Three dimensional view of the SYK/AdS duality

Journal of High Energy Physics, Sep 2017

Abstract We show that the spectrum of the SYK model can be interpreted as that of a 3D scalar coupled to gravity. The scalar has a mass which is at the Breitenholer-Freedman bound of AdS2, and subject to a delta function potential at the center of the interval along the third direction. This, through Kaluza-Klein procedure on AdS2 × (S 1)/Z 2, generates the spectrum reproducing the bi-local propagator at strong coupling. Furthermore, the leading 1/J correction calculated in this picture reproduces the known correction to the poles of the SYK propagator, providing credence to a conjecture that the bulk dual of this model can be interpreted as a three dimensional theory.

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Three dimensional view of the SYK/AdS duality

JHE Three dimensional view of the SYK/AdS duality Sumit R. Das 0 1 3 Antal Jevicki 0 1 2 Kenta Suzuki 0 1 2 0 182 Hope Street, Providence, RI 02912 , U.S.A 1 Lexington , KY 40506 , U.S.A 2 Department of Physics, Brown University 3 Department of Physics and Astronomy, University of Kentucky , USA 4 V (y) with eigenvalue We show that the spectrum of the SYK model can be interpreted as that of a 3D scalar coupled to gravity. The scalar has a mass which is at the Breitenholer-Freedman bound of AdS2, and subject to a delta function potential at the center of the interval along the third direction. This, through Kaluza-Klein procedure on AdS2 the spectrum reproducing the bi-local propagator at strong coupling. Furthermore, the leading 1=J correction calculated in this picture reproduces the known correction to the poles of the SYK propagator, providing credence to a conjecture that the bulk dual of this model can be interpreted as a three dimensional theory. AdS-CFT Correspondence; Gauge-gravity correspondence; 1/N Expansion - HJEP09(217) 1 Introduction 2 3 3D interpretation thus providing an example of the butter y e ect [40{49]. Like vector models, the SYK model is solvable at large N . Vector models, in general, at large N can be expressed in terms of bi-local elds, and it was proposed in [50] that these bi-local elds in fact provide a bulk construction of the dual higher spin theory [51], with the pair of coordinates in the bi-local combining to provide the coordinates of the emergent AdS space-time. The simplest proposal of [50] was implemented nontrivially in three dimensions giving an understanding of bulk higher spin elds [52{54]. In the one dimensional SYK case [10, 11] such bulk mapping is realized in its simplest form, with the bi-local times mapped to AdS2 space-time, thus providing an elementary example (in addition to the c = 1 matrix model [55]) of how a Large N quantum mechanical model grows an additional dimension. { 1 { Nevertheless, and despite great interest, the precise bulk dual of the SYK model is still ununderstood. It has been conjectured in [56{59] that the gravity sector of this model is the Jackiw-Teitelboim model [60, 61] of dilaton-gravity with a negative cosmological constant, studied in [62], while [63] provides strong evidence that it is actually Liouville theory. (See also [64{69]) It is also known that the matter sector contains an in nite tower of particles [8{10]. Recently, couplings of these particles have been computed by calculating six point functions in the SYK model [ 70 ]. In this paper, we provide a three dimensional interpretation of the bulk theory. The zero temperature SYK model corresponds to a background AdS2 I, where I = S1=Z2 is a nite interval whose size needs to be suitably chosen. There is a single scalar eld coupled to derivative of the eld at the other end.1 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 [57]. The strong coupling limit of the SYK model corresponds to a trivial metric in the third direction, while at nite 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 function2 with both points at the center of the interval is in precise agreement with the strong coupling bi-local propagator, using the simplest identi cation of the AdS coordinates proposed in [50]. For nite coupling, we adopt the proposal of [57, 58], and show that to order 1=J , the poles of the propagator shift in a manner consistent with the explicit results in [9]. In section 2, we review relevant aspects of the bilocal formulation of the model. In section 3, we discuss the three dimensional interpretation. Section 4 contains some concluding remarks. 2 Overview of SYK In this section, we will give a brief review of the Large N formalism and results along [10, 11]. The Sachdev-Ye-Kitaev model [5] is a quantum mechanical many body system with all-to-all interactions on fermionic N sites (N 1), represented by the Hamiltonian H = Jijkl i j k l ; (2.1) 1 N X 4! i;j;k;l=1 { 2 { 1We thank Edward Witten for a clari cation on this point. 2Note that this two point function is not the same as the standard AdS2 propagator. We thank Juan Maldacena for discussions about this point. where i are Majorana fermions, which satisfy f i; j g = ij . The coupling constant Jijkl are random with a Gaussian distribution with width J . The generalization to analogous q-point interacting model is straightforward [5, 9]. After the disorder averaging for the random coupling Jijkl, there is only one e ective coupling J in the e ective action. The model is usually treated by replica method. One does not expect a spin glass state in this model [7] so that we can restrict to the replica diagonal subspace [10]. The Large N theory is simply represented through a (replica diagonal) bi-local collective eld: where we have suppressed the replica index. The corresponding path-integral is 1 N X N i=1 where Scol is the collective action: N 2 + Tr log J 2N Z 2q dt1dt2 q(t1; t2) : Here 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. One also has an appropriate order O(N 0) measure . This action being of order N gives a systematic G = 1=N expansion, while the measure found as in [ 72 ] begins to contribute at one-loop level (in 1=N ). Other formulations can be employed using two bi-local elds. These can be seen to reduce to Scol after elimination. In the above action, the rst linear term represents a conformal breaking term, while the other terms respect conformal symmetry. In the IR limit with strong coupling jtjJ 1, the collective action is reduced to the critical action which exhibits an emergent reparametrization symmetry Sc[ ] = Tr log N 2 J 2N Z 2q dt1dt2 q(t1; t2) ; (t1; t2) ! f (t1; t2) = f 0(t1)f 0(t2) (f (t1); f (t2)) ; 1 q with an arbitrary function f (t). This symmetry is responsible for the appearance of zero modes in the strict IR critical theory. This problem was addressed in [10] with analog of the quantization of extended systems with symmetry modes [73]. The above symmetry mode representing time reparametrization can be elevated to a dynamical variable introduced according to [ 74 ] through the Faddeev-Popov method, leading to a Schwarzian action for this variable [11] proposed by Kitaev, and established rst at quadratic level in [9]: S[f ] = N 12 B1 , with B1 representing the strength of the rst order correction, established in numerical studies by Maldacena and Stanford [9]. The details of the non-linear evaluation are give in [11]. For the rest of this paper we proceed with q = 4. Fluctuations around the critical IR background can be studied by expanding the bi-local eld as [10] tj . With a simple coordinate transformation the bi-local eld (t1; t2) in a complete orthonormal basis can be then considered as a eld in two dimensions (t; z). Expand the uctuation eld as t = (t1 + t2) ; z = (t1 t2) ; u ;!(t; z) = sgn(z) ei!t Z (j!zj) ; kernel [8]. Then, the quadratic action can be written as S(2) = 3J 32p X ;! N ~ ;! g~( ) 1 ~ ;! ; where the normalization factor N is and the kernel is given by After a eld rede nition [10] the e ective action can be written as N = 8<(2 ) 1 :2 1 sin g~( ) = for for = 3=2 + 2n = ir ; 2 3 cot 2 : e Sm = 1 3J Z (t; z) hge(pDB) 1 i (t; z) ; { 4 { featuring the Bessel operator The operator DB is in fact closely related to the laplacian on AdS2, DB rAdS2 = pzDB 1 p z 1 4 where t and z in (2.10) are the Poincare coordinates in AdS2. This, therefore, realizes the naive form of the proposal of [50]. However the action for (t; z) is non-polynomial in To understand the implications of this, consider the bi-local propagator, rst evaluated in [8]. From the above e ective action, one has that the poles are determined as solutions of g~( ) = 1, they represent a sequence denoted by pm as = tan ; 2m + 1 < pm < 2m + 2 (m = 0; 1; 2; ) (2.20) Therefore, the bi-local propagator is written as residues of where z>(z<) is the greater (smaller) number among z and z0. The residue function is derivatives. 2pm Since that pm are zeros of ge( ) 1 g~( ) 1 e g( ) =pm 1 = 3p2m 1, near each pole pm, we can approximate as where fm can be determined from residue of 1=(g( ) 1) at = pm. Explicitly evaluating these residues, the inverse kernel is written as an exact expansion m=1 [p2m + (3=2)2][ pm 14 , (m > 0) in AdS2: The e ective action near a pole labelled by m is that of a scalar eld with mass, M m2 = g d2x where the metric g is given by g = diag( 1=z2; 1=z2). 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 eld. In other words the bi-local representation e ectively packs an in nite product of AdS Laplacians with growing masses. An illustration of how this can happen is given in the appendix A, relating to the scheme of Ostrogradsky. It is this feature which leads to the suggestion that the theory should be represented by an enlarged number of elds, or equivalently by an extra Kaluza-Klein dimension. For nite coupling, the poles of the propagator is shifted. This has been calculated by [9] in a 1=J expansion. 2 (pm)2 fm ; e 6 p3m { 5 { 2 1 p2m 1 4 2m ; : : (2.18) (2.19) (2.22) (2.23) (2.24) (2.25) 3D interpretation ary terms) where 0 is a constant, and by AdS2 with a metric and a dilaton According to [57] and [58], the bulk dual of the SYK model involves Jackiw-Teitelboim theory of two dimensional dilaton gravity, whose action is given by (upto usual boundSJT = 1 is a dilaton eld. The zero temperature background is given where a is a parameter which scales as 1=J . 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 extremal black holes, and the AdS2 is the near horizon geometry [57]. The three dimensional metric, with the dilaton being the third direction, is given by ds2 = 1 z2 dt2 + dz2 + 1 + dy2 : a 2 z This is in fact the near-horizon geometry of a charged extremal BTZ black hole. 3.1 Kaluza-Klein decomposition We will now show that the in nite sequence of poles in the previous section from the KaluzaKlein tower of a single scalar in a three dimensional metric (3.4) where the direction y is an interval L < y < L. The action of the scalar is S = where V (y) = V (y), with the constant V and the size L to be determined. This is similar to Horava-Witten compacti cation on S1=Z2 [75] with an additional delta function potential.3 The scalar satis es 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: one can rewrite the action (3.5) in the form of 3See also [76, 77]. We are grateful to Cheng Peng for bringing this to our attention. (t; z; y) = e i!t !(z; y) ; { 6 { where D0 is the a-independent part and D1 is linear in a: z2 + ! 2 D1 = a z 2 m20 z2 V (y) ; 1 : Here, we neglected higher order contributions of a. The eigenfunctions of D0 can be clearly written in the form !(z; y) = !(z) fk(y) : Then fk(y) is an eigenfunction of the Schrodinger operator k2. This is a well known Schrodinger problem: the eigenfunctions and the eigenvalues are HJEP09(217) presented in detail in appendix B. After solving this part, the kernels are reduced to 2 m20 + p2m z2 ; D1 = a z 2 m20 2 qm z2 where pm are the solutions of (2=V )k = tan(kL) while qm are the expectation values of V (y) operator respect to fpm . If we choose V = 3 and L = 2 the solutions of (3.11) agree precisely with the strong coupling spectrum of the SYK model given by g~( ) = 1, as is clear from (2.14) and (2.16). 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. 3.2 Evaluation of G(0) We start by determining the leading, zero-th order G(0) propagator obeying D0 G(!0;)!0 (z; y; z0; y0) = (z z0) (y y0) (! + !0) : We rst separate the scaling part of the propagator by G(0) = p Expanding in a basis of eigenfunctions fk(y), z Ge(0) and multiplying z2. e G(0)(z; y; !; z0; y0; !0) = X fk(y)fk0 (y0)Ge!;k;!0;k0 (z; z0) (0) k;k0 The Green's function Ge(!0;)k;!0;k0 (z; z0) is clearly proportional to (k k0) and satis es the equation 2 i G(0) 0 e!;k;!0;k0 (z; z0) = z 32 (z z0) (! + !0) (k k0) : (3.14) where we have de ned 2 0 k2 + m02 + 1=4: { 7 { (3.8) (3.9) ; (3.10) (3.11) (3.12) (3.13) (3.15) e Then, substituting this expansion into the Green's equation (3.12) and using eqs. (C.4) and (C.2), one can x the coe cient g(0). Finally, the -integral form of the propagator is 1=4, which is the BF bound of AdS2, we have 02 = p2m, and the equation which determine pm, (3.11) is precisely the equation which determines the spectrum of the SYK theory found in [8, 10]. With this choice, the real space zeroth order propagator in three dimensions is G(0)(t; z; y; t0; z0; y0) = m=0 jzz0j 21 X1 fpm (y)fpm (y0) Z d! 2 e i!(t t0)Z d N Z (j!zj) Z (j!z0j) : 2 p2m We now show that the above propagator with y = y0 = 0 is in exact agreement with the bi-local propagator of the SYK model. The Green's function with these end points is The operator which appears in (3.14) 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 and the integral over the continuous values can be now performed exactly as in the calculation of the SYK bi-local propagator [10]. Closing the contour for the continuous integral in Re( )! 1, one of this contour. (1): = 2n + 3=2, (n = 0; 1; 2; nds that there are two types of poles inside ), 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 4This represents a modi ed set of wavefunctions with boundary conditions at z ! 1 in contrast to the standard AdS wavefunctions. { 8 { are presented in appendix D. Therefore, the nal remaining contribution is just written as residues of 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. (2.21). 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. In this section, we study the rst order eigenvalue shift due to D1 by treating this operator as a perturbation onto the D0 operator. The result will con rm the duality a = 1=J , where a is de ned in the dilaton background (3.3) 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. (3.10). The eigenfunction of D0 operator is and using the orthogonality condition (C.3), its matrix element in the space is found as = a dz jzj 2 Z 0 (j!zj)D1 jzj 2 Z (j!zj) 1 0 m02 aj!j qm2 + dz Z 1 0 z Z 0 (j!zj) h dz Z 0 (j!zj)Z (j!zj) z2 J +1(j!zj) J 1(j!zj)i : (3.28) { 9 { 1 jzj 2 Z (j!zj) ; N h 2 (m02 + p2m + 14 )i ; 0 : Now following the rst order perturbation theory, we are going to determine the rst order eigenvalue shift. Using the Bessel equation, the action of D1 on the D0 eigenfunction (3.23) is found as 1 D1 jzj 2 Z (j!zj) = a 1 jzj 2 " For the derivative term, we use the Bessel function identity (for example, see 8.472 of [78]) to obtain J 1(x) x Z (j!zj) j!j J +1(j!zj) h J 1(j!zj)i : Therefore, now the matrix element is determined by integrals (3.23) (3.24) (3.25) (3.26) (3.27) 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, the linear combination of the Bessel function is reduced to a single Bessel function as Z (x) = J (x). Since sin 2 ( ( + )2 we have now found the matrix element for the discrete mode is given by Now, we compare this result with the 1=J rst order eigenvalue shift of the SYK model, which is for the zero mode found in [9] as 2aj!j sin 2 ( 0 1) " 2 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 = eigenvalue shift is found as 1=4 as before, the zero mode rst order k(2; !) = 1 ; (zero temperature) (3.32) where K 2:852 for q = 4. The !-dependence of our result (3.31) thus agrees with that of the SYK model. Furthermore, this comparison con rms 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 rst O(a) order shift for the pole as = + 6 aj!j 2 + q02 + O(a2) : For the zero mode part (m = 0) of the on-shell propagator in eq. (D.6), the leading order is O(1=a). This contribution comes from the coe cient 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 G(z0e)ro mode(t; z; 0; t0; z0; 0) = 4a (2 + q0) This agrees with the order O(J ) contribution of the SYK bi-local propagator of Maldacena/Stanford [9]. 2 aj!j (2 + q02) : K j!j + 2 J 3 2 9 (3.30) (3.31) (3.33) (3.34) In this paper we have provided 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 eld can be exactly reproduced by that of a scalar eld living in AdS2 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 S1=Z2 of the propagator are consistent with the results of the SYK model [9]. We would like to emphasize that there are two aspects of this 3d perspective. The rst concerns the agreement of the strong coupling spectrum and the form of the leading nite J correction. This agreement may very well follow from more general considerations [79].5 The second aspect is that the exact large-J propagator agrees, and the form of the leading enhanced correction for large but nite J agrees as well. We believe that this second aspect is rather non-trivial and intruiging and the implications are yet to be fully understood. This three dimensional view is a good way of re-packaging the in nite tower of states of the SYK model. Our analysis was done at the linearized level and the 3D gravity is only used to x the background, as we did not treat them dynamically.6 Demonstrating full duality at the nonlinear level is an open problem. In particular it would be interesting if the three point function of bi-locals [ 70 ] has a related 3d interpretation. Acknowledgments We acknowledge useful conversations with Robert de Melo Koch, Animik Ghosh, Juan Maldacena, Cheng Peng, Al Shapere, Edward Witten and Junggi Yoon on the topics of this paper. We also thank Wenbo Fu, Alexei Kitaev, Grigory Tarnopolsky and Jacobus Verbaarschot for relevant discussions on the SYK model. This work is supported by the Department of Energy under contract DE-SC0010010. The work of SRD is partially supported by the National Science Foundation grant NSF-PHY-1521045. AJ would like to thank the Galileo Galilei Institute for Theoretical Physics (GGI) for the hospitality and INFN for partial support during the completion of this work, within the program \New Developments in AdS3/CFT2 Holography". We also learned of possibly related work by Marika Taylor [80]. A Actions non-polynomial in derivatives To illustrate how an action which is non-polynomial in derivatives can arise let us start with the example of N decoupled elds 5We thank the referee for bringing this paper to our attention. (A.1) N X n=1 L = 'nDn'n : One can then introduce elds ' = which represents a transformation preserving the determinant: Integrating 's out, one eventually ends up with the e ective Lagrangian N Y n=1 Dn = Db + Dbn : N 1 Y n=1 L' = ' PN QnN=1 Dn n1< <nN 1 Dn1 DnN 1 ! ' : (A.2) (A.3) (A.4) (A.5) (A.6) (B.1) (B.2) HJEP09(217) n N 1 X DnN 1 Dnp Dnp 1 ; ; Here all the poles are contained in the higher-order laplacian, as in eq. (2.16). The opposite procedure of going from this e ective action with the N -th order Laplacian to the rst one, requires introducing N 1 extra elds, which would correspond to the scheme introduced by Ostrogradsky. B Schrodinger equation In this appendix, we consider the equation of f (y), which is the Schrodinger equation: h where E is an eigenvalue of the equation. Since we con ned the eld 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 Schrodinger equation (B.1) over ( "; ") and taking limit " ! 0 as f 0(+0) f 0( 0) = V f (0) : Since the potential of the Schrodinger 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 = where k2 = E. For odd parity solution, to satisfy the boundary condition f (+0) = f ( 0), we need f ( 0) = 0. This implies that k = n L ; (n = 1; 2; 3; ) (B.4) Finally, let us prove the orthogonality of the parity even wave function (B.5): Using the solution (B.5) and evaluating the integral in the left-hand side, one obtains Z L L dy fm(y)fm0 (y) = m;m0 : B2 sin(L(k k k0 k0)) sin(L(k + k0)) k + k0 : constant is xed as A = 1=pL. Then, the continuity condition (B.2) is automatically satis ed. The normalization (ii) Even: for even parity case, a solution satisfying the boundary conditions at y = L is given by f (y) = (B sin(k(y L)) (0 < y < L) B sin(k(y + L)) ( L < y < 0) where k2 = E. The evenness of the parity guarantees f ( 0) = f (+0). So, we only need to impose the condition (B.2) on this solution. This condition gives an equation k = tan(kL) : 2 V s 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 bilocal propagator (2.20). We denote the solutions of (2=3)k = tan( k=2) by pm, (2m + 1 < pm < 2m + 2), (m = 0; 1; 2; ). The normalization constant is xed as B = 2k 2kL sin(2kL) : (B.3) (B.5) (B.6) (B.7) (B.8) (B.9) Now let's assume k 6= k0. Then, the integral result can be rearranged to the form of k2 B2 k02 cos(Lk) cos(Lk0) h k0 tan(Lk) k tan(Lk0) i = 0 ; (B.10) where the nal equality is due to the relation tan(Lk) = 2k=3. Next, we consider k = k0 case. In this case, due to the delta function identity, the result (B.9) is reduced to B2 L sin(2Lk) 2k k;k0 = k;k0 ; where for the equality we used eq. (B.7). Therefore, now we have proven the orthogonality (B.8). C Completeness condition of Z In this appendix, we summarize some properties of the Bessel function Z , which are used to determine the zero-th order propagator (3.18). The linear combination of the Bessel functions is de ned by [8] Z (x) = J (x) + From this orthogonality condition, one can x the normalization for the completeness condition of Z . Namely, dividing each Z by pN , nally we nd the completeness condition as Z (jxj) Z (jx0j) = x (x x0) : D Evaluation of the contour integral In this appendix, we give a detail evaluation of the continuous and the discrete sums appearing in eq. (3.19). As we de ned 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, which satis es the Bessel equation Z (C.1) is given by 2 Z (j!zj) : In [8], the orthogonality condition of the linear combination of the Bessel function with N 2 p2m Z d Z (j!zj) Z (j!z0j) = I1 + I2 ; I1 I2 1 X n=0 2 2 ; 0 2 sinh( r) r2 + p2m Zir(j!zj) Zir(j!z0j) : Now, one can notice that the second term exactly cancels with the contribution from I1. One can also repeat the above discussion for z0 > z case. Therefore, combining these two cases the total contribution is now I1 + I2 = 2 sin( pm) J pm (j!jz>) + Jpm (j!jz>) Jpm (j!jz<) ; (D.5) where z>(z<) is the greater (smaller) number among z and z0. Then, the propagator is reduced to Let us evaluate the continuous sum I2 rst. Using the symmetry of the integrand, one can rewrite the integral as I2 = i Z i1 2 d i1 sin( 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 > z0. Inside of this contour, we have two types of the poles. (i) at = pm coming from the coe cient factor. (ii) at = 3=2 + 2n, (n = 0; 1; 2; ) coming from , where = 1. After evaluating residues at these poles, one obtains I2 = h J pm (j!zj) + pm Jpm (j!zj)i Jpm (j!z0j) This agrees with the result given in eq. (3.22). Open Access. 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Sumit R. Das, Antal Jevicki, Kenta Suzuki. Three dimensional view of the SYK/AdS duality, Journal of High Energy Physics, 2017, 17, DOI: 10.1007/JHEP09(2017)017