Investigations on the SYK Model and its Dual Gravity Theory Kenta Suzuki Department of Physics Brown University A dissertation submitted for the degree of Doctor of Philosophy May 2018 c Copyright 2018 by Kenta Suzuki All Right Reserved This dissertation by Kenta Suzuki is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy Date Professor Dr. Antal Jevicki, Advisor Recommended to the Graduate Council Date Professor Dr. David Lowe, Reader Date Professor Dr. Chung-I Tan, Reader Approved to the Graduate Council Date Professor Dr. Andrew G. Campbell, Dean of the Graduate School iii Curriculum Vitae Education B.A., Physics: Chiba University, Japan, 2008 2012 Department of Physics, Graduate School of Science, Chiba University, Japan, 2012 2013 Ph.D., Physics, Brown University, 2013 2018. Advisor: Prof. Antal Jevicki Academic Honors Galkin Foundation Fellowship Award, Brown University, 2017 2018 Publications (i). Journal Articles 1. “Space-Time in the SYK Model,” arXiv:1712.02725 [hep-th], with S. R. Das, A. Ghosh and A. Jevicki. 2. “Three Dimensional View of Arbitrary q SYK models,” JHEP 1802, 162 (2018), [arXiv:1711.09839 [hep-th]], with S. R. Das, A. Ghosh and A. Jevicki. 3. “Three Dimensional View of the SYK/AdS Duality,” JHEP 1709, 017 (2017), [arXiv:1704.07208 [hep-th]], with S. R. Das and A. Jevicki. 4. “Bi-Local Holography in the SYK Model: Perturbations,” JHEP 1611, 046 (2016), [arXiv:1608.07567 [hep-th]], with A. Jevicki. 5. “Bi-Local Holography in the SYK Model,” JHEP 1607, 007 (2016), [arXiv:1603.06246 [hep- th]], with A. Jevicki and J. Yoon. iv 6. “Thermofield Duality for Higher Spin Rindler Gravity,” JHEP 1602, 094 (2016), [arXiv:1508.07956 [hep-th]], with A. Jevicki. 7. “Physical unitarity for a massive Yang-Mills theory without the Higgs field: A perturba- tive treatment,” Phys. Rev. D 87, no. 2, 025017 (2013), [arXiv:1209.3994 [hep-th]], with K. I. Kondo, H. Fukamachi, S. Nishino and T. Shinohara. (ii). Conference Proceedings 1. “Finite Temperature Maps in Vector/Higher Spin Duality,” Proceedings, International Work- shop on Higher Spin Gauge Theories : Singapore, Singapore, November 4-6, 2015, with A. Jevicki and J. Yoon. 2. “Physical unitarity of a massive Yang-Mills theory without the Higgs field from a viewpoint of confinement,” Proceedings, 10th Conference on Quark Confinement and the Hadron Spec- trum (Confinement X) : Munich, Germany, October 8-12, 2012, [arXiv:1301.2480 [hep-th]], with K. I. Kondo, H. Fukamachi, S. Nishino and T. Shinohara. Service Referee for Physical Review D and Physical Review Letters, 2016 present Teaching Fall 2013: Phys 0030, Classical Mechanics, Teaching Assistant Spring 2014: Phys 0040, Electromagnetism, Teaching Assistant Spring 2015: Phys 1100, Introduction to General Relativity, Teaching Assistant Spring 2016: Phys 0060, Electromagnetism, Teaching Assistant Spring 2017: Phys 2340, Group Theory, Teaching Assistant v Acknowledgements First of all, I would like to thank my advisor Prof. Antal Jevicki for his extremely helpful guidances and patience for discussions at all times throughout my Ph.D. program. He always surprised me with his broad knowledge of physics, but still he was willing to discuss various materials at any time. It was always a pleasure to work together, regardless of whatever topic we were working on. I would also like to thank my thesis committee members: Prof. Chung-I Tan and Prof. David Lowe, for reading the draft of this dissertation carefully and their constructive comments, on top of their teaching during my Ph.D. program. Many parts of my work resulted from discussions with my collaborators. Prof. Sumit R. Das always guided our research forward and broadened my understanding by his inspiring ideas. Prof. Robert de Mello Koch patiently taught me many details of his work and always encouraged me with his helpful guidance. Dr. Junggi Yoon provided me with his thoughtful and helpful guides for physics and Ph.D. life from the very beginning of my graduate study. It was very pleasure for me to discuss with Dr. Animik Ghosh for various topics. Also, my gratitude goes to all Faculties, Postdocs and Students at the Department of Physics, especially in the high energy theory group for their helpful and inspiring discussions and teaching. I also thank Mary Ann Rotondo for all her supports. It was my pleasure to spend my Ph.D. life in the high energy theory group at Brown University. Finally, I am very grateful to the Galkin Foundation for its support during the academic year 2017-2018. The work is also supported by the Department of Energy under contract DE- SC0010010. Kenta Suzuki Brown University April 2018 vi Contents 1 Introduction 2 2 The Model 4 2.1 Bi-local method 4 2.2 Relation to Zero Mode Dynamics 7 3 Shift of the Classical Solution 9 3.1 Evaluation of 1 9 3.2 Evaluation of 2 12 3.3 All Order Evaluation in q > 2 13 4 Finite Temperature 15 4.1 Classical Solutions 15 4.2 Tree-Level Free Energy 18 5 Bi-local Propagator and Spectrum 20 5.1 Zero Mode Contribution 23 6 3D Interpretation 25 6.1 Kaluza-Klein Decomposition 26 6.2 Evaluation of G(0) 27 6.3 First Order Eigenvalue Shift 29 7 Question of Dual Spacetime 31 7.1 “i” Problem 31 7.2 Transformations and Leg Factors 33 7.3 Green’s Functions and Leg Factors 37 8 Conclusion 41 A ✏-Expansion 42 B s-Regularization and Schwarzian Action 43 C Schrodinger Equation 47 D Completeness Condition of Z⌫ 48 E Evaluation of the Contour Integral 49 –1– F EAdS Scalar Propagators 50 F.1 p-Integral Form 51 F.2 ⌫-Integral Form 52 F.3 p-Integral 53 F.4 ⌫-Integral 54 G Unit Normalized EAdS/dS Wave Functions 55 H Completeness and Orthogonality of Ki⌫ 56 1 Introduction The Sachdev-Ye-Kitaev (SYK) model was proposed as a simpler, yet non-trivial example of the AdS/CFT correspondence, which was based on the earlier model by Sachdev and Ye (SY) [1–5]. Detailed investigations of the SYK model [6–18] have shed light on interesting, highly non-trivial aspects of the AdS/CFT correspondence and provide a potential framework for quantum black holes. The model, which can be studied at Large N , features an emergent conformal reparametrization symmetry at the IR critical point, which triggers out-of-time- order correlators exhibiting quantum chaos, with a maximal Lyapunov exponent characteristic of quantum black holes [7], providing an example of the butterfly e↵ect [19–25]. Connections of the model to Random Matrix Theory have further elaborated this quantum black hole interpretation of the model [26–37]. Related models have been studied also [38–43] as well as various generalizations [44–58]. The solution and properties in leading order in Large N are shared with tensor type models [59–85]. Despite great interest on the model, the precise bulk dual of the SYK model is still not understood. It has been conjectured in [86–89] that the gravity sector of this model is the Jackiw-Teitelboim model [90, 91] of dilaton-gravity with a negative cosmological constant studied in [92], while [93–97] provide strong evidences that it is actually Liouville theory. (Various other aspects of this dilaton-gravity sector have also been studied [98–105].) On the other hand, it is also known that the matter sector contains an infinite tower of particles [9–11]. Couplings of these particles have been computed by calculating higher-point functions in the SYK model [15, 16], and the spectrum of the matter sector can be understood from 3D gravity theory [106–108]. In this thesis, we describe the development of systematic Large N representation of the model given in [11, 12], through a nonlinear bi-local collective field theory. This representation systematically incorporates arbitrary n-point bi-local correlators through a set of 1/N vertices and propagator and as such gives the bridge to a dual description. It naturally provides a holographic interpretation along the lines proposed more generally in [109, 110], with the center of mass and relative coordinates of the two points in the bi-local fields. The Large –2– N SYK model represents a highly non-trivial non-linear system. At the IR critical point (strong coupling limit) there appears a zero mode problem which at the outset prevents a perturbative expansion. We treat this mode through introduction of collective “time” coordinate as a dynamical variable as in quantization of extended systems [111]. Its Faddeev- Popov quantization was seen to systematically project out the zero modes, providing for a well defined propagator and expansion around the IR point. What one has is a fully nonlinear interacting system of bi-local matter with a discrete gravitational degree of freedom governed by a Schwarzian action. We will demonstrate the non-linear derivation of the action to be exact at all orders [12], which leads to the so called “enhanced” contributions at the linearized quadratic level originally described in [10]. We will also describe the three dimensional interpretation of the bulk theory [106, 107]. The zero temperature SYK model with four point interactions corresponds to a background AdS2 ⇥ I, where I = S 1 /Z2 is a finite interval whose size needs to be suitably chosen. There is a single scalar field coupled to gravity, whose mass is equal to the Breitenlohner-Freedman bound [112] of AdS2 . The scalar field satisfies Dirichlet boundary conditions at the ends and feels an external delta function potential at the middle of the interval. The background can be thought of as coming from the near-horizon geometry of an extremal charged black hole which reduces the gravity sector to Jackiw-Teitelboim model with the metric in the third direction becoming the dilaton of the latter model [87]. The strong coupling limit of the SYK model corresponds to a trivial metric in the third direction, while at finite coupling this acquires a dependence on the AdS2 spatial coordinate. With a suitable choice of the size of the interval L and the strength of the delta function potential V , we show that at strong coupling, (i) the spectrum of the Kaluza-Klein (KK) modes of the scalar is precisely the spectrum of the SYK model and (ii) the two point function with both points at the center of the interval is in precise agreement with the strong coupling bi-local propagator, using the simplest identification of the AdS coordinates proposed in [109]. For finite coupling, we adopt the proposal of [87, 88], and show that to order 1/J, the poles of the propagator shift in a manner consistent with the explicit results in [10]. We will finally address the so called “i” problem discussed in [108]. If one considers the Euclidean partition function of the model, changing variables to the center of mass and relative coordinates of the bi-local, one reaches a solution (the propagator and quadratic fluctuations) with a Lorentzian signature, coming from the fact that the two points of the bi-local become coordinates of a Lorentzian signature. On the other hand, we expect that the dual theory of the Euclidean SYK model should live in Euclidean spacetime [10]. One issue which is detrimental to a potential Lorentzian identification associated with this data comes from the factors of “ i ” which inevitably appears in a Lorentzian dual theory, but absent in the SYK propagator. Secondly, the radial part of the AdS2 wave functions which appear in the SYK eigenfunctions (whether or not we write this in the 3D language) are not the usual normalizable AdS wave functions, but satisfy di↵erent boundary conditions. These unusual wave functions are, however, required since these are the ones which diagonalize the SYK kernel [9, 11]. This suggests that they might be better thought of as dS2 wave functions [10] –3– 1. The resolution of both issues are given as follows. We will show that a non-local transform relates the bi-local field to a field whose underlying dynamics is in Euclidean AdS2 . We will arrive at this transform following the same principles underlying the derivation of the corresponding transform for the O(N ) model in d = 3 [114, 115]: the idea is to find a canonical transformation in the four dimensional phase space of the two points in the bi-local such that the symmetries of EAdS2 are realized correctly. This suggests a simple transformation kernel for the momentum space fields. It turns out that the corresponding position space kernel is a H 2 Radon transform. Radon transforms have appeared (explicitly or implicitly) in discussions of AdS/CFT, most notably in [116–118] where this is used to go from the bulk to the kinematic space of the boundary field theory on a time slice. Indeed the space on which the bi-locals live is a version of kinematic space. However, unlike these papers we are not working on a time slice in the bulk - rather our transform takes unequal Euclidean time fields on EAdS2 to bi-locals. Though mathematically identical, our transform is conceptually somewhat di↵erent. The necessity of a Radon transform in this context has been in fact mentioned in [10]. This transformation takes the particular combinations of Bessel functions which appear in the SYK propagators to the modified Bessel functions which appear in the standard EAdS2 propagator. However there are additional factors which morally resemble the leg pole factors of the c = 1 matrix model necessary to relate the collective field [119] to the usual tachyon field of the dual 2D string theory and reproduce the S-Matrix [120–124] (for a recent improved understanding see [125]). In this latter case these factors are believed to arise from the discrete states of the 2D string. For our case it is tempting to speculate that the leg pole factors also arise from similar bulk degrees of freedom, which remain to be identified. In fact we find an intriguing analogy between the SYK propagator and the propagator of macroscopic loop operators [120]. 2 The Model 2.1 Bi-local method In this subsection, we will give a brief review of our formalism [11, 12]. The Sachdev-Ye-Kitaev model [7] is a quantum mechanical many body system with all-to-all random interactions on fermionic N sites (N 1), represented by the Hamiltonian N 1 X H = Jijkl i j k l , (2.1) 4! i,j,k,l=1 where i are Majorana fermions, which satisfy { i , j } = ij . The coupling constant Jijkl are random with a Gaussian distribution. The original model is given by this four-point inter- action; however, it is a simple generalization to analogous q-point interacting model [7, 10]. 1 This has been suggested by J. Maldacena [113]. –4– In this thesis, we follow the more general q model, unless otherwise specified. Nevertheless, our main interest represents the original q = 4 model. After the disorder averaging for the random coupling Jijkl , there is only one e↵ective coupling J and the e↵ective action is written as Z N X n Z n N !q 1 X J2 X X a a a b Sq = dt i @t i dt1 dt2 i (t1 ) i (t2 ) , (2.2) 2 2qN q 1 i=1 a=1 a,b=1 i=1 where a, b are the replica indexes. Throughout this thesis, we only consider this Euclidean time model. We do not expect a spin glass state in this model [8] and we can restrict to replica diagonal subspace [11]. Therefore, introducing a (replica diagonal) bi-local collective field: N 1 X (t1 , t2 ) ⌘ i (t1 ) i (t2 ) , (2.3) N i=1 the model is described by a path-integral Z Y Scol [ ] Z = D (t1 , t2 ) µ[ ] e , (2.4) t1 ,t2 with an appropriate order O(N 0 ) measure µ and the collective action: Z h i Z N 0 N J 2N q Scol [ ] = dt @t (t, t ) + Tr log dt1 dt2 (t1 , t2 ) , (2.5) 2 t0 =t 2 2q where the trace term comes from a Jacobian factor due to the change of path-integral variable, and the trace is taken over the bi-local time. This action being of order N gives a systematic 1/N expansion, while the measure µ found as in [126] begins to contribute at one loop level (in 1/N ). Here the first linear term represents a conformal breaking term, while the other terms respect conformal invariance. This naive expression of the breaking term represents a product at the same point, which will be receiving regularization in our perturbation. In the IR with the strong coupling limit |t|J 1, the collective action is reduces to the critical action Z N J 2N Sc [ ] = Tr log dt1 dt2 q (t1 , t2 ) , (2.6) 2 2q which exhibits the emergent conformal reparametrization symmetry 1 0 0 q (t1 , t2 ) ! f (t1 , t2 ) = f (t1 )f (t2 ) (f (t1 ), f (t2 )) , (2.7) with an arbitrary function f (t). The critical saddle-point solution is given by p !2 |f 0 (t1 )f 0 (t2 )| q 0,f (t1 , t2 ) = b , (2.8) |f (t1 ) f (t2 )| –5– where b is a time-independent constant. This symmetry is responsible for the appearance of zero modes in the strict IR critical theory. This problem was addressed in [11, 12] with analog of the quantization of extended systems with symmetry modes [111]. The above symmetry mode representing time reparametrization can be elevated to a dynamical variable through the Faddeev-Popov method which we summarize as follows: we insert into the partition function (2.4), the functional identity: R Z Y Y ✓Z ◆ u· f Df (t) u· f = 1, (2.9) t t f so that after an inverse change of the integration variable, it results in a combined represen- tation Z Y ✓Z ◆ Y Z = Df (t) D (t1 , t2 ) µ(f, ) u · f e Scol [ ,f ] , (2.10) t t1 ,t2 with an appropriate Jacobian. After separating the critical classical solution 0 from the bi-local field: = 0 + , the total action is now given by Z N ⇥ ⇤ Scol [ , f ] = S[f ] + f s + Sc [ ] . (2.11) 2 Here [ ]s represents a regularized expression for the breaking operator, that we will specify in Section 3.1. The action of the time collective coordinate is given by Z N ⇥ ⇤ S[f ] = 0,f s . (2.12) 2 We have in [11] given the explicit evaluation of the nonlinear action S[f ] for the case of q = 2 demonstrating the Schwarzian form [11] conjectured by Kitaev and constructed at quadratic level by Maldacena and Stanford [10]. For general q, the naive form of the composite operator in (2.5) generates again a Schwarzian action, which we exhibited through an "-expansion presented in Appendix A. Taking into account the regularized breaking term we confirm the Schwarzian form (in Appendix B) Z " ✓ ◆ # N↵ f 000 (t) 3 f 00 (t) 2 S[f ] = dt , (2.13) 24⇡J f 0 (t) 2 f 0 (t) with a coefficent ↵ = 12⇡B1 , (2.14) where " # tan( ⇡q ) 2⇡(q 1)(q 2) (q) = (q 2 6q + 6) . (2.15) 12⇡bq q sin( 2⇡ q ) and B1 representing the coefficient of first order shift of the saddle-point solution which will be summarized in Section 3.1. All together our result for the prefactor of the Schwarzian action –6– comes out in agreement with the value obtained first by Maldacena and Stanford through evaluation of zero mode dynamics [10]. Summarizing in the above construction we have an interacting picture of the emergent Schwarzian mode f (t), and a bi-local matter field combined in the nonlinear collective action (2.11). It is important to emphasize that this action exhibits reparametrization symmetry both at and also away from the IR point. For this, the delta constraint condition projecting out the state associated with wave function u(t1 , t2 ) represents a gauge fixing condition with an corresponding Faddeev-Popov measure. This formulation then allows systematic perturbative calculations around the IR point. 2.2 Relation to Zero Mode Dynamics Before we proceed with our perturbative calculations it is worth comparing the above exact treatment of the reparametrization mode (2.13) with a linearized determination of the zero mode dynamics, as considered in [10]. We will be able to see that the latter follows from the former. Expanding the critical action around the critical saddle-point solution 0 , we have the quadratic kernel (which defines the propagator) and a sequence of higher vertices and so on. This expansion is schematically written as h p i Z Z 1 1 Sc 0 + 2/N ⌘ = N Sc [ 0 ] + ⌘·K·⌘ + p V(3) · ⌘ ⌘ ⌘ + · · · , (2.16) 2 N where the kernel is 2S c[ 0] K(t1 , t2 ; t3 , t4 ) = 0 (t1 , t2 ) 0 (t3 , t4 ) 1 1 q 2 = 0 (t1 , t3 ) 0 (t2 , t4 ) + (q 1)J 2 (t13 ) (t24 ) 0 (t1 , t2 ) , (2.17) with tij = ti tj . Then, the bi-local propagator D is determined as a solution of the following Green’s equation: Z dt3 dt4 K(t1 , t2 ; t3 , t4 ) D(t3 , t4 ; t5 , t6 ) = (t15 ) (t26 ) . (2.18) In order to inverse the kernel K in the Green’s equation (2.18) and determine the bi-local propagator, let us first consider an eigenvalue problem of the kernel K: Z dt3 dt4 K(t1 , t2 ; t3 , t4 ) un,t (t3 , t4 ) = kn,t un,t (t1 , t2 ) , (2.19) where n and t are labels to distinguish the eigenfunctions. The zero mode, whose eigenvalue is k0 = 0 is given by 0,f (t1 , t2 ) u0,t (t1 , t2 ) = . (2.20) f (t) f (t)=t –7– Now, we consider the zero mode quantum fluctuation around a shifted classical background Z (t1 , t2 ) = cl (t1 , t2 ) + dt0 "(t0 ) u0,t0 (t1 , t2 ) , (2.21) with cl = 0 + 1 where 1 is a first 1/J shift of the classical field from the critical point. Then, the quadratic action of " in the first order of the shift is given by expanding Sc [ cl + " · u0 ]. This quadratic action can be written in terms of the shift of the kernel K as Z Z N 0 0 S2 ["] = dtdt "(t) "(t ) dt1 dt2 dt3 dt4 u0,t (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) u0,t0 (t3 , t4 ) , (2.22) 4 where Z 3S c[ 0] K(t1 , t2 ; t3 , t4 ) = dt5 dt6 1 (t5 , t6 ) . (2.23) 0 (t1 , t2 ) 0 (t3 , t4 ) 0 (t5 , t6 ) Let us formally denote the t1 - t4 integrals in Eq.(2.22) by Z kt (t t0 ) = dt1 dt2 dt3 dt4 u0,t (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) u0,t0 (t3 , t4 ) , (2.24) because this is related to the eigenvalue shift due to K up to normalization. Then, we can write the quadratic action (2.22) as Z N S2 ["] = dt kt "2 (t) . (2.25) 4 We now give a formal proof that the quadratic action (2.25) is equivalent to the quadratic action of Eq.(2.13). This statement can be easily seen from the following identity: Z 3S [ c 0] dt1 dt2 dt3 dt4 u0,t (t1 , t2 ) u0,t0 (t3 , t4 ) 0 (t1 , t2 ) 0 (t3 , t4 ) 0 (t5 , t6 ) Z 2 0,f (t3 , t4 ) = dt3 dt4 K(t3 , t4 ; t5 , t6 ) . (2.26) f (t) f (t0 ) f (t)=t R This identity is derived as follows. In the zero mode equation K · u0 = 0, rewriting the kernel as derivatives of Sc as in the first line of Eq.(2.17), and taking a derivative of this equation respect to f (t0 ), one finds Z 3S 0,f (t1 , t2 ) c[ 0,f ] 0,f (t3 , t4 ) 0 = dt1 dt2 dt3 dt4 · · f (t) f (t)=t 0,f (t1 , t2 ) 0,f (t3 , t4 ) 0,f (t5 , t6 ) f (t0 ) f (t0 )=t0 Z 2S 2 c[ 0,f ] 0,f (t3 , t4 ) + dt3 dt4 · , (2.27) 0,f (t3 , t4 ) 0,f (t5 , t6 ) f (t) f (t0 ) f (t)=t where we used the zero mode expression (2.20). Since Sc is invariant under the reparametriza- tion, we can change the argument of Sc from 0,f to 0 . Then, we get the identity (2.26). –8– We note that at next cubic level, one will have disagreement and the zero mode dynamics will not give the Schwarzian derivative. This follows from the further identity: Z 2S [ 3 c 0] 0,f (t5 , t6 ) dt5 dt6 · 0 (t5 , t6 ) 0 (t7 , t8 ) f (t) f (t0 ) f (t00 ) f (t)=t Z 4S [ c 0] = dt1 dt2 dt3 dt4 dt5 dt6 u0,t (t1 , t2 ) u0,t0 (t3 , t4 ) u0,t00 (t5 , t6 ) 0 (t1 , t2 ) 0 (t3 , t4 ) 0 (t5 , t6 ) 0 (t7 , t8 ) Z 3S [ 2 c 0] 0,f (t3 , t4 ) 0,f (t5 , t6 ) 3 dt3 dt4 dt5 dt6 0 , 0 (t3 , t4 ) 0 (t5 , t6 ) 0 (t7 , t8 ) f (t) f (t ) f (t)=t f (t00 ) f (t)=t (2.28) where the second term in the right-hand side explains the expected discrepancy. 3 Shift of the Classical Solution In large N limit, the exact classical solution cl is given by the solution of the saddle-point equation of the collective action (2.5). This classical solution corresponds to the one-point function: ⌦ ↵ (t1 , t2 ) = cl (t1 , t2 ) . (3.1) At the strict strong coupling limit, the classical solution is given by the critical solution 0 , which is a solution of the saddle-point equation of the critical action (2.6). One can then develop a perturbative 1/J expansion for the full solution . 3.1 Evaluation of 1 Let us consider the first order shift 1 of the classical solution from the critical solution induced by the breaking term. We start with the naive delta function breaking term of the action Scol (2.5). Substitution of cl = 0 + 1 gives Z dt3 dt4 K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) = @1 (t12 ) , (3.2) where the kernel is given in Eq.(2.17). It is useful to separate the J dependence from the bi-local field by 2 cl (t1 , t2 ) = J q 0 (t1 , t2 ) + ··· , (3.3) so that the critical solution 0, now reads sgn(t12 ) 0 (t1 , t2 ) = b 2 , (3.4) |t12 | q with ⇣ ⌘ 2 3 1q tan ⇡ ✓ ◆ 4 q 2 5 b = 1 . (3.5) 2⇡ q –9– Now the kernel (2.17) does not have the explicit J 2 factor in the second term, and such rescaled kernel denoted by K will be used in the rest of the thesis. Since 1 sgn(t12 ) 0 (t1 , t2 ) = bq 1 2 , (3.6) 2 |t12 | q and the kernel has dimension K ⇠ |t| 4+4/q , from dimension analysis would need to be 1 the form of sgn(t12 ) 1 (t1 , t2 ) = A 4 , (3.7) |t12 | q where A is a t-independent coefficient. In checking this ansatz we have the following integral in the first term of the LHS of Eq.(3.2) Z sgn(t13 ) sgn(t24 ) sgn(t34 ) Ab2q 2 dt3 dt4 . (3.8) 2 2 2 2 4 |t13 | q |t24 | q |t34 | q This type of integral is already evaluated in Appendix A of [9]. In general, the result is Z  sgn(t13 ) sgn(t24 ) sgn(t34 ) 2 sin(2⇡↵) + 2 sin(2⇡(↵ + )) + sin(2⇡(↵ + 2 )) dt3 dt4 2 2 2↵ = ⇡ |t13 | |t24 | |t34 | sin(2⇡↵) sin(2⇡ ) sin(2⇡(↵ + )) sin(2⇡(↵ + 2 )) h i sin(2⇡ ) + sin(2⇡(↵ + )) (1 2 ) sgn(t ) 12 ⇥ 2↵+4 2 . (2↵) (2 ) (3 2↵ 4 ) |t12 | (3.9) Our interest is = 1 1/q. For this case, the result is inversely proportional to (4/q 2↵ 1). If we plug ↵ = 2/q into this equation, we can see that the Gamma function in the denominator gives infinity: (4/q 2↵ 1) = ( 1) = 1, while other part is finite. Therefore, the first term of the LHS of Eq.(3.2) vanishes. The second term is trivial to evaluate; however the resulting form does not agree with the naive -function source in RHS. Hence, we conclude that the 0 -source is only matched in the non-perturbative solution level, where all the 1/J corrections are summed over. To proceed, consider a more general ansatz for 1 : sgn(t12 ) 1 (t1 , t2 ) = B1 2 , (3.10) +2s |t12 | q where B1 is a t-independent coefficient. The parameter s has to be s > 0, because the dimension of 1 needs to be less than the scaling dimension of 0 . Now using this ansatz, we are going to evaluate Eq.(3.2). The integral of the first term of LHS of Eq.(3.2) is evaluated from Eq.(3.9) with = 1 1/q and ↵ = s + 1/q as ⇣ ⌘ ⇣ ⌘ B1 b2q 2 ⇡ 2 cot ⇡q 2 q 1 sgn(t12 ) ⇣ ⇣ ⌘⌘ ⇣ ⇣ ⌘⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ . (3.11) 2 2q +2s sin ⇡ 1q + s cos ⇡ s 1q 2 q + 2s 2 2 q 2 q 2s 1 |t 12 | – 10 – Hence, after a slight manipulation the LHS of Eq.(3.2) becomes Z sgn(t12 ) dt3 dt4 K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) = (q 1)B1 bq 2 (s, q) , (3.12) 2 2 +2s |t12 | q where we used Eq.(3.5) and we defined ⇣ ⌘ 2 ⇡ q (s, q) = 1 ⇣ ⇣ ⌘⌘ ⇣ ⇣ ⌘⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘. q sin ⇡ 1q + s cos ⇡ s 1 q 2 q + 2s 3 2 q 2 q 2s 1 (3.13) Now we note that for s = 1/2, (1/2, q) = 0, so that the ansatz (3.10) would be the homoge- neous equation associated with Eq.(3.2). This limit s ! 1/2 therefore leads to the following first order shift of the background: 2 h i 1 cl (t1 , t2 ) = J q 0 (t1 , t2 ) + J 1 (t1 , t2 ) + · · · , (3.14) with sgn(t12 ) sgn(t12 ) 0 (t1 , t2 ) = b 2 , 1 (t1 , t2 ) = B1 2 . (3.15) +1 |t12 | q |t12 | q We will however keep the parameter s infinitesimally away from 1/2 as a regularization. Then, 6q ⇣ ⌘ 1 1 2 (s, q) = s 2 + O (s 2 ) , (3.16) (q 1) bq 1 where is defined in Eq.(2.15), and the RHS in Eq.(3.12) can be interpreted as a regularized non-zero source term of the form 1 1 sgn(t12 ) 1 2 Qs (t1 , t2 ) ⌘ (s 2 ) 6qB1 b 2 + O (s 2) . (3.17) 2 +2s |t12 | q The is obtained by expanding (s, q) (3.13) around s = 1/2 so that (q 1)bq 1 0 = (s = 12 , q) . (3.18) 6q Here, the prime denotes a derivative respect to s. We use this regularized source to define the regularized breaking term by Z Z ⇥ ⇤ f s ⌘ lim dt1 dt2 f (t1 , t2 ) Qs (t1 , t2 ) . (3.19) s! 12 Finally, the coefficient B1 can be deduced from the numerical result found in [10]. Com- parison of the two results gives the relation: B1 ↵G = , (3.20) bJ J – 11 – with the numerical approximated value of ↵G established in [10] 2(q 2) ↵G ⇡ , (3.21) 16/⇡ + 6.18(q 2) + (q 2)2 p q and J = q 1 J. 2 2 3.2 Evaluation of 2 Now we would like to go further higher order term in the expansion of the classical solution. This term is given by 2 h i 1 2 (t , cl 1 2 t ) = J q (t , 0 1 2 t ) + J (t , 1 1 2 t ) + J (t , 2 1 2 t ) + · · · , (3.22) with sgn(t12 ) 2 (t1 , t2 ) = B2 2 , (3.23) +2 |t12 | q where B2 is a t-independent coefficient. The dimension of 2 is already fixed by 1 , so what we need to do is just to fix the coefficient B2 . Substituting the above expansion of the classical field into the critical action Sc (2.6) and expanding it, one finds that the equation determining 2 is given by Z dt3 dt4 K(t1 , t2 ; t3 , t4 ) 2 (t3 , t4 ) 1 1 1)(q 2) q 3 1 2 (q = [ 0 ? 1 ? 0 ? 1 ? 0 ](t1 , t2 ) 0 (t1 , t2 ) 1 (t1 , t2 ) , (3.24) 2 R where the star product is defined by [A ? B](t1 , t2 ) ⌘ dt3 A(t1 , t3 )B(t3 , t2 ). Now, we are going to evaluate each term of this equation. For the first term in the LHS is again given by Eq.(3.9) with = 1 1/q and ↵ = 1/q + 1 as q(q 1)(3q 2) sgn(t12 ) (LHS 1st) = 2⇡ B2 b2q 2 . (3.25) (q 2 4) tan( ⇡q ) |t |4 2q 12 For the first term of the RHS, we need to use Eq.(3.9) twice. First for the middle of the term: 1 1 1 ? 0 ? 1 , and then for the result sandwiched by the remaining 0 ’s. Then, we have 2⇡ 2 q 2 (q 1)(3q 2) sgn(t12 ) (RHS 1st) = B12 b3(q 1) 2 . (3.26) (q 2)2 |t12 | 4 q The second terms in the LHS and RHS are trivially evaluated. Therefore, now one can see that all terms have the same t12 dependence. Then, comparing their coefficients, we finally fix B2 as ✓ ◆ ✓ ◆ B12 q + 2 2 ⇡ B2 = (q 2) + (3q 2) tan . (3.27) b 8q q – 12 – 3.3 All Order Evaluation in q > 2 In this subsection, we extend our previous perturbative expansion of the classical solution to all order contributions in the 1/J expansion. Because of the dimension of 1 (3.15), the time-dependence is already fixed for all order as in Eq.(3.32). Therefore, we only need to determine the coefficient Bn , and in this subsection we will give a recursion relation which fixes the coefficients. However, we will not use this subsection’s result in the rest of the thesis, so readers who are interested only in the first few terms in the 1/J expansion (3.22) may skip this subsection and move on to Section 4. As we saw in Section 3.1, the structure of the classical solution in q = 2 model is di↵erent from q > 2 case. In this subsection, we focus on q > 2 case. We generalize the expansion (3.22) to all order by 1 X 2 m cl (t1 , t2 ) = J J m (t1 , t2 ) . (3.28) q m=0 Now, we substitute this expansion into the critical action Sc (2.6). As we saw before, the kinetic term does not contribute to the perturbative analysis when q > 2; therefore, we discard the kinetic term here. The contribution of the kinetic term will be recovered in the full classical solution with correct UV boundary conditions. Hence, the saddle-point equation is now formally written as " 1 # 1 " 1 #q 1 X X m m 0 = J m (t1 , t2 ) + J m (t1 , t2 ) . (3.29) m=0 m=0 Using the multinomial theorem, each term can be reduced to polynomials of m ’s. Substitut- ing these results into Eq.(3.29) leads the saddle-point equation written in terms polynomials with all order of 1/J expansion. From this equation, one can further pick up order O(J n ) terms. For n = 0, it is the equation of 0 . Therefore, we consider n 1 case, which is given by X  ⇣ ⌘ k1 ⇣ ⌘ k2 k1 +k2 +··· (k1 + k2 + · · · )! 1 1 1 0 = ( 1) ⇥ 0 ? 1 ? 0 ? 2 ? 0 ? · · · (t1 , t2 ) k1 !k2 !k3 ! · · · k1 +2k2 +···=n X (q 1)! k0 k1 k2 + ⇥ 0 (t1 , t2 ) 1 (t1 , t2 ) 2 (t1 , t2 ) · · · , (3.30) k0 !k1 !k2 ! · · · k1 +2k2 +···=n with k0 = q (1 + k1 + · · · + kn 1 ). Let us consider this order O(J n ) equation more. Because of the constraint k1 + 2k2 + · · · = n, we know that kn+1 = kn+2 = · · · = 0. Also the same constraint implies that kn = 0 or 1, and when kn = 1, then k1 = k2 = · · · = kn 1 = 0. Therefore, it is useful to separate kn = 1 terms from kn = 0 ones. After this separation, the – 13 – order O(J n ) equation is reduced to a more familiar form: Z dt3 dt4 K(t1 , t2 ; t3 , t4 ) n (t3 , t4 ) X (k1 + · · · + kn 1 )! = ( 1)k1 +···+kn 1 k1 ! · · · kn 1 ! k1 +2k2 +···+(n 1)kn 1 =n  ⇣ ⌘ k1 ⇣ ⌘ kn 1 1 1 1 ⇥ 0 ? 1 ? 0 ? ··· ? n 1 ? 0 (t1 , t2 ) X (q 1)! kn 1 k0 k1 ⇥ 0 (t1 , t2 ) 1 (t1 , t2 ) ··· n 1 (t1 , t2 ) , k0 !k1 ! · · · kn 1! k1 +2k2 +···+(n 1)kn 1 =n (3.31) where k0 = q (1 + k1 + · · · + kn 1 ). This is the equation which determines n from { 0 , 1 , · · · , n 1 } sources. However, we already know the t12 dependence of n (t1 , t2 ). Namely, sgn(t12 ) n (t1 , t2 ) = Bn 2 . (3.32) +n |t12 | q Therefore, we only need to determine the coefficient Bn . Probably it is hard to evaluate the star products in the RHS of Eq.(3.31) by direct integrations of t’s, and it is better to use momentum space representations. Z d! i!t12 m (t1 , t2 ) = Bm e m (!) , (3.33) 2⇡ where we excluded the coefficient Bm from m (!) for later convenience, and m (!) = 2 +m 1 Cm |!| q sgn(!), with 1 m 1 m 2 p (1 q 2) Cm ⌘ i 2 q ⇡ . (3.34) ( 1q + m 2 + 12 ) With this definition of Cm , we can write the inverse of the critical solution as Z Z 1 d! i!t12 1 q 1 d! i!t12 1 2q 0 (t , 1 2t ) = e 0 (!) = b C 4 2 q e |!| sgn(!) . (3.35) 2⇡ 2⇡ Now, we can evaluate each term in Eq.(3.31) using these Fourier transforms. Then, every term has the same ! integral; therefore, comparing the coefficients, one obtains h i bq 2 (q 1) C2+n 4 bq C22 4 Cn Bn q q X (k1 + · · · + kn 1 )! = ( 1)k1 +···+kn 1 k1 ! · · · kn 1 ! k1 +2k2 +···+(n 1)kn 1 =n ⇣ ⌘k1 +···+kn 1 +1 ⇣ ⌘k1 ⇣ ⌘ kn 1 ⇥ bq 1 C 2 4 B1 C 1 · · · Bn 1 Cn 1 q X (q 1)! k ⇥ bk0 B1k1 · · · Bnn 11 C2+n 4 , (3.36) k0 !k1 ! · · · kn 1! q k1 +2k2 +···+(n 1)kn 1 =n – 14 – with k0 = q (1 + k1 + · · · + kn 1 ). This is the recursion relation which determines Bn from {B1 , B2 , · · · , Bn 1 }. Note that Cm ’s are a priori known numbers as defined in Eq.(3.34). 4 Finite Temperature Up to here, we have been considering only zero-temperature solutions in the SYK model. In this section, we will consider the finite-temperature solutions 1, and 2, and the tree-level free energy in the low temperature region. 4.1 Classical Solutions As we saw in Section 3, the 1/J expansion of the classical solution in the strongly coupling region is given by 2 h i 1 2 cl (t1 , t2 ) = J q 0 (t1 , t2 ) + J 1 (t1 , t2 ) + J 2 (t1 , t2 ) + · · · , (4.1) where sgn(t12 ) sgn(t12 ) sgn(t12 ) 0 (t1 , t2 ) = b 2 , 1 (t1 , t2 ) = B1 2 , 2 (t1 , t2 ) = B2 2 . (4.2) +1 +2 |t12 | q |t12 | q |t12 | q In order to evaluate tree-level free energy, we first need finite-temperature versions of these classical solutions. 0 is the solution of the strict strong coupling limit, where the model exhibits an emergent conformal reparametrization symmetry: t ! f (t) with the 0 trans- formation (2.7). Therefore, to obtain the finite-temperature version of 0 , we just need to use f (t) = ⇡ tan( ⇡t ) with the above transformation [7]. This map maps the infinitely long zero-temperature time to periodic thermal circle. Thus, this gives us " #2 q ⇡ 0, (t1 , t2 ) = b sgn(t12 ) . (4.3) sin( ⇡t12 ) Since 1 and 2 are the shifts of the classical solution from the strict IR limit, they do not enjoy the reparametrization symmetry. Therefore, we cannot use the above method to get their finite-temperature counterparts. However, we can approximate finite-temperature solutions by mapping the zero-temperature solutions onto a thermal circle and summing over all image charges: X1 (t12 ) = ( 1)m =1 (t12 + m) . (4.4) m= 1 In this approximation, the finite-temperature solutions (two-point function in terms of the fundamental fermions) trivially satisfy the KMS condition. This approximation also works order by order in the 1/J expansion. Therefore, after separating positive m and negative m and changing the labeling, one finds " 1 1 # X ( 1)m X ( 1)m 1, (t12 ) = B1 2 2 . (4.5) +1 +1 m=0 ( m + t12 ) q m=1 ( m t12 ) q – 15 – The summations of m can be evaluated to give the Hurwitz zeta functions. In the same way, we can approximate 2 in terms the Hurwitz zeta functions. In [10], Maldacena and Stanford obtained a first order shift of the classical solution in finite-temperature through a numerical solution of the exact Schwinger-Dyson equation. The estimation of 1, by the above “image charge” method can be seen to agree well with the numerical ansatz. The solution of [10] is shown in their Eq.(3.122) reading: 2⇡|t12 | G(t1 , t2 ) ↵G ⇡ = f0 (t12 ) , f0 (t12 ) = 2 + . (4.6) Gc (t1 , t2 ) J tan | ⇡t12 | with the notation, Gc = 0, and G = 1, . We can see in Figure 1 that our approximated solution for 1, is pretty close to this solution. It is more convenient to introduce a new variable ✓ ◆ |t12 | 1 1 1 y ⌘ . y (4.7) 2 2 2 Then, we have f0 (y) = 2 + 2⇡y tan(⇡y). On the other hand for the figure, we rewrite our approximated solution by  ⇣ ⌘ ⇣ ⌘ 1, (t12 ) B1 2 1 2 3 = 2 ⇣ q + 1, 4 ⇣ q + 1, 4 ⇥ F0 (y, q) , (4.8) 0, (t12 ) 2(2⇡) q b where 2 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘3 2 1 y 2 1 y 2 y y 2 ⇣ q + 1, 4 + 2 + ⇣ q + 1, 4 2 ⇣ + 1, 34 + q 2 ⇣ 2 q + 1, 34 2 F0 (y, q) ⌘ (cos ⇡y) q 4 ⇣ ⌘ ⇣ ⌘ 5. ⇣ 2q + 1, 14 ⇣ 2q + 1, 34 (4.9) Here, we adjusted the normalization of F0 so that F0 (y = 0, q) = 2 = f0 (y = 0). In Figure 1, we plotted f0 (y) and F0 (y, q) with q = 2, 4, 1000. We can see that for any value of q, F0 is pretty close to f0 in all range of y. We will now develop a small temperature expansion which will give further useful infor- mation about the finite temperature solution and also the free energy. For this one expands the equation iteratively starting from 0, as sources. We develop this method for 1, in the rest of this subsection. The expansion of 0, solution (4.3) in the small temperature region is given by " # sgn(t12 ) ⇡ 2 t12 2 (q + 5)⇡ 4 t12 4 0, (t12 ) = b 2 1+ + + ··· . (4.10) |t | q 3q 90q 2 12 We then expand the finite-temperature solution 1, by " # 2 3 sgn(t12 ) t12 t12 t12 1, (t1 , t2 ) = B1 2 1 + c1,1 + c1,2 + c1,3 + ··· , (4.11) +1 |t12 | q – 16 – 1 Figure 1. f0 (y) and F0 (y, q) with q = 2, 4, 1000 in the range of 2  y  12 . and then, using the equation of motion for 1 (3.2) we iteratively determine the coefficients c1,i starting from the lower order ones. As we will see in the next subsection, to evaluate its free energy contribution, we need a1 ⌘ c1,3 . First we consider O( 1 ) order. The equation in this order reads Z sgn(t34 ) B1 c1,1 1 dt3 dt4 K(t1 , t2 ; t3 , t4 ) 2 = 0, (4.12) |t34 | q where K denotes the zero temperature kernel. Using the formula in Eq.(3.9), one can evaluate the left-hand side integrals. In general, the integral does not vanish. Therefore, to satisfy the equation, we need c1,1 = 0. Next for O( 2 ) order, we have an equation Z sgn(t34 ) B1 c1,2 2 dt3 dt4 K(t1 , t2 ; t3 , t4 ) 2 1 |t34 | q Z " ! # ⇡ 2 (q 1)B1 bq 2 q sgn(t13 )sgn(t24 ) sgn(t13 )sgn(t24 ) (t13 ) (t24 ) sgn(t34 ) = dt3 dt4 b 2 + + (q 2) . 3q 2 2 2 2 2 2 4 2 +1 |t13 | q |t24 | q |t13 | q |t24 | q |t12 | q |t34 | q (4.13) Again one can evaluate the integrals and find c1,2 = (q 1)⇡ 2 /3q. Finally we consider O( 3 ) order. The equation of this order reads Z 3 sgn(t34 ) B1 c1,3 dt3 dt4 K(t1 , t2 ; t3 , t4 ) 2 = 0. (4.14) 2 |t34 | q The LHS integral identically vanishes. Hence, we cannot determine the coefficient c1,3 from this equation. Nevertheless, this iterative method precisely recovers the expansion of (4.6) up to the third order: " # sgn(t12 ) (q 1)⇡ 2 t12 2 2⇡ 2 t12 3 (2q 1)(q + 5)⇡ 4 t12 4 G(t1 , t2 ) = B1 2 1 + + ··· . |t | q +1 3q 3 90q 2 12 (4.15) – 17 – where we used the relation (3.20). Using this 1, expansion as source together with 0, , we can also apply this method to determine low temperature expansion of 2, . 4.2 Tree-Level Free Energy Now we evaluate the tree-level free energy through the regularized breaking term. The order ( J)0 contribution to the tree-level free energy, which comes from Sc [ 0, ], was already evaluated in [7, 10, 86]. Therefore in this section, we will evaluate higher order contributions of the 1/ J expansion to the tree-level free energy. The action of the collective time coordinate was evaluated in Appendix B from the regularized breaking term, which leads to the Schwarzian action given in Eq.(2.13). Now, we use the classical solution: f (t) = ⇡ tan( ⇡t ). Then, the integral can be evaluated to give 2⇡ 2 / . Therefore, the S[f ] contribution to the tree-level free energy is N B1 ⇡ 2 F0 = . (4.16) J This contribution can actually be evaluated directly from the regularized breaking term by Z N F0 = lim dt1 dt2 0, (t1 , t2 ) Qs (t1 , t2 ) , (4.17) 2 s! 12 where the finite temperature critical solution 0, and the regularized source Qs are given in Eq.(4.3) and (3.17), respectively. Since the regularized source Qs has a factor (s 1/2), in order to obtain non-vanishing contribution after the limit, we only need to extract a single pole (s 1/2) 1 term from the integral. For this purpose, we expand the finite temperature 2 2 critical solution 0, by power series of |t12 | up to |t| q order, which is responsible for a single pole term. This leads to  Z N B1 ⇡ 2 dt F0 = (q 1)bq 1 (s, q) . (4.18) 6q J |t|2s s! 1 2 Hence, using the expansion of (s, q) in Eq.(3.16) and taking the limit s ! 1/2, we obtain the final result. This result agrees with the result found in Eq.(4.16) from the Schwarzian action. Now we consider the next ( J) 2 order contribution. The contribution from the breaking term to such order is given by Z N F1 = lim dt1 dt2 1, (t1 , t2 ) Qs (t1 , t2 ) . (4.19) 2 s! 12 2 2 Again to compute this free energy, we only need to extract the |t| q order term from 1, . 2 2 From the expansion in Eq.(4.15), one can read o↵ the |t| q order term as 2B1 ⇡ 2 sgn(t12 ) 1, (t1 , t2 ) = 2 2 + ··· . (4.20) 3 3 J 1+ q |t12 | q 2 – 18 – Following the same process as in the previous subsection, one can evaluate the contribution from the breaking term to this order free energy. However, this is not the all contributions to this ( J) 2 order free energy. The critical action Sc part also gives a contribution to this ( J) 2 order, which is half of the breaking term contribution with opposite sign. Therefore, combining these two contributions, the final answer for the ( J) 2 order free ernrgy is given by N B12 F1 = ⇡ 2 q . (4.21) b( J)2 In the following, we discuss the general ( J) n order contribution of the tree-level free energy. For this purpose, let us first look at the collective action (2.11). After rescaling the bi- local field by ! J 2/q , one sees the explicit J-dependence appearing only in the breaking term. Hence, from the J-derivative trick, the tree-level free energy is solely determined by the breaking term by Z @ N J Fn = lim dt1 dt2 n, (t1 , t2 )Qs (t1 , t2 ) . (4.22) @J 2J s! 12 We know that any order of 1/J correction for the zero temperature classical solution is given by Eq.(3.32). Even though we don’t know exact finite-temperature version of these corrections, we nevertheless expect the finite-temperature solution can be expanded in low temperature region as " # Bn sgn(t12 ) t12 n+2 n, (t1 , t2 ) = 1 + · · · + an + ··· , (4.23) J n |t | 2q +n 12 where an is a q-dependent constant, but independent of J, or t. As we saw in the previous 2 2 sections, the |t| q order term is only needed to extract the (s 1/2) 1 poles. Hence, sub- stituting this order term into Eq.(4.22), one can perform the integrals and the limit together with Eq.(3.17). This result is given by @ 3qan N B1 Bn J Fn = . (4.24) @J b ( J)n+1 After the integration of J, the free energy is given by 3qan N B1 Bn Fn = . (4.25) (n + 1)b ( J)n+1 We can check the consistency of this formula with previous results. For n = 0, we have B0 = b and a0 = ⇡ 2 /(3q). Then the formula gives the result we found above. For n = 1, we have a1 = 2⇡ 2 /3, and then the formula again leads to the result found above. For general order we only need to determine an to evaluate the free energy. We note that in principle the coefficient of the zero temperature solution Bn can be determined from the recursion relation (3.36). – 19 – In summary, we have obtained the following 1/J corrections to the tree-level free energy ✓ ◆ F B1 ⇡ 2 B12 ⇡ 2 q 3qa2 B13 q+2 h ⇣ ⌘i 2 ⇡ = (q 2) + (3q 2) tan q + · · · . (4.26) N J b( J)2 b2 ( J)3 8q For q = 4, we can compute the coefficients as F 1 2 3 = 0.197 ( J) + 0.208 ( J) + 0.038 ⇥ a2 ( J) + ··· , (4.27) N with a2 to be determined. These results agree with the recent numerical results of [27, 28]. 5 Bi-local Propagator and Spectrum In this section, we consider the bi-local two-point function: D E (t1 , t2 ) (t3 , t4 ) , (5.1) where the expectation value is evaluated by the path integral (2.4). After the Faddeev-Popov procedure and changing the integration variable as we discussed in Section 1, this two-point function becomes D E f (t1 , t2 ) f (t3 , t4 ) , (5.2) where now the expectation value is evaluated by the gauged path integral (2.10). Now, we expand the bi-local field around a classical (large N ) background solution cl . Namely, r 2 (t1 , t2 ) = cl (t1 , t2 ) + ⌘(t1 , t2 ) , (5.3) N where ⌘ are quantum fluctuations, but the zero mode is eliminated from its Hilbert space. We will discuss the zero mode contribution in the following subsection. Therefore, the two-point function is now decomposed as D E D E 2 D E f (t1 , t2 ) f (t3 , t4 ) = cl,f (t1 , t2 ) cl,f (t3 , t4 ) + ⌘(t1 , t2 )⌘(t3 , t4 ) . (5.4) N The second term in the RHS is the bi-local propagator D determined by Eq.(2.18), which was evaluated in [11] for q = 4 (and also in [9, 10]) as 1 Z 0 8 X e i!(t+ t+ ) p2m D(t1 , t2 ; t3 , t4 ) = sgn(t t0 ) p d! J ⇡ m=1 sin(⇡pm ) p2m + (3/2)2 " # pm + 32 0 ⇥ J pm (|!t |) + 3 Jpm (|!t |) Jpm (|!t |) , pm 2 (5.5) where pm are the solutions of 2pm /3 = tan(⇡pm /2), and t± = (t1 ±t2 )/2 and t0± = (t3 ±t4 )/2. – 20 – Let us now describe the derivation of this bi-local propagator D with q = 4 for notational simplicity, but everything can be generalized to any q by small modifications. Fluctuations around the critical IR background can be studied by expanding the bi-local field as in Eq.(5.3) with cl (t1 , t2 ) = 0 (t1 , t2 ) , (5.6) where the critical IR background solution is given by ✓ ◆1 1 4 sgn(t12 ) 0 (t1 , t2 ) = p . (5.7) 4⇡J 2 |t12 | At the quadratic level, we have the quadratic kernel K. The diagonalization of this quadratic kernel is done by the eigenfunction u⌫,! and the eigenvalue ge(⌫) as Z dt01 dt02 K(t1 , t2 ; t01 , t02 ) u⌫,! (t01 , t02 ) = ge(⌫) u⌫,! (t1 , t2 ) . (5.8) The quadratic kernel K is in fact a function of the bi-local SL(2, R) Casimir ˆ1 + D ˆ2 2 1 ˆ 1 ˆ C1+2 = D P1 + Pˆ2 K ˆ1 + K ˆ2 ˆ 2 Pˆ1 + Pˆ2 K1 + K 2 2 = (t1 t2 ) 2 @ 1 @ 2 , (5.9) with the SL(2, R) generators D ˆ = t@t , Pˆ = @t , and Kˆ = t2 @t . The common eigenfunctions of the bi-local SL(2, R) Casimir (5.9) are, due to the properties of the conformal block, given by the three-point function of the form D E sgn(t12 ) Oh (t0 ) O (t1 ) O (t2 ) = . (5.10) |t10 |h |t h 20 | |t12 | 2 h Since the SYK quadratic kernel K is a function of this bi-local SL(2, R) Casimir, this three- point function is also the eigenfunction of the SYK quadratic kernel. For the investigation of dual gravity theory, it is more useful to Fourier transform from t0 to ! by D E Z D E f Oh (!) O (t1 ) O (t2 ) ⌘ dt0 ei!t0 Oh (t0 ) O (t1 ) O (t2 ) p sgn(t12 ) t1 +t2 = ⇡ cot(⇡⌫) ( 12 ⌫) |!|⌫ 1 ei!( 2 ) Z⌫ (| !t212 |) , 2 |t12 | 2 (5.11) where we used h = ⌫ + 1/2 and defined tan(⇡⌫/2) + 1 Z⌫ (x) = J⌫ (x) + ⇠⌫ J ⌫ (x) , ⇠⌫ = . (5.12) tan(⇡⌫/2) 1 The t0 integral in the Fourier transform can be performed by decomposing the integration region into three pieces. The complete set of ⌫ can be understood from the representation – 21 – theory of the conformal group, as discussed recently in [132]. We have the discrete modes ⌫ = 2n + 3/2 with (n = 0, 1, 2, · · · ) and the continuous modes ⌫ = ir with (0 < r < 1). Adjusting the normalization, we define our eigenfunctions by 1 u⌫,! (t, z) ⌘ sgn(z) z 2 ei!t Z⌫ (|!z|) , (5.13) which have normalization condition Z 1 Z dt 1 dz ⇤ u (t, z) u⌫ 0 ,!0 (t, z) = N⌫ (⌫ ⌫ 0 ) (! !0) , (5.14) 1 2⇡ 0 z 2 ⌫,! with ( (2⌫) 1 for ⌫ = 3/2 + 2n N⌫ = (5.15) 2⌫ 1 sin ⇡⌫ for ⌫ = ir . Here we used the change of the coordinates by t1 + t2 t1 t2 t ⌘ , z ⌘ , (5.16) 2 2 and then the bi-local field ⌘(t1 , t2 ) ⌘(t1 , t2 ) ⌘ (t, z) , (5.17) can be then considered as a field in two dimensions (t, z). Expand the fluctuation field as X (t, z) = ˜ ⌫,! u⌫,! (t, z) , (5.18) ⌫,! the quadratic action can be written as 3J X S(2) = p N⌫ ˜ ⌫,! g˜(⌫) 1 ˜ ⌫,! , (5.19) 32 ⇡ ⌫,! where the kernel is given by ⇣ ⇡⌫ ⌘ 2⌫ g˜(⌫) = cot . (5.20) 3 2 Now this leads to the bi-local propagator p Z Z 0 0 16 ⇡ 1 dr u⇤ir,w (t, z) uir,w (t0 , z 0 ) D(t, z; t , z ) = dw 3J 0 Nir ge(ir) 1 p X 1 Z 16 ⇡ 1 u (t, z) u⌫n w (t0 , z 0 ) ⇤ + dw ⌫n w . (5.21) 3J N ⌫n ge(⌫n ) 1 ⌫n =2n+ 32 n=0 The r-integral is evaluated as explained in Appendix E which picks up poles determined by solutions of g˜(⌫) = 1, they represent a sequence denoted by pm as 2pm ⇣ ⇡p ⌘ m = tan , 2m + 1 < pm < 2m + 2 (m = 0, 1, 2, · · · ) (5.22) 3 2 – 22 – Therefore, the bi-local propagator is written as residues of ⌫ = pm poles as 3 Z 1 1 X > )J <) 32⇡ 2 i!(t t0 ) Z pm (|!|z pm (|!|z D(t, z; t0 , z 0 ) = d! e R(pm ) , (5.23) 3J 1 Np m m=1 where z > (z < ) is the greater (smaller) number among z and z 0 . The residue function is defined by ✓ ◆ 1 3p2m R(pm ) ⌘ Res = 2 . (5.24) g˜(⌫) 1 ⌫=pm [pm + (3/2)2 ][⇡pm sin(⇡pm )] Since that pm are zeros of ge(⌫) 1, near each pole pm , we can approximate as ⇥ ⇤ ge(⌫) 1 ⇡ ⌫2 (pm )2 fm , (5.25) where fm can be determined from residue of 1/(e g (⌫) 1) at ⌫ = pm . Explicitly evaluating these residues, the inverse kernel is written as an exact expansion 1 X ✓ ◆ 1 6 p3m 1 = . (5.26) g˜(⌫) 1 [p2m + (3/2)2 ][⇡pm sin(⇡pm )] ⌫2 p2m m=1 2 = p2 1 The e↵ective action near a pole labelled by m is that of a scalar field with mass, Mm m 4, (m > 0) in AdS2 : Z  ✓ ◆ e↵ 1 p 2 µ⌫ 2 1 2 Sm = g d x g @µ m @⌫ m pm m , (5.27) 2 4 where the metric gµ⌫ is given by gµ⌫ = diag( 1/z 2 , 1/z 2 ). It is clear from the above analysis that a spectrum of a sequence of 2D scalars, with growing conformal dimensions is being packed into a single bi-local field. In other words the bi-local representation e↵ectively packs an infinite product of AdS Laplacians with growing masses. It is this feature which leads to the suggestion that the theory should be represented by an enlarged number of fields, or equivalently by an extra Kaluza-Klein dimension, which we will explain in the next section. 5.1 Zero Mode Contribution In the above discussion, we have excluded the zero mode (m = 0) contribution which corre- sponds to the pole p0 = 3/2. If we had this mode in Eq.(5.23), indeed Z p0 = Z 3/2 leads to a divergence of the propagator because of ⇠ 3/2 = 1. This divergence can be treated by shifting the classical solution slightly away from the critical IR fixed point as first discussed by [10] in a 1/J expansion. Therefore, for the first term in the RHS of Eq.(5.4), expanding the classical field up to the second order 2 1 cl (t1 , t2 ) = 0 (t1 , t2 ) + 1 (t1 , t2 ) , (5.28) J 2 Here we have rescaled the entire field by J 2/q to separate out all J dependencies from . – 23 – one has D E cl,f (t1 , t2 ) cl,f (t3 , t4 ) D E  ✓ ◆ 1 D E t1 $ t 3 = 0,f (t1 , t2 ) 0,f (t3 , t4 ) + 0,f (t1 , t2 ) 1,f (t3 , t4 ) + + · · · , (5.29) J t2 $ t 4 where 1 0 0 q 0,f (t1 , t2 ) = f (t1 )f (t2 ) 0 (f (t1 ), f (t2 )) , 1 q + 12 0 0 1,f (t1 , t2 ) = f (t1 )f (t2 ) 1 (f (t1 ), f (t2 )) . (5.30) Now, we consider an infinitesimal reparametrization f (t) = t + "(t). Then, the classical fields are expanded as Z 0,f (t1 , t2 ) = 0 (t1 , t2 ) + dt "(t) u0,t (t1 , t2 ) + · · · , Z 1,f (t1 , t2 ) = 1 (t1 , t2 ) + dt "(t) u1,t (t1 , t2 ) + · · · , (5.31) where @ 0,f (t1 , t2 ) @ 1,f (t1 , t2 ) u0,t (t1 , t2 ) ⌘ , u1,t (t1 , t2 ) ⌘ . (5.32) @f (t) f (t)=t @f (t) f (t)=t Therefore, in the quadratic order of ", the classical field two-point function is now written in term of the two-point function of ". For later convenience, it is better to write down this as momentum space integral as D E cl,f (t1 , t2 ) cl,f (t3 , t4 ) Z  ✓ ✓ ◆◆ d! ⇤ 1 ⇤ t 1 $ t3 = h"(!)"( !)i u0,! (t1 , t2 )u0,! (t3 , t4 ) + u0,! (t1 , t2 )u1,! (t3 , t4 ) + + ··· . 2⇡ J t 2 $ t4 (5.33) Let us first evaluate the " two-point function. The collective coordinate action is given in Eq.(2.13). Expanding f (t) = t + "(t), the quadratic action of " can be obtained from this action. Hence, the two-point function in momentum space is 24⇡J 1 h"(!)"( !)i = . (5.34) ↵N ! 4 One can also Fourier transform back to the time representation to get 2⇡J h"(t1 )"(t2 )i = |t12 |3 . (5.35) ↵N – 24 – Next, we evaluate u0 and u1 . Taking the derivative respect to f (t), one obtains  ✓ ◆ 1 0 0 (t1 t) (t2 t) u0,t (t1 , t2 ) = (t1 t) + (t2 t) 2 0 (t1 , t2 ) , q t1 t2  ✓ ◆ 2+q 0 (t1 t) (t2 t) u1,t (t1 , t2 ) = (t1 t) + 0 (t2 t) 2 1 (t1 , t2 ) 2q t1 t2 (2 + q)B1 u0,t (t1 , t2 ) = . (5.36) 2b |t12 | After some manipulation, one can show that the momentum space expressions are given by p 3 ib ⇡ |!| 2 sgn(!t ) i!t+ u0,! (t1 , t2 ) = 2 1 e J 3 (|!t |) , q |2 t | q 2 2 (2 + q)B1 u0,! (t1 , t2 ) u1,! (t1 , t2 ) = . (5.37) 4b |t | Using the two-point function of " and above u0 and u1 expressions, finally the two-point function (5.4) up to order J 0 is given by D E  ✓ ◆ Z 12 (2 + q)B1 1 1 d! ⇤ f (t1 , t2 ) f (t3 , t4 ) = J + + u (t1 , t2 )u0,! (t3 , t4 ) ↵N 4b |t | |t0 | ! 4 0,! + D(t1 , t2 ; t3 , t4 ) . (5.38) What we have established therefore is the following. What one has is first the leading “classical” contribution to the bi-local two-point function which usually factorizes, due to the dynamics of the reparametrization symmetry mode. It now represents the leading “big” contribution, as in [10], and a sub-leading one. 6 3D Interpretation In this section, we describe the spectrum of matter fields predicted by the q = 4 SYK bi-local propagator (5.22) can be understood as a Kaluza-Klein of a single scalar field in 3-dimensional space-time. A generalization to arbitrary even integer q is also constructed in [107], but here we restrict ourselves to q = 4 case just for notational simplicity. According to [87, 88], the bulk dual of the SYK model involves Jackiw-Teitelboim theory of two dimensional dilaton gravity, whose action is given by (up to usual boundary terms) Z 1 p h i SJT = g (R + 2) 2 0 , (6.1) 16⇡G where 0 is a constant, and is a dilaton field. The zero temperature background is given by AdS2 with a metric dt2 + dz 2 ds2 = , (6.2) z2 – 25 – and a dilaton a (z) = + ··· , 0 + (6.3) z where a is a parameter which scales as 1/J and the ellipsis denotes higher order corrections. In the following we will choose, without loss of generality, 0 = 1. This action can be thought as arising from a higher dimensional system which has ex- tremal black holes, and the AdS2 is the near horizon geometry [87]. The three dimensional metric, with the dilaton being the third direction, is given by 1⇥ ⇤ ⇣ a ⌘2 2 ds2 = 2 dt2 + dz 2 + 1 + dy . (6.4) z z This is in fact the near-horizon geometry of a charged extremal BTZ black hole. 6.1 Kaluza-Klein Decomposition We will now show that the infinite sequence of poles in the previous section from the Kaluza- Klein tower of a single scalar in a three dimensional metric (6.4) where the direction y is an interval L < y < L. The action of the scalar is Z 1 p h i S = d3 x g g µ⌫ @µ @⌫ m20 2 V (y) 2 , (6.5) 2 where V (y) = V (y), with the constant V and the size L to be determined. This is similar to Horava-Witten compactification on S 1 /Z2 [127] with an additional delta function potential. 3 The scalar satisfies Dirichlet boundary conditions at the ends of the interval. We now proceed to decompose the 3D theory into 2 dimensional modes. Using Fourier transform for the t coordinate: Z d! i!t (t, z, y) = e ! (z, y) , (6.6) 2⇡ one can rewrite the action (6.5) in the form of Z Z 1 d! S = dzdy ! (D0 + D1 ) ! , (6.7) 2 2⇡ where D0 is the a-independent part and D1 is linear in a: m20 1⇣ 2 ⌘ D0 = @z2 + ! 2 + @ V (y) ,  z2 z2 y a 1 m20 1⇣ 2 ⌘ D1 = @z2 @z + ! 2 @ + V (y) . (6.8) z z z2 z2 y Here, we neglected higher order contributions of a. The eigenfunctions of D0 can be clearly written in the form ! (z, y) = ! (z) fk (y) . (6.9) 3 See also [128, 129]. We are grateful to Cheng Peng for bringing this to our attention. – 26 – Then fk (y) is an eigenfunction of the Schr¨odinger operator @y2 + V (y) with eigenvalue k 2 . This is a well known Schrodinger problem: the eigenfunctions and the eigenvalues are pre- sented in detail in Appendix C. After solving this part, the kernels are reduced to ✓ 2 ◆  ✓ 2 ◆ 2 2 m0 + p2m a 2 1 2 m0 qm 2 D0 = @ z + ! , D1 = @z @z + ! , (6.10) z2 z z z2 where pm are the solutions of (2/V )k = tan(kL) (6.11) while qm are the expectation values of @y2 V (y) operator respect to fpm . If we choose V = 3 and L = ⇡2 the solutions of (6.11) agree precisely with the strong coupling spectrum of the SYK model given by g˜(⌫) = 1, as is clear from (5.19) and (5.20). This is our main observation. For these values of V , L, the propagator G is determined by the Green’s equation of D. We now use the perturbation theory to evaluate it. This will then be compared with the corresponding propagator of the bi-local SYK theory. 6.2 Evaluation of G(0) We start by determining the leading, zero-th order G(0) propagator obeying (0) D0 G!,!0 (z, y; z 0 , y 0 ) = (z z 0 ) (y y 0 ) (! + ! 0 ) . (6.12) p e (0) We first separate the scaling part of the propagator by G(0) = z G and multiplying z 2 . Expanding in a basis of eigenfunctions fk (y), X e (0) (z, y, !; z 0 , y 0 , ! 0 ) = G e (0) 0 0 (z; z 0 ) fk (y)fk0 (y 0 )G (6.13) !,k;! ,k k,k0 The Green’s function G e (0) 0 0 (z, z 0 ) is clearly proportional to (k k 0 ) and satisfies the !,k;! ,k equation h i z 2 @z2 + z @z + ! 2 z 2 ⌫02 G e (0) 0 0 (z; z 0 ) = z 32 (z z 0 ) (! + ! 0 ) (k k 0 ) . (6.14) !,k;! ,k where we have defined ⌫02 ⌘ k 2 + m20 + 1/4. (6.15) The operator which appears in (6.14) is the Bessel operator. Thus the Green’s function can be expanded in the complete orthonormal basis. For this, we use the same basis form Z⌫ as in the SYK evaluation 4 : Z e (0) G (z; z 0 ) = d⌫ ge⌫(0) (z 0 ) Z⌫ (|!z|) . (6.16) !,k; !,k 4 This represents a modified set of wave functions with boundary conditions at z ! 1 in contrast to the standard AdS wave functions. Some basic aspects of Euclidean AdS scalar propagators are summarized in Appendix F – 27 – Then, substituting this expansion into the Green’s equation (6.12) and using Eqs.(D.6) and (0) (D.2), one can fix the coefficient ge⌫ . Finally, the ⌫-integral form of the propagator is given by Z (0) 0 0 12 d⌫ Z⌫⇤ (|!z|) Z⌫ (|!z 0 |) G!,k; !,k (z; z ) = |zz | . (6.17) N⌫ ⌫ 2 ⌫02 We now note that if we choose m20 = 1/4, which is the BF bound of AdS2 , we have ⌫02 = p2m , and the equation which determine pm , (6.11) is precisely the equation which determines the spectrum of the SYK theory found in [9, 11]. With this choice, the real space zeroth order propagator in three dimensions is 1 X Z Z (0) 0 0 0 0 1 d⌫ Z⌫⇤ (|!z|) Z⌫ (|!z 0 |) 0 d! i!(t t0 ) G (t, z, y; t , z , y ) = |zz | 2 fpm (y)fpm (y ) e . N⌫ ⌫ 2 p2m 2⇡ m=0 (6.18) We now show that the above propagator with y = y 0 = 0 is in exact agreement with the bi-local propagator of the SYK model. The Green’s function with these end points is 1 X Z Z (0) 0 0 0 1 d! i!(t t0 ) d⌫ Z⌫⇤ (|!z|) Z⌫ (|!z 0 |) G (t, z, 0; t , z , 0) = |zz | 2 C(pm ) e , (6.19) 2⇡ N⌫ ⌫ 2 p2m m=0 where we have defined 2 p2m 2p3m C(pm ) ⌘ fpm (0)fpm (0) = Bm = . (6.20) p2m + (3/2)2 [p2m + (3/2)2 ][⇡pm sin(⇡pm )] Now we note that Kaluza-Klein wave function coefficient coincides in detail with the SYK one, namely: 2pm C(pm ) = R(pm ) , (6.21) 3 where R(pm ) was given in Eq.(5.24). As in Eq.(D.6), the integration of ⌫ is a short-hand notation which denotes a summation of ⌫ = 3/2 + 2n, (n = 0, 1, 2 · · · ) and an integral of ⌫ = ir, (0 < r < 1). The sum over these discrete values of ⌫ and the integral over the continuous values can be now performed exactly as in the calculation of the SYK bi-local propagator [11]. Closing the contour for the continuous integral in Re(⌫)! 1, one finds that there are two types of poles inside of this contour. (1): ⌫ = 2n + 3/2, (n = 0, 1, 2, · · · ), and (2): ⌫ = pm , (m = 0, 1, 2, · · · ). The contributions of the former type of poles precisely cancel with the contribution from the discrete sum over n. Details of the evaluation which explicitly shows the cancelation are presented in Appendix E. Therefore, the final remaining contribution is just written as residues of ⌫ = pm poles as 1 Z X 1 > )J <) (0) 0 0 1 0 12 i!(t t0 ) Z pm (|!|z pm (|!|z G (t, z, 0; t , z , 0) = |zz | d! e R(pm ) . (6.22) 3 1 Npm m=0 – 28 – Altogether we have shown that y = 0 mode 3D propagator is in precise agreement with the q = 4 SYK bi-local propagator at large J given in Eq.(5.23). The propagator is a sum of non- standard propagators in AdS2 . While it vanishes on the boundary, the boundary conditions at the horizon are di↵erent from that of the standard propagator in AdS. 6.3 First Order Eigenvalue Shift In this section, we study the first order eigenvalue shift due to D1 by treating this operator as a perturbation onto the D0 operator. The result will confirm the duality a = 1/J, where a is defined in the dilaton background (6.3) and J is the coupling constant in the SYK model. Since the t and y directions are trivial, let us start with the kernels already solved for these two directions given in Eq.(6.10). The eigenfunction of D0 operator is 1 |z| 2 Z⌫ (|!z|) , (6.23) and using the orthogonality condition (D.3), its matrix element in the ⌫ space is found as h i N⌫ ⌫ 2 (m20 + p2m + 14 ) ⌫,⌫ 0 . (6.24) Now following the first order perturbation theory, we are going to determine the first order eigenvalue shift. Using the Bessel equation, the action of D1 on the D0 eigenfunction (6.23) is found as " !# 3 1 a @z m20 qm2 + 4 D1 |z| Z⌫ (|!z|) = 2 1 Z⌫ (|!z|) . (6.25) |z| 2 z z2 For the derivative term, we use the Bessel function identity (for example, see 8.472 of [130]) ⌫ @x J⌫ (x) = ± J⌫⌥1 (x) ⌥ J⌫ (x) , (6.26) x to obtain h i ⌫ @z Z⌫ (|!z|) = Z⌫ (|!z|) |!| J⌫+1 (|!z|) ⇠⌫ J ⌫ 1 (|!z|) . (6.27) |z| Therefore, now the matrix element is determined by integrals Z 1 h iZ 1 1 1 Z⌫⇤0 (|!z|)Z⌫ (|!z|) dz |z| 2 Z⌫⇤0 (|!z|)D1 |z| 2 Z⌫ (|!z|) = a ⌫ m20 2 qm + 3 4 dz 0 0 z2 Z 1 Z ⇤0 (|!z|) h i a|!| dz ⌫ J⌫+1 (|!z|) ⇠⌫ J ⌫ 1 (|!z|) . 0 z (6.28) For the continuous mode (⌫ = ir), the integrals might be hard to evaluate. In the following, we restrict ourself to the real discrete mode ⌫ = 3/2 + 2n. In such case, ⇠⌫ = 0. Therefore, – 29 – the linear combination of the Bessel function is reduced to a single Bessel function as Z⌫ (x) = J⌫ (x). Since Z ⇥ ⇤ 1 J↵ (x)J (x) 2 sin ⇡2 (↵ ) dx = , [Re(↵), Re( ) > 0] 0 x ⇡ ↵2 2 Z 1 ⇥ ⇤ J↵ (x)J (x) 4 sin ⇡2 (↵ 1) dx = ⇥ ⇤⇥ ⇤, [Re(↵), Re( ) > 1] (6.29) 0 x2 ⇡ (↵ + )2 1 (↵ )2 1 we have now found the matrix element for the discrete mode is given by ⇥ ⇤" ⇥ ⇤ # 2a|!| sin ⇡2 (⌫ ⌫ 0 1) 2 ⌫ (m20 qm 2 + 3) 4 1 . (6.30) ⇡ (⌫ + 1)2 ⌫ 02 (⌫ 1)2 ⌫ 02 Next, let us focus on the zero mode (⌫ = ⌫ 0 = 3/2) eigenvalue. In the above formula, taking the bare mass to the BF bound: m20 = 1/4 as before, the zero mode first order eigenvalue shift is found as a|!| (2 + q02 ) . (6.31) 2⇡ Now, we compare this result with the 1/J first order eigenvalue shift of the SYK model, which is for the zero mode found in [10] as ↵K |!| k(2, !) = 1 + ··· , (zero temperature) (6.32) 2⇡J where ↵K ⇡ 2.852 for q = 4. The !-dependence of our result (6.31) thus agrees with that of the SYK model. Furthermore, this comparison confirms the duality a = 1/J. Finally, we can now complete our comparison by showing agreement for the m = 0 mode contribution to the propagator. We include the first O(a) order shift for the pole as 3 a|!| ⌫ = + 2 + q02 + O(a2 ) . (6.33) 2 6⇡ For the zero mode part (m = 0) of the on-shell propagator in Eq.(E.6), the leading order is O(1/a). This contribution comes from the coefficient factor of the Bessel function, which was responsible for the double pole at ⌫ = 3/2. For other p0 setting them to 3/2, we obtain the leading order contribution from the zero mode as Z 1 (0) 9⇡ B02 1 d! i!(t t0 ) Gzero mode (t, z, 0; t0 , z 0 , 0) = 2 |zz 0 | 2 e J 3 (|!z|)J 3 (|!z 0 |) . 4a (2 + q0 ) 1 |!| 2 2 (6.34) This agrees with the order O(J) contribution of the SYK bi-local propagator of Malda- cena/Stanford [10]. – 30 – 7 Question of Dual Spacetime 7.1 “i” Problem In this section, we clarify the question regarding the signature of the SYK dual gravity theory. The bi-local SL(2, R) Casimir (5.9) can be seen to take the form of a Laplacian of Lorentzian two dimensional Anti de-Sitter or de-Sitter space-time (in this two dimensional case they are characterized by the same isometry group SO(2,1) or SO(1,2)). Under the canonical identification with AdS dt2 + dˆ z2 ds2 = , (7.1) zˆ2 it equals C1+2 = z 2 ( @t2 + @zˆ2 ) . (7.2) Consequently the SYK eigenfunctions should be compared with known AdS2 or dS2 basis wave functions. Note that the Bessel function Z⌫ (5.12) are not the standard normalizable modes used in quantization of scalar fields in AdS2 : in particular they have rather di↵erent boundary conditions at the Poincare horizon. Another important property of this basis is that when viewed as a Schrodinger problem as in [9] it has a set of bound states, in addition to the scattering states. This will be discussed in detail in Section 7.2 (see the left picture of FIG. 2). This leads one to try an identification with de-Sitter basis functions 5 . In fact as we will argue, the bi-local SYK wave functions can be realized as a particular ↵-vacuum of Lorentzian dS2 with a choice of ↵ = i⇡h = i⇡(⌫ + 1/2). This is seen as follows. We consider the dS2 background with a metric given by d⌘ 2 + dt2 ds2 = . (7.3) ⌘2 This can be obtained by the coordinate change (5.16) by replacing z ! ⌘. The Euclidean (Bunch-Davies [133]) wave function of a massive scalar field is given by E i!t ! (⌘) e , (7.4) with r E 1 1 ! (⌘) = ⌘ 2 H⌫(2) (|!|⌘) , ⌫ = m2 , (7.5) 4 (2) where H⌫ is the Hankel function of the second kind. Since the t-dependence is always like ei!t , in the following we will focus only on the ⌘ dependence. The ↵-vacuum wave function 5 This possibility has been emphasized to us by J. Maldacena [113]. – 31 – is defined by Bogoliubov transformation from this Euclidean wave function [134, 135] as h i ↵ E ↵ E⇤ ! (⌘) ⌘ N ↵ ! (⌘) + e ! (⌘) 1 h i = N↵ ⌘ 2 H⌫(2) (|!|⌘) + e↵ H⌫(1) (|!|⌘) , (7.6) where 1 ,N↵ = p (7.7) 1 e↵+↵⇤ and ↵ is a complex parameter. Now let us consider a possibility of ↵-vacuum with ✓ ◆ 1 ↵ = i⇡ ⌫ + = i⇡h . (7.8) 2 With this choice of ↵, using the definition of the Hankel functions J ⌫ (x) e i⇡⌫ J⌫ (x) J ⌫ (x) ei⇡⌫ J⌫ (x) H⌫(1) (x) = , H⌫(2) (x) = , (7.9) i sin(⇡⌫) i sin(⇡⌫) one can rewrite the ↵-vacuum wave function as 1 ! ↵ 2⌘2 ! (⌘) = i⇡⌫ Z⌫ (|!|⌘) , (7.10) 1 + ⇠⌫ e where Z⌫ is defined in Eq.(5.12). After excluding the ⌘-independent part of the wave function, we can write the ⌘-dependent part as 1 ↵ ! (⌘) = ⌘ 2 Z⌫ (|!|⌘) . (7.11) This wave function agrees with the eigenfunction of the SYK quadratic kernel (5.13) after the identifications of ⌘ = (t1 t2 )/2 and t = (t1 + t2 )/2. Due to this observation, one might attempt to claim that the dual gravity theory of the SYK model is given by Lorentzian dS2 space-time. However, there is a critical issue in this claim. Apart from the Lorentzian signature in this metric (7.3), we still have a discrepancy in the exponent of the partition function (2.4) with a factor of “i”. Namely, if the dual gravity theory (higher spin gravity or string theory) is Lorentzian dS2 , it must have Z ⇣ ⌘ Z = Dhn D m exp i Sgrav [h, ] + Smatter [h, ] , (7.12) where we collectively denote the graviton and other “higher spin” gauge fields by hn and the dilaton and other matter fields by m . Hence the agreement of the SYK bi-local propagator D E 1 X DSYK (t1 , t2 ; t01 , t02 ) = (t1 , t2 ) (t01 , t02 ) = Gpm (t1 , t2 ; t01 , t02 ) , (7.13) m=0 with a dS2 propagator 1 XD E 1 1 1 X DdS (⌘, t; ⌘ 0 , t0 ) = m (⌘, t) m (⌘ 0 , t) = Gm (⌘, t; ⌘ 0 , t0 ) , (7.14) i i m=0 m=0 – 32 – is only up to the factor i. Namely, even if we have a complete agreement of Gpm with Gm by identifying the coordinates by (5.16) (with a replacement of z ! ⌘), there is a problem with the signature (i.e. the discrepancy of the factor i). For higher point functions, the same i-problem proceeds due to the i factors coming from the propagator and each vertex. To conclude, for the Euclidean SYK model under consideration, one needs a dual gravity theory to be in the hyperbolic plane H2 (i.e. Euclidean AdS2 ) for the matching of n-point functions. We will set the basis for the EAdS2 realization in the next subsection. 7.2 Transformations and Leg Factors As we have commented in the Introduction in order to identify an Euclidean bulk dual description (rather than a Lorentzian), we will need a transformation which brings the SYK eigenfunctions (as given on bi-local space-time) to the standard eigenfunctions of the EAdS2 Laplacian. We will arrive at this transformation by considering the bi-local map described in [114, 115] for higher dimensional case. In our current d = 1 case, the map is even simpler. It will be seen to take the form of a H 2 Radon transform (a related suggestion was made in [10]). The need for a non-local transform on external legs appears to be characteristic of collective theory (which as a rule contains a minimal set of physical degrees of freedom). The first appearance of Radon type transforms in identifying holographic space-time was seen in the c = 1 / D = 2 string correspondence. This is seen precisely in the form of what is known as the regular Radon transform. Let us describe procedure formulated in [114, 115] for constructing the bi-local to space- time map. The method is based on construction of canonical transformations in phase space: bi-local (t1 , p1 ), (t2 , p2 ) and EAdS2 (⌧, p⌧ ), (z, pz ). We consider the Poincare coordinates for the Euclidean AdS2 space d⌧ 2 + dz 2 ds2 = . (7.15) z2 One way to obtain the bi-local map is to equate the SL(2, R) generators. Jˆ1+2 = JˆEAdS . (7.16) The one-dimensional bi-local conformal generators are ˆ 1+2 = t1 p1 + t2 p2 , D Pˆ1+2 = p1 p2 , ˆ 1+2 = K t21 p1 t22 p2 , (7.17) and the EAdS2 generators are given by ˆ EAdS = ⌧ p⌧ + z pz , D PˆEAdS = p⌧ , ˆ EAdS = (z 2 K ⌧ 2 ) p⌧ 2⌧ z pz , (7.18) where we defined p1 ⌘ @t1 , p2 ⌘ @t2 , p⌧ ⌘ @⌧ , pz ⌘ @z . Equating the generators, we can determine the map. From the Pˆ generators, we have p⌧ = p1 + p2 . Using this result for the other generators, we get two equations to solve: z pz = (t1 ⌧ )p1 + (t2 ⌧ )p2 z 2 p⌧ = (t1 ⌧ )2 p1 + (t2 ⌧ ) 2 p2 . (7.19) – 33 – These are solved by ✓ ◆2 t1 p1 t2 p2 2 t1 t2 ⌧ = , p⌧ = p 1 + p 2 , z = p1 p2 , p2z = 4p1 p2 . (7.20) p1 p2 p1 p2 One can see that the canonical commutators are preserved under the transform (at least classically, i.e. in terms of the Poisson bracket). Namely, [⌧, p⌧ ] = [z, pz ] = 1 and others vanish provided that [ti , pj ] = ij , with (i, j = 1, 2). Hence, we conclude the map is canonical transformation, which is also a point transformation in momentum space. For the kernel which implements this momentum space correspondence we can take (p⌧ (p1 + p2 )) R(p1 , p2 ; p⌧ , pz ) = p . (7.21) p2z + 4p1 p2 Through Fourier transforming all momenta to corresponding coordinates, the associated co- ordinate space kernel becomes 6 R(t1 , t2 ; ⌧, z) = (⌘ 2 (⌧ t)2 z2) . (7.22) With a multiplicative factor of additional of ⌘ this is known as the Circular Radon transform (7.25) which has a simple relationship to Radon transform on H 2 . There is another construction of the Radon transform which is used in [116–118] and is based on integration over geodesics. For the Euclidean AdS2 space-time (7.15), a geodesic is given by a semicircle 1 (⌧ ⌧0 )2 + z 2 = 2 , (7.23) E where ⌧ = ⌧0 is the center of the semicircle and 1/E is the radius. The Radon transform of a function of the bulk coordinates f (⌧, z) is a function of the parameters of a geodesic (E, ⌧0 ) defined by Z ⇥ ⇤ Rf (E, ⌧0 ) ⌘ ds f (⌧, z(⌧ )) , (7.24) where the integral is over the geodesic. From the geodesic equation (7.23), this transform is explicitly written as Z t+⌘ Z 1 ⇥ ⇤ dz ⇣ 2 ⌘ Rf (⌘, t) = 2⌘ d⌧ ⌘ (⌧ t)2 z 2 f (⌧, z) , (7.25) t ⌘ 0 z where we have used the identifications 1/E = ⌘ and ⌧0 = t; the resulting function [Rf ](⌘, t) is understood as a function on the Lorentzian dS2 (7.3). We will now explicitly evaluate the Radon transformation of (unit-normalized) EAdS2 wave functions (see Appendix G) 1 i!⌧ EAdS2 (⌧, z) = ↵⌫ z 2 e K⌫ (|!|z) (7.26) 6 Here, we have ignored possible issues related to the range of variables. – 34 – From the above formula of the Radon transform (7.25), we get h i Z t+⌘ p d⌧ 1 R EAdS2 (⌘, t) = ↵⌫ ⌘ 2 2 (⌘ 2 (⌧ t)2 ) 4 e i!⌧ K⌫ (|!| ⌘2 (⌧ t)2 ) . t ⌘ ⌘ (⌧ t) (7.27) Now shifting the integral variable ⌧ ! ⌧ + t and using the symmetry of the integrand, one can rewrite this integral as h i Z ⌘ ✓ ◆3 p i!t 1 4 R EAdS2 (⌘, t) = 2 ↵⌫ ⌘ e d⌧ cos(!⌧ ) K ⌫ (|!| ⌘2 ⌧ 2) . (7.28) 0 ⌘2 ⌧ 2 Further rewriting the cos(!⌧ ) in terms of J 1/2 (!⌧ ) and changing the integration variable to ⌧ = ⌘ sin ✓, we find r h i 1 ⇡ 3 ↵⌫ |!| 2 ⌘ i!t R EAdS2 (⌘, t) = e 2 sin(⇡⌫) Z ⇡ h i 2 1 ⇥ d✓ (tan ✓) 2 J 1 (|!|⌘ sin ✓) I ⌫ (|!|⌘ cos ✓) I⌫ (|!|⌘ cos ✓) , (7.29) 2 0 where we decomposed the modified Bessel function of the second kind into two first kinds. This ✓ integral is indeed given in Eq.(4) of 12 · 11 of [131], which leads to " # h i p ( 14 + ⌫2 ) 1 tan ⇡⌫ + 1 R EAdS2 (⌘, t) = 2i ⇡ 3 ⌫ ⌫ ⌘ 2 e i!t J⌫ (|!|⌘) + 2 J ⌫ (|!|⌘) , (7.30) (4 + 2) tan ⇡⌫ 2 1 where we also used Eq.(G.7). The inside of the square bracket precisely agrees with the particular combination of Bessel functions, Z⌫ (|!|⌘) function defined in Eq.(5.12). When ⌫n = 3/2 + 2n the second term in this square bracket vanishes. As will be clear soon, we need the radon transform of the modified Bessel function I⌫n with. This can be likewise evaluated to yield R[↵⌫0 n z 1/2 e ik⌧ I⌫n (|k|z)] = (2⌫n ⌘)1/2 e ikx J⌫n (|k|⌘) (7.31) where ✓ ◆1 ⌫n 2⌫n 2 ( 34 + 2 ) ↵⌫0 n = ⌫n (7.32) ⇡ ( 14 + 2 ) The extra ⌫-dependent factor in (7.30) which appears in front of the unit-normalized dS2 wave function described in Appendix G should be understood as a leg factor (7.34). As we will see later, this is analogous to what happens in the c = 1 matrix model [120–124]. In summary, we have the Radon transform (EAdS2 ) (dS2 ) R !,⌫ (⌧, z) = L(⌫) !,⌫ (⌘, t) , (7.33) where EAdS2 and dS2 are the unit-normlized wave functions defined in Eq.(G.1) and Eq.(G.4), respectively, while the leg factor is defined by p (1 + ⌫ ) L(⌫) ⌘ (Leg Factor) = 2i ⇡ 43 ⌫2 . (7.34) (4 + 2) – 35 – VdS2 VEAdS2 y y Figure 2. The de Sitter potential VdS2 has bound states and scattering states. On the other hand, the Euclidean AdS potential VAdS2 has only scattering modes. The inverse transformations are 1 (dS2 ) 1 (EAdS2 ) R !,⌫ (⌘, t) = L (⌫) !,⌫ (⌧, z) . (7.35) for ⌫ 6= 3/2 + 2n, while for ⌫ = 3/2 + 2n we have instead (dS2 ) R 1 !,⌫n (⌘, t) = ↵⌫0 n z 1/2 e ik⌧ I⌫n (|k|z) (7.36) Under the Radon transform R, the Laplacian of Lorentzian dS2 is transformed into that of Euclidean AdS2 : ⇤ds2 dS2 (⌘, t) = R ⇤EAdS2 EAdS2 (⌧, z) , (7.37) with ⇤ds2 = ⌘ 2 ( @⌘2 + @t2 ) , ⇤EAdS2 = z 2 (@⌧2 + @z2 ) . (7.38) Here, dS2 = R EAdS2 and wave functions are not normalized. In the rest of this section, we will show that the Radon transformation flips the sign of the potential appearing in the equivalent Schrodinger problem as formulated in [9]. We start from the Radon transformation (7.37). Expanding the wave functions by 1 X dS2 (⌘, t) = ⌘2 e i!t edS (⌘; k) , 2 ! 1 X EAdS2 (⌧, z) = z 2 e i!⌧ eEAdS (!; z) , (7.39) 2 ! we have corresponding Bessel equations for edS2 and eEAdS2 . By changing the coordinates by y ⌘ log(!⌘) or y ⌘ log(!z), these Bessel equations are reduced to the Schrodinger equations – 36 – as ⇣ ⌘ @y2 ey edS2 = ⌫ 2 edS2 , ⇣ ⌘ @y2 + ey eEAdS2 = ⌫ 2 eEAdS2 . (7.40) Therefore, the Radon transform flips the sign of the corresponding Schrodinger potential (see FIG. 2). The de Sitter potential VdS2 = ey has bound states as well as scattering states. On the other hand, the Euclidean AdS potential VAdS2 = ey has only scattering modes. 7.3 Green’s Functions and Leg Factors In this subsection, we start from the SYK bi-local propagator (5.21). Using the inverse Radon transformation (7.35), we will show that the resulting propagator can be written in terms of functions which appear in EAdS2 , except for the leg-factors. The SYK bi-local propagator is given by Z 1 X u⇤⌫,! (t1 , t2 )u⌫,! (t01 , t02 ) 0 0 1 G(t1 , t2 ; t1 , t2 ) / J d! , (7.41) 1 ⌫ N⌫ [e g (⌫) 1] where u⌫,! are the eigenfunctions defined in Eq.(5.13). Here the summation over ⌫ is a short- hand notation denotes the discrete mode sum and the continuous mode sum. Now, we identify ⌘ = (t1 t2 )/2 and t = (t1 + t2 )/2. Then, the propagator can be written in terms of the dS2 wave functions as Z 1 ( 1 X 4 sin ⇡⌫n ⇤ G(⌘, t; ⌘ 0 , t0 ) = 2⇡J 1 d! (⌘, t) !,⌫n (⌘ 0 , t0 ) 1 ge(⌫n ) 1 !,⌫n n=0 Z 1 ⇤ ) (⌘, t) (⌘ 0 , t0 ) !,⌫ !,⌫ + dr , (7.42) 0 e g (⌫) 1 ⌫=ir where ⌫n = 2n + 3/2. Next, we use the inverse Radon transform (7.35) to bring the dS wave functions into the EAdS wave functions. Z 1 ( 1 X 4 sin ⇡⌫n ⇤ 0 0 G(⌧, z; ⌧ , z ) = 2⇡J 1 d! |L 1 (⌫n )|2 !,⌫n (⌧, z) !,⌫n (⌧ 0 , z 0 ) 1 e g (⌫ n ) 1 n=0 Z 1 ⇤ ) (⌧, z) (⌧ 0, z0) !,⌫ !,⌫ + dr |L 1 (⌫)|2 . (7.43) 0 ge(⌫) 1 ⌫=ir Here we have defined !,⌫n (⌧, z) by !,⌫n (⌧, z) = ↵⌫0 n z 1/2 e ik⌧ I⌫n (|k|z) (7.44) Note that !,⌫n (⌧, z) is not really a EAdS wavefunction. We can directly evaluate the continuous mode summation for the full Green’s function including the leg factors in the integrand. However, we postpone this for a while and we – 37 – first demonstrate how to extract the leg factors from the integrand and evaluate the integral without the leg factors. Writing the ⌫’s appearing in the leg factors in terms of the Bessel di↵erential operators, we can pull out the leg factors from the summation over m or the integral of r as Z 1 ( 1 2 X 4 sin ⇡⌫n ⇤ G(⌧, z; ⌧ 0 , z 0 ) = 2⇡J 1 L 1 (ˆ pEAdS2 ) d! 0 0 !,⌫n (⌧, z) !,⌫n (⌧ , z ) 1 e g (⌫ n ) 1 n=0 Z 1 ⇤ ) 0 0 !,⌫ (⌧, z) !,⌫ (⌧ , z ) + dr . 0 ge(⌫) 1 ⌫=ir (7.45) with r 1 pˆEAdS2 ⌘ ⇤EAdS2 + , (7.46) 4 where the Laplacian of EAdS2 is defined in Eq.(7.38). The above expression for the leg factor di↵erential operators is slightly ambiguous. What we mean is that one of the leg factor di↵erential operator is acting on (⌧, z) and the other leg factor operator is acting on (⌧ 0 , z 0 ). To make this more explicit, we can introduce two delta functions (z z1 ) (z 0 z2 ) and integrals over z1 and z2 , where one of the leg factor is acting on (z z1 ) and the other is acting on (z 0 z2 ). Furthermore, we can rewrite the delta functions in terms of the completeness of the Bessel function (H.1). Therefore, now the propagator is written as Z 1 Z 0 0 dz1 1 dz2 ⇤ G(⌧, z; ⌧ , z ) = L (z; z1 ) GEAdS2 (⌧, z1 ; ⌧ 0 , z2 ) L(z2 ; z 0 ) , (7.47) 0 z1 0 z2 with Z 1 ( 1 X 4 sin ⇡⌫n ⇤ GEAdS2 (⌧, z; ⌧ 0 , z 0 ) = 2⇡J 1 d! (⌧, z) !,⌫n (⌧ 0 , z 0 ) 1 ge(⌫n ) 1 !,⌫n n=0 Z 1 ⇤ ) 0 0 !,⌫ (⌧, z) !,⌫ (⌧ , z ) + dr , (7.48) 0 ge(⌫) 1 ⌫=ir and we defined the leg factor integral kernel as Z 1 i ( 34 + ir 2) L(z1 ; z2 ) ⌘ 5/2 dr r sinh(⇡r) 1 ir Kir (z1 ) Kir (z2 ) . (7.49) ⇡ 0 (4 + 2) Let us now evaluate the continuous mode summation in the EAdS2 propagator (7.48). Eval- uating this integral as a contour integral as in [11, 106], there is only one set of poles coming from ⌫ = pm , with (m = 0, 1, 2, · · · ) which are the solutions of ge(pm ) = 1. Therefore, after this integral the EAdS2 propagator (7.48) becomes Z 1 ( 1 2 X pm 0 0 0 12 i!(⌧ ⌧ 0 ) GEAdS2 (⌧, z; ⌧ , z ) = |zz | d! e 0 (p ) Kpm (|!|z > ) Ipm (|!|z < ) J 1 e g m m=0 1 ) 4 X ⌫n + 2 K⌫n (|!|z) K⌫n (|!|z 0 ) . (7.50) ⇡ n ge(⌫n ) 1 – 38 – As we will see in the next section this form of the propagator is analogous to the Wilson loop (or macroscopic loop) operator propagators in the c = 1 matrix model (7.61). Now we go back to the o↵-shell expression of the propagator (7.43) and evaluate the continuous mode summation for the full Green’s function with including the leg factors in the integrand: Z 1 ⇤ 0 0 !,⌫ (⌧, z) !,⌫ (⌧ , z ) Icont ⌘ dr |L 1 (⌫)|2 . (7.51) 0 ge(⌫) 1 ⌫=ir We evaluate this integral as a contour integral as before. We note that since the modified Bessel function K⌫ is regular on the entire ⌫-complex plane, we have two sets of poles: (i). ⌫ = pm , with (m = 0, 1, 2, · · · ). (ii). ⌫ = ⌫n = 2n + 3/2, with (n = 0, 1, 2, · · · ) where ( 34 ⌫2 ) = 1. After evaluating the residues at these poles, we find the integral as 1 ( 1 |zz 0 | 2 0 X ( 3 + p m ) ( 3 p m ) pm i!(⌧ ⌧ ) Icont = e 4 1 2 pm 4 1 2 pm 0 (p ) Kpm (|!|z > )Ipm (|!|z < ) 4⇡ 2 ( 4 + 2 ) ( 4 2 ) e g m m=0 ) 1 X 2 ( 3 + ⌫n ) ✓ ◆ 2 ⌫ n + 4 2 2 ( 1 + ⌫n ) K⌫n (|!|z > )I⌫n (|!|z < ) . (7.52) ⇡ 4 2 e g (⌫ n ) 1 n=0 The second line in the RHS looks similar to the discrete mode contribution to the propagator (7.43). However, these two contributions do not cancel each other. Hence there are two types of the contributions to the final result as G(⌧, z; ⌧ 0 , z 0 ) 1 Z ( 1 |zz 0 | 2 1 i!(⌧ ⌧ 0) X ( 34 + p2m ) ( 34 p2m ) pm = d! e Kpm (|!|z > )Ipm (|!|z < ) 2⇡J 1 m=0 ( 14 + p2m ) ( 14 p2m ) ge0 (pm ) ✓ ◆ ) 1 X 2( 3 + ⌫n h i 2 ) ⌫n + 4 2( 1 ⌫n I⌫n (|!|z < ) 2I⌫n (|!|z > ) I ⌫n (|!|z > ) . (7.53) 4 + 2 ) ge(⌫n ) 1 n=0 p0 Of course, here we still have the zero mode (p0 = 3/2) problem coming from ( 34 2 ) = 1. In this expression, the Bessel function part of the first contribution in the RHS is the standard form for EAdS propagator, while the extra factor coming from the leg-factors can be possibly understood as a contribution from the naively pure gauge degrees of freedom as in the c = 1 model (c.f. [121–124]), in which case the second contribution in RHS represents the contribution from these modes as in [120]. In section 6, we presented the 3D picture of the SYK theory, based on the fact that the non-trivial spectrum predicted by the model, which are solutions of ge(pm ) = 1 with (m = 0, 1, 2, · · · ) can be reproduced through Kaluza-Klein mechanism in one higher dimension. This picture is more natural in the AdS2 interpretation of the bi-local space. Now, we will point out a similarity between the 3D picture of the SYK model [106, 107] and the c = 1 Liouville theory (2D string theory) [119–124]. In the 3D description we have a scalar field Z 1 p h i S3D = dx3 g g µ⌫ @µ @⌫ m20 2 V (y) 2 , (7.54) 2 – 39 – with a background metric dt2 + dˆ z2 ⇣ a ⌘2 2 ds2 = + 1 + dy , (7.55) zˆ2 zˆ where a ⇠ J 1 , but here we only consider the leading in 1/J contribution and suppress the subleading contributions coming from the yy-component of the metric. The detail of the potential V (y) depends on q and for that readers should refer to [106, 107]. The propagator for the scalar field in this background in the leading order of 1/J is given by Z Z 1 X d! i!(t t0 ) d⌫ Z⌫⇤ (|!ˆ z 0 |) z |) Z⌫ (|!ˆ z , t, y; zˆ0 , t0 , y 0 ) = |ˆ G(0) (ˆ z zˆ0 | 2 fk (y)fk (y 0 ) e , 2⇡ N⌫ ⌫ 2 k2 k (7.56) where fk (y) is the wave function along the third direction y with momentum k. This is simply a rewriting the propagator (7.42) by treating the non-local kernel (eigenvalue) by an extra dimension. The identical procedure leads to the leg-factors. After the (inverse) Radon transform and the contour integral for the continuous mode sum, the propagator is reduced to (0) 0 G!; ! (z, y; z , y0) 1 ( |zz 0 | 2 X ( 34 + k2 ) ( 34 k 2) = fk (y)fk (y 0 ) Kk (|!|z > )Ik (|!|z < ) 4⇡ k ( 14 + k 2) ( 14 k 2) ✓ ◆ ) 1 X 2( 3 + ⌫n h i 2 ) ⌫n +2 4 2( 1 ⌫n I⌫n (|!|z < ) 2I⌫n (|!|z > ) I ⌫n (|!|z > ) . (7.57) 4 + 2 ) ⌫n2 k2 n=0 On the other hand, for the c = 1 matrix model / 2D string duality, the Wilson loop operator is related to the matrix eigenvalue density field by ⇣ ⌘ Z 1 `M (t) W (t, `) ⌘ Tr e = dx e `x (t, x) . (7.58) 0 The corresponding propagator was found by Moore and Seiberg [120] as D E Z 1 Z 1 ⇤ (t, ') 0 0 0 0 p E,p E,p (t , ' ) w(t, ')w(t , ' ) = dE dp , (7.59) 1 0 sinh ⇡p E 2 p2 with ` = e ' and the normalized wave function p iEt p ' E,p (t, ') = p sinh ⇡p e Kip ( µe ). (7.60) After evaluating the p-integral as a contour integral, we obtain the propagator as Z 1 ( D E ⇡E p p 0 0 iE(t t0 ) < > w(t, ')w(t , ' ) = ⇡ dE e KiE ( µe ' ) IiE ( µe ' ) 1 2 sinh ⇡E 1 ) X ( 1)n n2 p p '< '> + Kn ( µe ) In ( µe ) . (7.61) E 2 + n2 n=1 – 40 – The point we want to make here is that this 3D picture is completely parallel to the c = 1 Liouville theory (2D string theory) [119–124]. Namely, if we make a change of coordinate by z = e ' , then the '-direction becomes the Liouville direction, while the y-direction (at least in the leading order of 1/J) can be understood as the c = 1 matter direction. In this comparison, the ⌧ -direction serves as an extra direction. Finally, the ⌫ appearing in the SYK model is realized as a momentum k along the y-direction in the 3D picture (7.55). Therefore, we have the following correspondence between the c = 1 Liouville theory and the 3D picture of the SYK model. c=1 3D SYK ie ' z it y ip ⌫ iE k p µ |!| 8 Conclusion In this thesis we described the bi-local formulation of the SYK model by defining a reparametriza- tion invariant collective theory at the IR point and away from it. A regularized action repre- senting an interacting theory between a Schwarzian coordinate and bi-local matter is specified. We presented the non-linear derivation of the Schwarzian action, which is exact at all orders. It generates perturbative calculations in the SYK model around the conformal IR point which are systematic in the inverse of the strong coupling J. We gave the evaluation of the tree level free energy in this expansion. Even though, the present calculations are done at tree level in 1/N , the formalism given allows for loop level calculations with no difficulty: by projection of the zero mode the perturbation expansion is well defined, while the Jacobian(s) of the changes of variables provide exact counter terms which are expected to cancel infinities appearing in loop diagrams. Next, we have presented a three dimensional perspective of the bulk dual of the SYK model. At strong coupling we showed that the spectrum and the propagator of the bi-local field can be exactly reproduced by that of a scalar field living in AdS2 ⇥ S 1 /Z2 with a delta function potential at the center. The metric on the interval in the third direction is the dilaton of Jackiw-Teitelboim theory, which is a constant at strong coupling. We also calculated the leading 1/J correction to the propagator which comes from the corresponding term in the metric in the third direction, and showed that form of the poles of the propagator are consistent with the results of the SYK model [10]. Finally we addressed the question of what represents the bulk dual space-time of the SYK model. At the outset, the question seems simple since the small fluctuations of the (Euclidean) SYK model are completely given by a set of Lorentzian wave functions associated with the SL(2, R) isometry group. With a simple identification of space-time, these are seen to associated with eigenfunctions in de-Sitter (or Anti de-Sitter) space-time (as was discussed – 41 – in [7, 9–11]). Likewise the propagator and higher point n-functions continue to feature this Lorentzian space-time structure. Even though this Lorentzian bulk dual interpretation seems to be straightforwardly as- sociated with the SYK bi-local data, we have stressed that there is a problem with this interpretation. Concentrating on the e↵ective bi-local Large N version of the model, we have provided a resolution, which follows from a further non-local redefinition of space-time. This comes in terms of Leg transformations of Green’s functions which places the theory in Eu- clidean AdS dual setting. Such transformations are actually characteristic of collective field representations of Large N theories. The leg transformations that we explicitly implement (apart from providing the EAdS2 space-time setting) also bring out the couplings of additional “discrete” states. We expect that all features of the SYK model we presented here will play a central role in full identification of the bulk dual for the SYK model. A ✏-Expansion In this appendix, we will exemplify how to obtain the non-linear Schwarzian action (2.13) associated with the naive form of the breaking term in Eq.(2.5). This is done by using "- expansion with q = 2/(1 ") and treating " as a small parameter. We note that for any q in the range of 2  q  1, the value of " is 0  "  1. Therefore, the convergence of this ✏-expansion is guaranteed. Even though we use the "-expansion, we can nevertheless calculate all order contributions of " as we will see below. We first rewrite the critical solution in the following way: p ! 1 |f 0 (t1 )f 0 (t2 )| 0,f (t1 , t2 ) = ⇡J |f (t1 ) f (t2 )| 2 p ! p !2 3 |f 0 (t )f 0 (t )| 1 2 "2 0 0 |f (t1 )f (t2 )| ⇥ 41 " log + log + ···5 , |f (t1 ) f (t2 )| 2 |f (t1 ) f (t2 )| (A.1) where the first term is the contribution from q = 2 case, which leads to the result Eq.(2.13) with ↵ = 1. To evaluate higher order " contributions, we use the following expansions of the logarithm in the t1 ! t2 limit: p ! |f 0 (t1 )f 0 (t2 )| 1 |f 00 (t2 )|2 1 |f 000 (t2 )| log = log |t1 t2 | |t1 t2 | 2 + |t1 t2 | 2 + · · · . |f (t1 ) f (t2 )| 8 |f 0 (t2 )|2 12 |f 0 (t2 )| (A.2) The first log term gives an f -independent divergent term which we will eliminate in the following. One also expands the factor representing q = 2 reparametrized critical solution – 42 – and then one finds O(") = 0. For order O("2 ) contribution, from Eq.(A.2), one can find Z ✓ ◆ 2 N "2 1 |f 00 (t2 )|2 1 |f 000 (t2 )| O(" ) = dt1 @1 |t1 t2 | log |t1 t2 | 4⇡J 4 |f 0 (t2 )|2 6 |f 0 (t2 )| t2 !t1 Z " ✓ 00 ◆2 # N" 2 000 f (t1 ) 3 f (t1 ) = dt1 , (A.3) 24⇡J f 0 (t1 ) 2 f 0 (t1 ) where we again eliminated the divergence term and used integration by parts. Hence, the total contribution up to O("2 ) for q = 2/(1 ") action is given by Z " ✓ ◆ # N↵ f 000 (t) 3 f 00 (t) 2 S[f ] = dt , (A.4) 24⇡J f 0 (t) 2 f 0 (t) where ↵(") = 1 "2 + O("3 ) . (A.5) In fact, there is no higher order contributions from O("3 ). This can be seen from an expansion p !n |f 0 (t1 )f 0 (t2 )| log |f (t1 ) f (t2 )| ⇣ ⌘n ✓ ◆ ⇣ ⌘n 1 1 |f 00 (t2 )|2 1 |f 000 (t2 )| 2 = log |t1 t2 | n |t 1 t 2 | log |t 1 t 2 | + ··· . 8 |f 0 (t2 )|2 12 |f 0 (t2 )| (A.6) This expansion together with the expansion of q = 2 reparametrized critical solution does not give any non-zero finite contribution to the action after the limit when n 3. Namely, the (log |t1 t2 |)n factor gives a strong divergence when n is large. However, if one wants to lower the power of this logarithm, then one gets a higher power of |t1 t2 |n , which strongly vanishes after setting t2 = t1 . This naive form of ↵ will turn out to be renormalized with our regularization of the breaking term. We evaluate this renormalized coefficient in the next appendix. B s-Regularization and Schwarzian Action In this appendix, we will directly evaluate the collective coordinate action with the regularized breaking term: Z Z N ⇥ ⇤ N S[f ] = 0,f s = lim dt1 dt2 0,f (t1 , t2 ) Qs (t1 , t2 ) , (B.1) 2 2 s! 12 and confirm that the result is given by Z " ✓ ◆2 # N↵ f 000 (t) 3 f 00 (t) S[f ] = dt , (B.2) 24⇡J f 0 (t) 2 f 0 (t) – 43 – with a coefficient ↵ = 12⇡B1 . (B.3) For this purpose, we expand the reparametrized critical solution with f (t) = t + "(t) as Z Z 1 (t , 0,f 1 2 t ) = (t 0 12 ) + dt a "(t a ) u (t , 0,ta 1 2 t ) + dta dtb "(ta )"(tb ) u(1),ta ,tb (t1 , t2 ) 2 Z 1 + dta dtb dtc "(ta )"(tb )"(tc ) u(2),ta ,tb ,tc (t1 , t2 ) + · · · , (B.4) 6 where we defined @ 0,f (t12 ) u0,ta (t1 , t2 ) ⌘ , @f (ta ) f (t)=t @ 2 0,f (t12 ) u(2),ta ,tb (t1 , t2 ) ⌘ , @f (ta )@f (tb ) f (t)=t @ 3 0,f (t12 ) u(3),ta ,tb ,tc (t1 , t2 ) ⌘ , @f (ta )@f (tb )@f (tc ) f (t)=t @n 0,f (t12 ) u(n),{ta ,··· ,tn } (t1 , t2 ) ⌘ . (B.5) @f (ta ) · · · @f (tn ) f (t)=t Let us first consider the quadratic and cubic order contributions. Taking derivatives and expressing in the momentum space, the quadratic and cubic coefficients are given by  2 i(!a +!b )t+ 1 u(2),!a ,!b (t1 , t2 ) = e |!a !b | cos((!a + !b )t ) sin |!a t | sin |!b t | 0 (t12 ) q |t |2 2 i(!a +!b )t+ 1 2 e |!a !b | 2 |t | J 3 (|!a t |) J 3 (|!b t |) 0 (t12 ) , (B.6) ⇡q 2 2 and u(3),!a ,!b ,!c (t1 , t2 )  4i i(!a +!b +!c )t+ 1 = e |!a !b !c | cos((!a + !b + !c )t ) sin |!a t | sin |!b t | sin |!c t | 0 (t12 ) q |t |3 r  4i ⇡|!c3 t | i(!a +!b +!c )t+ 1 e |!a !b | cos((!a + !b )t ) sin |!a t | sin |!b t | J 3 (|!c t |) 0 (t12 ) q2 2 |t |2 2 3 8i ⇡t 2 3 + 3 ei(!a +!b +!c )t+ |!a !b !c | 2 J 3 (|!a t |) J 3 (|!b t |) J 3 (|!c t |) 0 (t12 ) . (B.7) q 2 2 2 2 In fact, there are two more terms in the second line of RHS in u(3) obtained by permuta- tions of (!a , !b , !c ), but we omitted these terms in the above expression. Substituting these expressions into the action (B.1) and performing the t1 , t2 integrals, one finds single poles – 44 – (s 1/2) 1 coming from the double sine term in u(2) and from the triple sine term in u(3) . Namely for the quadratic contribution, we have Z dt1 dt2 dt3 dt4 u(2),!a ,!b (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) ✓ ◆ Z 1 q 1 dt = 22 2s ⇡ B1 bq 1 (s, q) (!a + !b ) 4+2s sin2 |!a t | , (B.8) q 0 |t | with Z 1 sin2 x 1 1 1 0 dx 4+2s = 1 + O((s 2) ) . (B.9) 0 x 6s 2 Also for the cubic contribution Z dt1 dt2 dt3 dt4 u(3),!a ,!b ,!c (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) ✓ ◆ Z 1 3 2s q 1 q 1 dt = 2 ⇡i B1 b (s, q) (!a + !b + !c ) sin |!a t | sin |!b t | sin |!c t | , q 0 |t |5+2s (B.10) with Z 1 dt sin |!a t | sin |!b t | sin |!c t | 0 |t |5+2s 1 = |!a !b !c | |!a |2 + |!b |2 + |!c |2 + O((s 1 0 2) ) . (B.11) 12(s 12 ) There are no other terms giving such (s 1/2) 1 pole. Such single pole factor (s 1/2) 1 cancels with the (s 1/2) factor in the regularized source Qs (3.17), and lead to Z dt1 dt2 dt3 dt4 u(2),ta ,tb (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) = B1 @t2a @t2b (tab ) , (B.12) and Z dt1 dt2 dt3 dt4 u(3),ta ,tb ,tc (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) = B1 @ta @tb @tc @t2a + @t2b + @t2c (tac ) (tbc ) . (B.13) With the experience of quadratic and cubic order computations, now we would like to evaluate all order contributions. As we saw above the poles associated to the limit s ! 1/2 only come from the double and triple sine terms. Therefore, we expect this structure is also true for any higher order contributions. Taking derivatives of the reparameterized critical solution, we find such term in n-th order is given by n ! 2 Y 0 (t12 ) u(n),{!a ,··· ,!n } (t1 , t2 ) = ( i)n (n 1)! ei(!a +···+!n )t+ sin |!i t | + · · · , (B.14) q |t |n i=a – 45 – where the ellipsis denotes non-singular terms in the limit s ! 1/2. Now, using the result (3.17), one obtains the contribution from the n-th order as Z dt1 dt2 dt3 dt4 u(n),{!a ,··· ,!n } (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) Z 1 n ! dt Y n 1 = 12⇡( i) (n 1)! B1 (s 2 ) (!a + · · · + !n ) 2+2s+n sin |!i t | + O(s 12 ) . 0 |t | i=a (B.15) The t -integral is given by Z 1 n n ! n ! dt Y Y X 1 sin |!i t | = |!i | |!i |2 1 + O((s 1 0 2) ) . (B.16) 0 |t |2+2s+n 12(s 2) i=1 i=1 i=1 Now, using this result and Fourier transforming back to {ta , · · · , tn } from {!a , · · · , !n }, we get Z dt1 dt2 dt3 dt4 u(n),{ta ,··· ,tn } (t1 , t2 ) K(t1 , t2 ; t3 , t4 ) 1 (t3 , t4 ) n ! n ! (n 1)! Y X = B1 @ ti @t2i (tan ) · · · (tn 1,n ) , (B.17) 2 i=a i=a where we have already taken s ! 1/2 limit. Finally, together with the expansion (B.4), one can see that the n-th order contribution to the collective coordinate action (2.13) is given by Z n ! n ! N B1 Y X 2 S[f ] = dt1 · · · dtn "(t1 ) · · · "(tn ) @ ti @ ti (t1n ) · · · (tn 1,n ) 4nJ i=1 i=1 Z N B1 ( 1)n 000 ⇣ 0 ⌘n 1 = dt " (t) " (t) . (B.18) 2J 2 This result can be summed over for all order to get Z N B1 S[f ] = dt Sch(f ; t) , (B.19) 2J where ✓ ◆ f 000 (t) 3 f 00 (t) 2 Sch(f ; t) = 0 . (B.20) f (t) 2 f 0 (t) To see this correspondence, one first rewrites the Schwarzian derivative by integration by parts as Z 1 f 000 (t) dt Sch(f ; t) = . (B.21) 2 f 0 (t) Then, we use f (t) = t + "(t) and expand the Schwarzian derivative by powers of " as Z Z ( 1)n 000 ⇣ 0 ⌘n 1 1 X dt Sch(f ; t) = dt " (t) " (t) . (B.22) 2 n=1 – 46 – This expansion completely agrees with the result found in Eq.(B.18). Finally as a reference, we give a relation of our coefficients to the coefficients ↵S and ↵G defined in [10]: ✓ ◆ ✓ ◆ ↵ J J = B1 = 2↵S = b ↵G , (B.23) 12⇡ J J p q where J = q 1 J. 2 2 C Schrodinger Equation In this appendix, we consider the equation of f (y), which is the Schr¨odinger equation: h i @y2 ± V (y) f (y) = E f (y) , (C.1) where E is an eigenvalue of the equation. Since we confined the field in L < y < L, we have boundary conditions: f (±L) = 0. The continuation conditions at y = 0 are f (+0) = f ( 0) and the other can be derived by integrating the Schr¨odinger equation (C.1) over ( ", ") and taking limit " ! 0 as f 0 (+0) f 0 ( 0) = ± V f (0) . (C.2) Since the potential of the Schr¨odinger equation is even function, the wave function is either odd or even function of y. (i) odd: For odd parity case, a solution satisfying the boundary conditions at y = ±L is given by ( A sin(k(y L)) (0 < y < L) f (y) = (C.3) A sin(k(y + L)) ( L < y < 0) where k 2 = E. For odd parity solution, to satisfy the boundary condition f (+0) = f ( 0), we need f (±0) = 0. This implies that ⇡n k = , (n = 1, 2, 3, · · · ) (C.4) L Then, the continuity condition (C.2) is automatically satisfied. The normalization constant p is fixed as A = 1/ L. (ii) even: For even parity case, a solution satisfying the boundary conditions at y = ±L is given by ( B sin(k(y L)) (0 < y < L) f (y) = (C.5) B sin(k(y + L)) ( L < y < 0) where k 2 = E. The evenness of the parity guarantees f ( 0) = f (+0). So, we only need to impose the condition (C.2) on this solution. This condition gives an equation 2 ⌥ k = tan(kL) . (C.6) V – 47 – Now we set L = ⇡/2 and V = 3, then we have ⌥(2/3)k = tan(⇡k/2), which is precisely the same transcendental equation determining poles of the q = 4 SYK bi-local propagator (5.22). We denote the solutions of (2/3)k = tan(⇡k/2) by pm , (2m 1 < pm < 2m), (m = 1, 2, 3, · · · ), and the solutions of (2/3)k = tan(⇡k/2) by qm , (2m 2 < qm < 2m 1), (m = 1, 2, 3, · · · ). The normalization constant is fixed as s 2k B = . (C.7) 2kL sin(2kL) Finally, let us prove the orthogonality of the parity even wave function (C.5): Z L dy fm (y)fm0 (y) = m,m0 . (C.8) L Using the solution (C.5) and evaluating the integral in the left-hand side, one obtains  2 sin(L(k k 0 )) sin(L(k + k 0 )) B . (C.9) k k0 k + k0 Now let’s assume k 6= k 0 . Then, the integral result can be rearranged to the form of B2 h i 0 cos(Lk) cos(Lk ) k 0 tan(Lk) k tan(Lk 0 ) = 0, (C.10) k 2 k 02 where the final equality is due to the relation tan(LK) = 2k/3. Next, we consider k = k 0 case. In this case, due to the delta function identity, the result (C.9) is reduced to  2 sin(2Lk) B L k,k0 = k,k0 , (C.11) 2k where for the equality we used Eq.(C.7). Therefore, now we have proven the orthogonality (C.8). D Completeness Condition of Z⌫ In this appendix, we give a derivation of the completeness condition (D.6), which is used to determine the zero-th order propagator (6.18). The linear combination of the Bessel functions is defined by [9] tan(⇡⌫/2) + 1 Z⌫ (x) = J⌫ (x) + ⇠⌫ J ⌫ (x) , ⇠⌫ = , (D.1) tan(⇡⌫/2) 1 which satisfies the Bessel equation ⇥ ⇤ z 2 @z2 + z @z + ! 2 z 2 Z⌫ (|!z|) = ⌫ 2 Z⌫ (|!z|) . (D.2) – 48 – In [9], the orthogonality condition of the linear combination of the Bessel function Z⌫ (D.1) is given by Z 1 dx ⇤ Z⌫ (x) Z⌫ 0 (x) = N⌫ (⌫ ⌫ 0 ) , (D.3) 0 x where N⌫ is defined in (5.15) Since Z⌫ is complete, one can expand any function on the basis of Z⌫ . In particular, we are interested in a delta function Z (x x0 ) = d⌫ ⌫ (x) Z⌫ (|x0 |) , (D.4) where ⌫ (x) is the coefficient of the expansion and the integral symbol of ⌫ is a short-hand notation of a combination of summation over ⌫ = 3/2 + 2n, (n = 0, 1, , 2 · · · ) and integration of ⌫ = ir, (r > 0). One can fix the coefficient ⌫ (x) by multiplying Z⌫⇤0 (|x0 |)/x0 to Eq.(D.4) and integrating over x0 with Eq.(D.3) as Z⌫⇤ (|x|) ⌫ (x) = . (D.5) N⌫ x Therefore, finally we find Z d⌫ ⇤ Z (|x|) Z⌫ (|x0 |) = x (x x0 ) . (D.6) N⌫ ⌫ E Evaluation of the Contour Integral In this appendix, we give a detail evaluation of the continuous and the discrete sums appearing in Eq.(6.19). As we defined before, the integral symbol d⌫ is a short-hand notation of a combination of summation over ⌫ = 3/2 + 2n, (n = 0, 1, 2, · · · ) and integration of ⌫ = ir, (r > 0). Namely, Z d⌫ Z⌫⇤ (|!z|) Z⌫ (|!z 0 |) = I1 + I2 , (E.1) N⌫ ⌫ 2 p2m with 1 X 2⌫ I1 ⌘ J⌫ (|!z|) J⌫ (|!z 0 |) , ⌫2 p2m ⌫= 32 +2n n=0 Z 1 dr r I2 ⌘ Z ⇤ (|!z|) Zir (|!z 0 |) . (E.2) 0 2 sinh(⇡r) r + p2m ir 2 Let us evaluate the continuous sum I2 first. Using the symmetry of the integrand, one can rewrite the integral as Z h i i i1 d⌫ ⌫ I2 = J ⌫ (|!z|) + ⇠ J ⌫ ⌫ (|!z|) J⌫ (|!z 0 |) . (E.3) 2 i1 sin(⇡⌫) ⌫ 2 p2m We evaluate this integral by a contour integral on the complex ⌫ plane by closing the contour in the Re(⌫)> 0 half of the complex plane if z > z 0 . Inside of this contour, we have two – 49 – types of the poles. (i) at ⌫ = pm coming from the coefficient factor. (ii) at ⌫ = 3/2 + 2n, (n = 0, 1, 2, · · · ) coming from ⇠ ⌫ , where ⇠ ⌫ = 1. After evaluating residues at these poles, one obtains ⇡ h i I2 = J pm (|!z|) + ⇠ pm Jpm (|!z|) Jpm (|!z 0 |) 2 sin(⇡pm ) X1 2⌫ 2 J⌫ (|!z|) J⌫ (|!z 0 |) 3 . (E.4) ⌫ p2m ⌫= 2 +2n n=0 Now, one can notice that the second term exactly cancels with the contribution from I1 . One can also repeat the above discussion for z 0 > z case. Therefore, combining these two cases the total contribution is now " ! # 3 ⇡ p m + I1 + I2 = J pm (|!|z > ) + 2 Jpm (|!|z > ) Jpm (|!|z < ) , (E.5) 2 sin(⇡pm ) pm 32 where z > (z < ) is the greater (smaller) number among z and z 0 . Then, the propagator is reduced to X1 Z 1 2 (0) 0 0 1 0 12 0 Bm p2m G (t, z, 0; t , z , 0) = |zz | d! e i!(t t ) 4 1 sin(⇡pm ) p2m + (3/2)2 m=0 " ! # 3 p m + ⇥ J pm (|!|z > ) + 2 Jpm (|!|z > ) Jpm (|!|z < ) . (E.6) pm 32 This agrees with the result given in Eq.(6.22). F EAdS Scalar Propagators In this Appendix, we summarize basic aspects of Euclidean AdS scalar propagators. Let us consider a scalar field Z 1 p h i S = dd+1 x g g µ⌫ @µ @⌫ + m2 2 , (F.1) 2 in Euclidean AdSd+1 Poincare coordinates dx2i + dz 2 ds2 = . (F.2) z2 A propagator satisfies the scalar Green’s equation 1 ( + m2 ) G(xi , z ; x0i , z 0 ) = p d (xi x0i ) (z z0) , (F.3) g where 1 p ⌘ p @µ ( g g µ⌫ @⌫ ) . (F.4) g – 50 – More explicitly, one can write down the equation as h i z 2 @z2 + (d 1)z @z z 2 @i2 + m2 G(xi , z ; x0i , z 0 ) = z d+1 d (xi x0i ) (z z0) . (F.5) d Excluding the scaling behavior of the Green’s function by G(xi , z ; x0i , z 0 ) = z 2 g(xi , z ; x0i , z 0 ), the equation is reduced to h i d+2 z 2 @z2 + z @z + z 2 @i2 ⌫ 2 g(xi , z ; x0i , z 0 ) = z 2 d (xi x0i ) (z z 0 ) , (F.6) with ⌫ 2 ⌘ m2 + d2 /4. In the following, we will give two di↵erent expressions of the o↵-shell scalar propagators and show that these two form of propagators are indeed equaivalent. F.1 p-Integral Form In this subsection, following the discussion in Appendix A of [136], we consider p-integral form of the propagator. First we rewrite the Green’s equation (F.6) as h i d 2 0 0 ⌫ + @ 2 i g(xi , z ; xi , z ) = z 2 d (xi x0i ) (z z 0 ) . (F.7) where 1 ⌫2 ⌫ ⌘ @z2 + @z . (F.8) z z2 The di↵erential operator ⌫ has eigenvalues ⌫ J⌫ (pz) = p2 J⌫ (pz) . (F.9) The Bessel function completeness condition is given by Z 1 0 0 (z z ) = z dp p J⌫ (pz) J⌫ (pz 0 ) . (F.10) 0 Now, we expand the rescaled Green’s function on the Bessel function basis with coefficients g˜p,~k as Z Z 1 dd k i~k·~x g(xi , z ; x0i , z 0 ) = e dp p J⌫ (pz) g˜p,~k (x0i , z 0 ) . (F.11) (2⇡)d 0 Plugging this expression into the Green’s equation (F.7), one can fix the coefficients as d z2 i~k·~ x0 g˜p,~k (xi , z) = e J⌫ (pz) . (F.12) p2 + (~k)2 Hence the o↵-shell form of the Green’s function is given by Z Z 0 0 0 d2 dd k i~k·(~x ~x0 ) 1 p G(xi , z ; xi , z ) = (zz ) d e dp J⌫ (pz) J⌫ (pz 0 ) . (F.13) (2⇡) 0 p 2 + (~k) 2 – 51 – The p integral is evaluated in Appendix F.3. The result is given by Z 1 p dp J⌫ (pz) J⌫ (pz 0 ) = K⌫ (kz > ) I⌫ (kz < ) , (F.14) 0 p2 + (~k)2 p where k ⌘ ~k 2 and z > (z < ) is the greater (smaller) number among z and z 0 . Therefore, the Green’s function is now Z d dd k i~k·(~x ~x0 ) G(xi , z ; x0i , z 0 ) = (zz 0 ) 2 e K⌫ (kz > ) I⌫ (kz < ) . (F.15) (2⇡)d F.2 ⌫-Integral Form In this subsection, we express the Green’s equation (F.6) as h i d+2 k ⌫ 2 g(xi , z ; x0i , z 0 ) = z 2 d (xi x0i ) (z z0) . (F.16) where k ⌘ z 2 @z2 + z @z z 2 k2 . (F.17) The di↵erential operator k has eigenvalues k Ki¯⌫ (kz) = ⌫¯2 Ki¯⌫ (kz) . (F.18) The modified Bessel function completeness condition is given by Z 1 ⇡2 d⌫ ⌫ sinh(⇡⌫) Ki⌫ (x) K i⌫ (y) = x (x y) . (F.19) 0 2 Now, we expand the rescaled Green’s function on the basis of the modified Bessel functions with coefficients g˜⌫¯,~k as Z Z 0 0 dd k i~k·~x 1 g(xi , z ; xi , z ) = e d¯ ⌫ ) Ki¯⌫ (kz) g˜⌫¯,~k (x0i , z 0 ) . ⌫ ⌫¯ sinh(⇡¯ (F.20) (2⇡)d 0 Plugging this expression back into the Green’s equation (F.16), one can fix the coefficients as d 2 z0 2 ~ 0 g˜⌫¯,~k (x0i , z 0 ) = 2 2 2 e ik·~x K i¯⌫ (kz 0 ) . (F.21) ⇡ ⌫¯ + ⌫ Hence the o↵-shell form of the Green’s function is given by Z Z 0 0 2 0 d2 dd k i~k·(~x ~x0 ) 1 ⌫¯ 0 G(xi , z ; xi , z ) = 2 (zz ) d e d¯ ⌫ 2 sinh(⇡¯ ⌫ ) Ki¯⌫ (kz) K ⌫ (kz i¯ ). ⇡ (2⇡) 0 ⌫¯ + ⌫ 2 (F.22) The ⌫¯ integral is evaluated in Appendix F.4. The result is Z 1 ⌫¯ 0 ⇡2 d¯⌫ 2 2 sinh(⇡¯ ⌫ ) K i¯ ⌫ (kz) K i¯ ⌫ (kz ) = K⌫ (kz > ) I⌫ (kz < ) . (F.23) 0 ⌫ ¯ + ⌫ 2 Therefore, the Green’s function is reduced to Z 0 0 0 d2 dd k i~k·(~x x0 ) ~ G(xi , z ; xi , z ) = (zz ) e K⌫ (kz > ) I⌫ (kz < ) . (F.24) (2⇡)d This agrees with the result of the previous subsection. – 52 – F.3 p-Integral In this subsection, we evaluate the p integral (F.14) Z 1 p I ⌘ dp J⌫ (pz) J⌫ (pz 0 ) . (F.25) 0 p 2 + (~k) 2 In order to use the contour integral, it is convenient to extend the integration region to Z 1 1 |p| I = dp J⌫ (|p|z) J⌫ (|p|z 0 ) . (F.26) 2 1 p + (~k)2 2 Here, the absolute value is understood in the sense of real variables, not in the sense of complex variables (i.e. |p| = p sgn(Re(p))). Since the large argument behavior (|z| 1) of the Bessel function is r ⇣ 2 ⇡⌫ ⇡⌘ J⌫ (z) = cos z , (F.27) ⇡z 2 4 we cannot close the contour neither upper nor lower half of the complex p plane. To manage this, we decompose the Bessel function into the Hankel functions: 1 ⇣ (1) ⌘ J⌫ (z) = H⌫ (z) + H⌫(2) (z) . (F.28) 2 This is because the Hankel functions have nice asymptotic behaviors in |z| 1 r h ⇣ 2 ⇡⌫ ⇡ ⌘i H⌫(1) (z) = exp i z , ⇡z 2 4 r h ⇣ 2 ⇡⌫ ⇡ ⌘i H⌫(2) (z) = exp i z . (F.29) ⇡z 2 4 Now, we have four terms in the p integral Z  1 1 |p| I = dp H⌫(1) (|p|z) H⌫(1) (|p|z 0 ) + H⌫(1) (|p|z) H⌫(2) (|p|z 0 ) 8 1 p2 + (~k)2 + H⌫(2) (|p|z) H⌫(1) (|p|z 0 ) + H⌫(2) (|p|z) H⌫(2) (|p|z 0 ) . (F.30) We label each term by {I1 , I2 , I3 , I4 } in pthe order of appearing in (F.30). In order to pick up a pole at p = ±ik + 0 , where k ⌘ ~k 2 , the contour of the complex p integral is closed + in upper or lower half of the complex plane for each term as ( ( upper (z > z 0 ) lower (z > z 0 ) I1 = upper I2 = I3 = I4 = lower . lower (z < z 0 ) upper (z < z 0 ) (F.31) – 53 – Evaluating the contour integral by residue theorem, each term gives ⇡i (1) I1 = H (ikz) H⌫(1) (ikz 0 ) , 8 ⌫ ( (1) (2) + ⇡i 8 H⌫ (ikz) H⌫ (ikz ) 0 (z > z 0 ) I2 = (1) (2) ⇡i 8 H⌫ ( ikz) H⌫ ( ikz 0 ) (z < z 0 ) ( (2) (1) ⇡i 8 H⌫ ( ikz) H⌫ ( ikz 0 ) (z > z 0 ) I3 = (2) (1) + ⇡i 8 H⌫ (ikz) H⌫ (ikz 0 ) (z < z 0 ) ⇡i (2) I4 = H ( ikz) H⌫(2) ( ikz 0 ) . (F.32) 8 ⌫ We note that we can combine I2 and I3 into more compact notation as ⇡i h (1) i I2 + I3 = H⌫ (ikz > ) H⌫(2) (ikz < ) H⌫(1) ( ikz < ) H⌫(2) ( ikz > ) , (F.33) 8 where z > (z < ) is the greater (smaller) number among z and z 0 . Therefore, now the entire p integral is written as ⇡i h (1) i I = H⌫ (ikz > ) J⌫ (ikz < ) H⌫(2) ( ikz > ) J⌫ ( ikz < ) . (F.34) 4 Finally, using ⌫ I⌫ (z) = i J⌫ (iz) = i⌫ J⌫ (iz) , (F.35) and ⇡ ⌫+1 (1) ⇡ ⌫ 1 K⌫ (z) = i H⌫ (iz) = i H⌫(2) ( iz) , (F.36) 2 2 one obtains I = K⌫ (kz > ) I⌫ (kz < ) . (F.37) F.4 ⌫-Integral In this Appendix, we evaluate the ⌫¯ integral (F.23) Z 1 ⌫¯ 0 I ⌘ d¯ ⌫ 2 sinh(⇡¯⌫ ) Ki¯⌫ (kz) K ⌫ (kz i¯ ). (F.38) 0 ⌫¯ + ⌫ 2 We use the same latter I as in previous section to denote di↵erent integrals, but it is obvious that I in Appendix F.3 defined in (F.25), while I in this section is defined in (F.38). We first consider z > z 0 case. It is convenient to decompose the second kind of modified Bessel functions into the first kind of modified Bessel functions by ⇡i Ii¯⌫ (kz) I i¯⌫ (kz) Ki¯⌫ (kz) = . (F.39) 2 sinh(⇡¯ ⌫) – 54 – Note that even though sinh(⇡¯ ⌫ ) in the denominator becomes zero at ⌫¯ = in, (n 2 Z), these points are not poles of Ki¯⌫ , because I n (x) = In (x) for n 2 Z. Now, the ⌫¯ integral is Z h ih i ⇡2 1 ⌫¯ 1 I = d¯ ⌫ 2 Ii¯⌫ (kz) I ⌫ (kz) i¯ Ii¯⌫ (kz 0 ) I ⌫ (kz i¯ 0 ) 8 1 ⌫¯ + ⌫ 2 sinh(⇡¯ ⌫) Z h i ⇡2 1 ⌫¯ 1 = d¯ ⌫ 2 Ii¯⌫ (kz) I ⌫ (kz) i¯ Ii¯⌫ (kz 0 ) 4 ⌫¯ + ⌫ 2 sinh(⇡¯ ⌫) Z 1 1 ⇡i ⌫¯ = d¯ ⌫ 2 2 Ki¯⌫ (kz) Ii¯⌫ (kz 0 ) . (F.40) 2 1 ⌫ ¯ + ⌫ We note that in this combination of Ki¯⌫ Ii¯⌫ is well defined in both z ! 1 and z 0 ! 0 limits. This can be seen from the asymptotic behaviors of the Bessel functions r ⇡ Ki¯⌫ (kz) ⇠ e kz , (z ! 1) (F.41) 2kz ✓ 0 ◆i¯⌫ 0 1 kz Ii¯⌫ (kz ) ⇠ , (z 0 ! 0) (F.42) (1 + i¯ ⌫) 2 The opposite combination is ill-defined because the limit Ii¯⌫ (z ! 1) leads to divergence. In order to evaluate the ⌫¯ integral (F.40) as a contour integral on the complex ⌫¯ plane, we need to decide whether we will close the contour in the upper half or lower half of the complex plane. This can be determined from (F.42). The limit z 0 ! 0 is converged only if we close the contour in the lower half of the complex ⌫¯ plane. Hence now the residue theorem at the pole ⌫¯ = i⌫ gives ⇡2 I = K⌫ (kz) I⌫ (kz 0 ) . (F.43) 2 This can be generalized to include z 0 > z case also as ⇡2 I = K⌫ (kz > ) I⌫ (kz < ) . (F.44) 2 where z > (z < ) is the greater (smaller) number among z and z 0 . G Unit Normalized EAdS/dS Wave Functions The unit-normalized Euclidean AdS2 wave function is given by 1 i!⌧ EAdS2 (⌧, z) = ↵⌫ z 2 e K⌫ (|!|z) , (G.1) where the normalization factor can be chosen as r ⌫ sin(⇡⌫) ↵⌫ = i . (G.2) ⇡3 – 55 – Then from the Bessel K⌫ orthogonality condition (H.2), the wave function is unit normalized: Z 1 Z 1 dz ⇤ d⌧ 2 !,⌫ (⌧, z) ! 0 ,⌫ 0 (⌧, z) = (! ! 0 ) (⌫ ⌫ 0 ) . (G.3) 1 0 z The Lorentzian dS2 wave function is given by 1 i!t dS2 (⌘, t) = ⌫ ⌘2 e Z⌫ (|!|⌘) . (G.4) Here, let us only consider the continuous modes (⌫ = ir). Now choosing the normalization factor as r ⌫ ⌫ = , (G.5) 4⇡ sin(⇡⌫) then the wave function is unit normalized for the continuous modes as Z 1 Z 1 d⌘ ⇤ dt (⌘, t) !0 ,⌫ 0 (⌘, t) = (! ! 0 ) (⌫ ⌫ 0 ) , for (⌫ = ir) (G.6) 1 0 ⌘ 2 !,⌫ We note that sin ⇡⌫ ↵⌫ = 2i ⌫ . (G.7) ⇡ H Completeness and Orthogonality of Ki⌫ In this appendix, we derive the completeness condition of modified Bessel function of the second kind Z 1 ⇡2 d⌫ ⌫ sinh(⇡⌫) Ki⌫ (x) Ki⌫ (y) = x (x y) , (H.1) 0 2 and the orthogonality condition Z 1 dx ⇡ 2 (⌫ ⌫ 0 ) Ki⌫ (x) Ki⌫ 0 (x) = , (H.2) 0 x 2 ⌫ sinh(⇡⌫) where x, y > 0. We start with defining Z 1 I ⌘ d⌫ ⌫ sinh(⇡⌫) Ki⌫ (x) Ki⌫ (y) . (H.3) 0 Using the integral representations of the modified Bessel function Z 1 1 Ki⌫ (x) = ds sin(x sinh s) sin(⌫s) , sinh( ⇡⌫ 2 ) 0 Z 1 1 Ki⌫ (y) = dt cos(y sinh t) cos(⌫t) , (H.4) cosh( ⇡⌫ 2 ) 0 one can rewrite I as Z Z Z 1 1 1 1 I = d⌫ ⌫ ds dt sin(⌫s) cos(⌫t) sin(x sinh s) cos(y sinh t) , (H.5) 4 1 1 1 – 56 – where using the symmetry of ⌫, s and t, we extended the integration regions to ( 1, 1). The ⌫ integral is evaluated as Z 1 @ ⇣ ⌘ d⌫ ⌫ sin(⌫s) cos(⌫t) = ⇡ (s + t) + (s t) . (H.6) 1 @s Therefore, now Z Z ⇡ 11 @ ⇣ ⌘ I = ds dt (s + t) + (s t) sin(x sinh s) cos(y sinh t) 4 1 @s Z 1 Z 11 ⇣ ⌘ @ ⇡ = ds dt (s + t) + (s t) sin(x sinh s) cos(y sinh t) 4 1 1 @s Z 1 ⇡x = ds cosh s cos(x sinh s) cos(y sinh s) . (H.7) 2 1 Changing the integration variable to u = sinh s, Z ⇡x 1 I = du cos(xu) cos(yu) 2 1 ⇡2 h i = x (x + y) + (x y) . (H.8) 2 Since x, y > 0, we keep only (x y) term and obtain the completeness condition Eq.(H.1). 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