comes about. For charge/FI combinations of the form (7.28) the solution of equation (7.22), consistent with the conditions (7.23), reads s 1 1 2 3 0 p 0 −1p 1 B = B = B = λ1 sgn(g ) ,B0 = λ2 |g | p g1. (7.29) g1

Here λ1 and λ2 are known positive constants, and the components of B that are omitted p 1 vanish. The upshot is that B is proportional to p . Therefore, since l2 in (7.24) is invariant under rescalings of B, it is clear that the p1 dependence drops out.

Now we ﬁx the overall scale l4, and study the behavior of S∗. In this context, changing p1 gives rise to a one parameter family of solutions with constant l2 and varying σ0. Interestingly, we ﬁnd that S∗ is constant along this family, which implies that S∗ is a function of l2 only.

Notice that in the regime R = 0 to R = R∗ the REE decreases monotonically from the pure AdS4 value to a constant which can be determined solely from the AdS2 radius. This is reminiscent of the c-function discussed in [23]. It would be interesting to investigate this connection further.

Then we explore how S∗ varies with l2. The variation is displayed in Fig (7.4). This plot clearly shows that S∗ increases with l2. On the other hand, R∗ depends on both l2 and σ0.

Moreover, it increases with l2 and decreases with σ0.

7.5 Discussion

In this chapter we have computed the renormalized entanglement entropy, S for 4d, N = 2

BPS black brane solutions. These solutions interpolate between AdS4 in the UV and a space with an AdS2 factor in the IR. Speciﬁcally, we have investigated the behavior of S as a CHAPTER 7. CODING HOLOGRAPHIC RG FLOW USING ENTANGLEMENT ENTROPY 139

0.45

0.40

0.35

0.30 l2 0.07 0.08 0.09 0.10 0.11 0.12

Figure 7.4: Variation of S∗ with l2

function of the size of the entangling region. We have found that in this context S first decreases monotonically with R, reaches an extremum and then increases again. This is the key finding of our investigation. We have already mentioned in the main text that all the parameters of the brane solutions enter by three independent symplectic invariant combinations that can be identified as the three independent scales of the system, namely l4, l2 and σ0. In our inspection we found that starting from the UV the S monotonically decreases until it reaches a minimum that is determined completely by the radius of AdS2. Then it starts to increase again and approaches the black brane entropy density. Furthermore, as pointed out in the previous section, when we increase the entropy of the brane this transition occurs closer to the UV. The fact that the S decreases monotonically for that region in R resembles the behavior of a c-function for vacuum states. Then the chemical potential starts to dominate once we go deeper in the IR. Entanglement entropy measures the entropy due to tracing out part of the total system. If the total system is in a mixed state this quantity receives contributions both from entanglement and from the mixedness of the original system. Since black branes correspond to mixed states we expect our computation to be influenced by both of these factors. In light of that, we are inclined to interpret our result in the following way–the REE is driven predominantly by entanglement close to the UV before reaching R∗, where contributions due to the mixedness of the branes take over. 140 7.5. DISCUSSION

Appendix

A: Duality transformations for N = 2 gauged supergravity

Supergravity solutions can be easily written in terms of symplectic vectors. These are vectors with 2nv + 2 components of which the ﬁrst nv + 1 components are labeled with an upper index and the remaining ones with a lower index, e.g., ! V I V = . (7.30) VI

I = 0, . . . , nv. These vectors are acted upon by symplectic transformations ! ! ! V˜ I UZ V I = , (7.31) ˜ VI WV VI where

U T V − W T Z = VU T − WZT = 1 U T W = W T U,ZT V = V T Z. (7.32)

We refer to these reparametrizations as duality transformations [25]. In N = 2 models, physical quantities must be invariant under duality transformations. Duality invariant quantities can be succinctly expressed in terms of the symplectic quartic invariant I4 [27] 1 I (V ) = tMNPQV V V V 4 4! M N P Q 2 2 2 = − V I V + V c V iV jV k − V 0 cijkV V V I 3 0 ijk 3 i j k lmn i j + cijk c V V VlVm. (7.33)

In this expression and the ones to follow, the lower-case indices run from 1 to nv only. The constant coeﬃcients cijk encode the underlying special geometry prepotential

1 XiXjXk F = − c . (7.34) 6 ijk X0 For future convenience, we also deﬁne

∂I4(V ) dI4(V ) = ΩMN , (7.35) ∂VN REFERENCES 141 with ! 0 I ΩMN = , (7.36) −I 0 the canonical symplectic matrix. Moreover, given four symplectic vectors we deﬁne

(1) (2) (3) (4) MNPQ (1) (2) (3) (4) I4 V ,V ,V ,V = t VM VN VP VQ . (7.37)

Notice the absence of the overall symmetrization factor with respect to (7.33). In practice, we can obtain the t-tensor by hitting (7.33) with four derivatives. The black hole solution is determined by a set of electric/magnetic charges and ﬂuxes.

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In this thesis we have tried to understand the interplay between quantum information and geometry through entanglement entropy (EE) in the framework of AdS/CFT correspondence. It has provided us with a nice geometrical problem and has the merit to shed light on how holography works. Ryu-Takayanagi (RT) proposal has provided us a way to compute EE from holography for Einstein gravity. In this thesis we have discussed various generalization of RT proposal for higher derivative gravity theories. These higher derivative theories are interesting as they extend the AdS/CFT dictionary beyond supergravity limit and incorporate the effect of finite coupling. Recently Lewkowycz and Maldacena (LM) have proposed to set this calculation in a self consistent way, solidifying its geometrical connection. Their idea stems from the fact that there exists a striking similarity between EE and the black hole entropy. Using LM technique one can formulate a proof for RT proposal and but unfortunately this cannot be generalized for more general theories of gravity, it seems that it only works for certain class of gravity theories. It will be nice find a way to overcome this difficulty in future. On the other hand there is a well defined way of deriving the black hole entropy for any diffeomorphism invariant theory based on the Nother charge method proposed by Iyer and Wald. Based on this Noether charge construction Wald has proposed a general formula for black hole entropy for any diffeomorphism invariant theory. For Einstein gravity both the black hole entropy and EE follows area law. Now the question is, can one simply use Wald entropy functional to compute EE for any general theories of gravity? The answer is no, as Wald entropy functional is valid only for bifurcation surface and for general surface one has to systematically modify it to get correct EE as predicted form the holography. Now it seems that one can connect EE with this Noether charge method and in this thesis we have partially achieved that, although still a rigorous proof is needed. Recently it has been shown that indeed one can connect EE with the Noether charge construction by demanding that the entropy functionals used to evaluate EE for general theories of gravity satisfy second law of thermodynamics (a generalized version of it). So it will be very nice to make this connection more rigorous in the future as it will provide a solid geometrical interpretation for EE . Further we have used holographic entanglement to constrain gravity in interesting ways. First, we have used positivity of relative entropy and analysed the inequality ∆S ≤ ∆H

145 146 for Gauss-Bonnet gravity for a given class of small perturbations around the vacuum state. We found that for all our examples, the positivity of the stress tensor two point function ascertained that this was respected. On the bulk side this corresponds to metric fluctuations having positive energy. The simplicity of the final result does cry out for a simpler explanation for our findings. Although the intermediate integrals involved appeared very complicated, the final result was simply proportional to the Weyl anomaly c. It would be nice to find a simple explanation for this finding. Unfortunately the relative entropy calculation is limited to the spherical entangling surface as for this kind of surface only, the modular hamiltonian (H) is known. Finally, we also considered other entangling surfaces and demanded that these close off smoothly in the bulk. In Gauss-Bonnet gravity, this led to the coupling being constrained. The spherical entangling surface did not lead to any constraints on the coupling while the cylindrical and slab entangling surfaces did. This leads to an interesting question. Suppose we knew how to extend the relative entropy results for the spherical entangling surface to other surfaces. Then the smoothness criteria above seems to constrain the coupling of the higher derivative interaction. This suggests that implicitly the relative entropy inequality knows about this. Since the positivity of relative entropy seems to rely only on the unitarity of the field theory, this raises the question if there is any conflict with unitarity if one is outside the allowed region for the coupling. It will be interesting to investigate this question since apriori there does not appear to be any such conflict in the dual gravity. It will also be interesting to find if there are other entangling surfaces which lead to a tighter bound and if the bounds are stronger than the causality constraints. Then at the end we have discussed how to code holographic RG flow using EE. We have investigated the behaviour of renormalized entanglement entropy (REE) in the context of a lorentz violating RG flow. We have found that REE first monotonically decreases and then increases smoothly along the RG flow, thus exhibiting a minima. It would be interesting to see if we can use the behavior of REE as a function of entangling surface in order to establish an order parameter for the phase transition between the vacuum to vacuum flow and vacuum to charged state flow. This is because the existence of the extremum for the REE tells us that the system is transiting from its vacuum behavior at that point. Form all this analysis it is quite evident that entropy functionals used to evaluate EE play a central role. In recent times, many different tools of quantum entanglement like, entanglement negativity, differential entropy, quantum error coding, relative entropy, cMERA etc, have been used in an attempt to build geometry from the field theory data, is some sense trying to prove the holographic principle. In the light of these recent advances, these holographic entropy functionals play a crucial role and a thorough understanding of these entropy functionals will teach us many important lessons about gravity.