Bulk from Bi-locals in Thermo Field CFT

We study the Large $N$ dynamics of the $O(N)$ field theory in the Thermo field dynamics approach. The question of recovering the high temperature phase and the corresponding $O(N)$ gauging is clarified. Through the associated bi-local representation we discuss the emergent bulk space-time and construction of (Higher spin) fields. We note the presence of `evanescent' modes in this construction and also the mixing of spins at finite temperature.


Introduction
The AdS/CFT correspondence with emergent Gravity from the boundary theory offers a framework for understanding deep quantum aspects of black holes [1]. Recently issues concerning the physics at the horizon and applicability of quantum mechanics have been vigorously debated [2]. Of central significance is the understanding of emergent Gravity [3] and its space-time [4].
A particular CFT scheme for understanding the space-time of eternal AdS black holes [5] is the so-called Thermo field dynamics (TFD) where identical copies of the CFT are suggested for right and left regions of a Penrose space-time [6]. In this scenario one at the outset has a question if these (decoupled CFT's) are capable of producing a connected [7] space-time characteristic of a black hole [8,9]. A further very relevant issue concerns the reconstruction of local bulk fields [10,11] from the two boundaries. The ability to accomplish this is central for a possible reconstruction of behind the horizon physics [12] for black holes.
A simple and calculable model of AdS/CFT correspondence is given by vector models [13,14] (in d-dimensions) and Vasiliev type Higher Spin Gravity theories [15][16][17][18] (in d + 1). For these tractable class of field theories an extensive study established [19] agreement for all spins [20,21]. Furthermore, BTZ type black holes dual to CFT 2 have been investigated in detail [22][23][24][25]. The reconstruction of Higher Spins and AdS space-time through bi-local [20,[26][27][28][29][30][31] fields was accomplished in a systematic 1/N expansion scheme. In this paper we study the Thermo field [32,33] theory of O(N ) vector models with the intent of understanding the associated dual bulk degrees of freedom [34]. For this we follow the earlier established framework of using collective bi-local degrees of freedom of the vector model and working out their Large N expansion. It will be shown that this leads to a linearized set of equations that produce modes which can be put in agreement with bulk (higher-spin) modes in d + 1 dimensional space-time. Of particular interest is the recovery of so-called 'evanescent' modes [35] that characterize black hole type backgrounds whose relevance and physical meaning was recently clarified in the works of [35][36][37].
We demonstrate that bi-local fields contain the information for generating all the physical modes in the dual space time, including the 'evanescent' modes. The question of bulk recovery from the disjoint CFT's is therefore further illuminated by present construction. It is seen in the collective construction that decoupled CFT's do not necessarily lead to decoupled collective fields. This issue is seen to be closely related to the issue of gauging the O(N ) symmetry as to produce singlet states in the spectrum as we explain. Finally the reconstructed bulk-fields involve mixing of all spins, and as such are more complex than in particular the construction suggested in [12]. We will elaborate on these differences in the Conclusions.
The outline of this paper is as follows. In Section 2 we develop the collective description of the TFD for (free) vector models. Fluctuations of bi-locals and their bulk interpretation is given systematically in Section 3. We give our conclusions in Section 4.

Thermofield Dynamics
Our CFT is a free O(N ) vector field theory (analogous constructions apply also for the interacting IR fixed point) The finite temperature theory,in the real time formalism of Schwinger (and Keldyish) [38,39] is based on a closed time path which implies doubling of degrees of freedom. This TFD (Thermo field dynamics) of the O(N ) vector model, is associated with the Hamiltonian where H and H is Hamiltonian of original and copied system.
In TFD formalism, one defines a new vacuum |0(β) defined in such a way that thermal average of all operators are fully reproduced [40]. Namely: The entangled vacuum state reads where the temperature T = 1/β dependence lies in θ( p) ≡ tanh −1 e −β| p| . The generator G equals Note that the operator a i ( p) and a( p) do not annihilate the new vacuum |0(β) . One can introduce Bogoliubov transformations generated by G : and similarly for a j † θ ( p), a j θ ( p) and a j † θ ( p). Then, a i † ( p) and a i † ( p) form Fock space and a( p) and a( p) annihilate the vacuum |0(β) . i.e.
For further details and proofs regarding the Thermo field formalism the reader should consult [41,42].

Bi-local Collective Field Representation
The basis of the AdS/CFT lies in the different manifestation of a theory when seen through the Large N expansion. Collective field theory is built to construct an exact all orders (in 1/N ) bulk representation. For the case of O(N ) vector models, it was suggested [26] that the bulk Higher spin theory is generated completely in terms of bi-local collective fields Ψ(t; x 1 , x 2 ) given in the canonical picture [27] as These bi-local operators generate all the spin-s primaries This representation is exactly canonical due to the existence of the conjugate field Π = ∂ ∂Ψ . For the bi-local case we have for all order in 1/N . The bi-local fields obey the Large N Schwinger-Dyson equations, which leads to a systematic 1/N expansion. It is implemented through the associated collective Hamiltonian: which is systematically given in powers of 1/N . 14) The 1/N series is obtained as follows: One first determines the Large N background field Ψ 0 ( x 1 , x 2 ) through minimization of the collective Hamiltonian. Expanding the bi-local field the collective Hamiltonian gives the series of higher vertices interaction vertices: with a natural star product defined as A B ≡ d x 2 A( x 1 , x 2 )B( x 2 , x 3 ) representing a matrix product in bi-local space. Here we assumed the scalar field interaction V to be at most quartic in ϕ, otherwise there is an additional interaction term generated from V . The only nontrivial issue with respect to exact duality is the re-interpretation of the bilocal space c. This represents a kinematical problem. When interpreted in physical terms, the collective space leads to extra emerging coordinates and emerging gravitational and higher spin degrees of freedom.
For the system at finite temperature, one might suggest doubling of collective fields, where J ij and J ij is O(N ) generator of φ and φ, respectively. This implies that we have bi-local fields To deal with these bi-local collective field, it is convenient to introduce new index i = 1, 2 which represent original vector field φ and tilde vector field φ, respectively. Then, one can define a new bi-local field Ψ(( x, i), ( y, j)) of which index is doubled bi-local space (i, x).
Note that we multiply i to each φ a in the definition of the bi-local field in order to have minus sign in front of H in the total Hamiltonian (2.2). As explained, this collective field Ψ(( x, i), ( y, j)) is invariant under the transformation We define star product A(( x, i), ( y, j)) B(( y, j), ( z, k)) ≡ C(( x, i), ( z, k)) representing a matrix product in doubled bi-local space. Moreover, Tr [A(( x, i), ( y, j))] is defined as a matrix trace in the doubled bi-local space. Now, one can use the result of the standard collective field theory. The collective Hamiltonian for the doubled O(N ) vector field is given by where we rescaled Π and Π by N . Note that the derivative ∇ 2 act on the first argument of the doubled bi-local space.

Thermal background
In Large N limit, one constructs the background field by minimizing the leading effective potential. This leads to an equation for the background field given by where I is identity of the doubled bi-local space. The background field is static and translationally invariant so that solution has the following form.
Note that the bi-local collective field Ψ(( x, i), ( y, j)) is symmetric in the doubled bi-local space. With this symmetric condition together with reality condition of the collective field, a solution of the equation for the background field is given by where F ( p) is an arbitrary function of p with F ( p) = F (− p). Hence, the static Large N background collective field is given by From the background field, one can evaluate the leading Hamiltonian H (0) Note that the background field is equal to equal-time two point function of the vector field with respect to the thermal vacuum |0(β) . i.e.
Using Bogoliubov transformation (e.g. (2.8)), one can easily evaluate the vacuum expectation value Ψ( x 1 , x 2 ; y 1 , y 2 ) β . Comparing to (2.26), one can determine F ( p) to be We comment that the above thermal ground state solution is not unique. Uniqueness could be accomplished by adding a term representing entropy to the Hamiltonian.

Collective modes and Bulk
To study the collective modes one expands the bi-local field around thermal background : where we include numerical factor i or −1 in the components of η(t; ( x, i), ( y, j)) in the same way as (2.20). Moreover, we also include complex conjugate of the numerical factors in the components of π(t; ( x, i), ( y, j)) in order to keep canonical commutation relations. By Large N expansion of the collective Hamiltonian, quadratic Hamiltonian reads This quadratic Hamiltonian gives equation of motion for η.
To study Hilbert space, it is convenient to express Hamiltonian into momentum space.
By Fourier transformation of η and π into the momentums space where we use e i p 1 · x−i p 2 · y as kernel, the quadratic Hamiltonian becomes where we define and . To analyze the constraint structure, we Legendre transform to a Lagrangian scheme. Towards this end one has to express momentum π in terms ofη from Hamilton's equation for η.
However, the matrix K is not always invertible because determinant of K is given by For | p 1 | = | p 2 |, the matrix K is invertible so that Legendre transformation to Lagrangian is possible.
By variation of Lagrangian, equation of motion are given by This agrees with (3.4) as expected.
On the other hand, for the case of | p 1 | = | p 2 |, the matrix K is not invertible. We diagonalize the matrix K to diagonalize kinetic term at first.
Under this transformation, the quadratic Hamiltonian density for | p 1 | = | p 2 | modes becomes Note that π 12 and π 21 do not appear in the Hamiltonian. In addition, the Hamiltonian contains a linear term in η 12 . Therefore, one can see that η 12 ( p 1 , p 2 ) is a Lagrangian multiplier and corresponding constraint C 1 is given by In addition, [H, C 1 ] give a secondary constraint C 2 .
Note that C 1 and C 2 form the first class constraints. e.g. [C 1 , C 2 ] = 0. We further analyze these constraints in Appendix B.
To find the fluctuations and to represent them in terms of of creation and annihilation operators, we need to calculate normal modes of the equation of motion (3.4). For this, we do not have to explicitly solve (3.4) because for the present problem we know the exact O(N ) singlet eigenstates in the Fock space. They are given in terms of bi-local creation operators in Appendix A. Utilizing these, through at the 1/N expansion we can easily obtain the eigenmodes for the linearized fluctuations . First, consider Ψ 11 (t; x, y). The fluctuation η 11 is defined by subtracting the background field, or equivalently, by normal ordering of Ψ 11 (t; x, y) with respect to the thermal vacuum |0(β) . Recall that the annihilation operators a i ( p) and a i ( p) do not annihilate the thermal vacuum, but Bogoliubov transformed annihilation operators a i θ ( p) and a i θ ( p) do. Therefore, for correct normal ordering with respect to the thermal vacuum, we have to use standard normal ordering rule for a i θ ( p), a i † θ ( p), a i θ ( p) and a i † θ ( p). Taking a rescaling of the bi-local field by N and the definition of the fluctuation (3.1) into account, one can express η 11 (t; x, y) in terms of normal ordered O(N ) invariant composite operators of a i θ ( p), a i † θ ( p), a i θ ( p) and a i † θ ( p): Adding the conjugate operators one can evaluate the full algebra of bi-local operators. Full details of the algebra are summarized in Appendix A together with a systematic 1/N procedure. This allows to express all the operators of the theory through 1/N series of a canonical set of operators which satisfy the commutation relations: At the linearized level, a leading term in Large N expansion is enough.
and other operators appearing in the fluctuation η 11 (t; x, y) such as a i † a i are subleading in 1 N expansion so that we ignore them. As a result, we obtain In the same way, one can also find the mode expansion of η 22 (t; x, y), η 12 (t; x, y) and η 21 (t; x, y). e.g.
Using these mode expansions, one can evaluate the two-point Green's function in the Large N limit. Moreover, one can confirm that Green's function satisfy the equation of motion (3.4) (See Appendix C).

Bulk Interpretation
At zero temperature, the bi-local fields were seen to map into bulk fields of higher spin theory in AdS. This is most simply accomplished in the light-cone gauge [20,28] and was furthermore extended to a timelike (canonical) quantization in [27]. At finite temperature the TFD is expected to generate a space-time similar to an eternal black hole, which asymptotically approaches AdS 4 . The fluctuation modes found in the linearized collective analysis are expected to fully reproduce bulk modes of higher spin in the modified space-time. Using the knowledge of the AdS/bi-local CFT map, we can work out their properties. Let us summarize the main features of the canonical bi-local map to AdS bulk fields given in [27] for the O(N ) vector model CFT. We have the following expression for the bulk higher spin field with the explicit one to one kernel f ( x, z; p, p z ; p 1 , p 2 ) was given in [27] and where A( p 1 , p 2 ) represents the bi-local annihilation operator. This map (and the kernel) directly follows from the following canonical relation between bi-local and AdS coordinates : Under this map (with the conjugates determined through a chain rule) the bi-local on-shell condition translates into an AdS on-shell condition Here p z is a canonical conjugate to the radial AdS coordinate z. Note that in this zero temperature case we have that p z is always real because (| p 1 | + | p 2 |, p 1 + p 2 ) is timelike.
Let us now apply the bi-local canonical map given above to analyze the finite temperature modes and their bulk interpretation. First we have the modes: and therefore the operators α † ( p 1 , p 2 ) and α † ( p 1 , p 2 ) create states in Hilbert space with timelike momentum. The map (3.25)∼(3.26) and the on-shell condition (3.28) produces real values for the momentum p z in the bulk.
On the other hand, our TFD collective analysis exhibits another set of modes corresponding to : These are characterized by spacelike momentum, which leads to imaginary values for the momentum in p z of the bulk.
We recognize the last set of modes as representing evanescent mode. In [35,37], the importance of such modes and their presence in black hole background asymptotic to AdS was discussed and emphasized. In particular their present was seen to prevent a straightforward extension of the bulk to boundary kernel in the case of black hole type backgrounds.
Regarding the full reconstruction of the bulk field for TFD we proceed as follows. In recent work [12,43], a construction was suggested based on the composite single trace operators, O and O of CFT R and CFT L , respectively, with the further argument that O can be reconstructed for Hilbert space of O's.
As we have seen that the full invariant Hilbert space is given in terms of three set of bi-local operators, α's, α's and γ's, their result should have been a linear combination of We have seen that the bi-local generate the spin-s primary operators. In the collective TFD we additionally have not only spin-s primary operator O 22 s (x; ζ) ∼: φ i φ i : but also the mixed one O 12 s (x; ζ) ∼ φ i φ i . It is additional set of mixed operators that is responsible for evanescent modes.
Free O(N ) vector model has spin-s conserved primary operator O µ 1 ···µs with conformal dimension ∆ = s + 1. In TFD of free O(N ) vector model, one can use the same formula to generate spin-s currents. Like the bi-local field Ψ(( x, i), ( y, j)) in TFD, we can consider not only conserved current related to : φ i φ i : or : φ i φ i : but also one with both of them. Hence, we construct spin-s conserved current, in which one can find a connection to bi-local fluctuation η ab , as follows. To find relation between bi-local oscillators and spin-s current, we transform the current into momentum space. We have seen that α † ( p 1 , p 2 ) and α † ( p 1 , p 2 ) create a state with timelike momentum (| p 1 | + | p 2 |, p 1 + p 2 ) and (−| p 1 | − | p 2 |, − p 1 − p 2 ). On the other hand, γ † ( p 1 , p 2 ) create a state with spacelike momentum (| p > | − | p < |, p > − p < ) where we define Hence, we have to separately Fourier transform O ab s (x; ζ) for timelike and spacelike momentum denoted by p timeline and p spacelike , respectively. After Fourier transformation, we solve constraints on the spin-s current in order to find physical degree of freedom because collective fields which we want to find a relation with contain only physical degree of freedom. In [27], we solved two constraints, conservation and traceless condition, of spin-s primary operators with timelike momentum to find physical operators. One can also repeat the same derivation for the case of spacelike momentum. A solution of the two conditions is given by where a null polarization vector (p) for timelike momentum (and ± (p) for spacelike momentum) is given by Using those polarization vector, the physical spin-s operators can be expressed in terms of bi-local oscillators. For p 0 > 0, where Θ( p 1 , p 2 ) and Φ( p 1 , p 2 ) are important functions to find the map between internal space coordinate θ of higher spin field and bi-local momentum space.
The physical operator with * polarization or hermitian conjugate of them have similar forms. For other spin-s current, we also have a similar result. One can easily calculate commutation relations of operators O ab s 's, with details given in Appendix D. One finds that these are non-diagonal, with mixing between different spins. It is clear that one has to diagonalize the commutation relations (D.10, D.11, D.15 and D.16), we define an operator A s (k) (for s ∈ Z) as follow.
where we K β,k is understood to be a matrix with indices s and s so that K − 1 2 β,k is the squareroot of inverse of the matrix K β,k . One can construct A † s (k), A s (k) and A † s (k) in a similar way. They satisfy canonical commutation relation.
and others vanish. Vacuum expectation value of A † s A s is and similar for A s (k)A † s (k) . Therefore, vacuum expectation value of number density operator of spin-s operator is We define bulk higher spin field H s for (s ∈ Z) from A's and A's. In the eternal black hole, one has four regions, I∼IV where region I and region III are right and left outside of black hole and region II (region IV) is inside of black hole contains future (past horizon), respectively (see the earlier work of [34,44]. The presence of evanescent modes indicates that the effective potential V (z) (in the language of [35]) is different from AdS case. To deduce the potential one would need the full bi-local map. This would define the wave functions f s,k (t, x, z) everywhere. The bulk field is given as follows where f (t, x, z) is a canonically normalized wave function in the black hole background. In region III, one has Here, we construct H III s in such a way that A † s (k) creates a mode f k (t, x, z) because the time direction in the region III is opposite to that of region I. Moreover, higher spin field inside of the black hole is s,k (t, x, z) and f (2) s,k (t, x, z) are two linearly independent solutions in the region II. To summarize we have studied the fluctuations of bi-local ('single trace') operators in O(N ) scalar CFT at finite temperature in the Hamiltonian Thermo field dynamics formalism. The invariant spectrum corresponding to bulk modes was evaluated and was seen to produce new modes (in comparison) with the zero temperature AdS case. An exact 1/N expansion for the composite higher spin operators was performed, and evaluated in terms of the bi-local modes. This allows a complete reconstruction of the bulk for all high spin fields.
The main features of the reconstructed bulk fields at finite temperature are the following: at zero temperature (and at leading Large N ) the spin s fields were directly given in terms of the spin-s composite operator of the CFT. At nonzero temperature this is not the case, we have mixing. In order the obtain spin-s bulk creation-annihilation modes with canonical properties we had a linear combination of all spin CFT operators. In this we find a notable difference from the construction of [12,43], whose basic premise is that the vacuum (AdS) correspondence persist with only doubling O and O of CFT R at finite temperature (the eternal black hole).
The mixing occurred due to the singlet constraint (2.18) that gets imposed on the doubled Hilbert space and the non-trivial thermal background that it induces. In particular the extra 'evanescent' modes are directly are associated with the modes to the mixed bi-local between the left and the right CFT. This phenomenon, which we have demonstrated in the Large N limit suggests that the ideas of reconstructing the bulk as suggested in various proposals involving 'subregion duality' likely do not hold. Some further implications on the reconstruction of bulk from bi-locals (precursors) were recently given in [45].

Conclusion
We have worked out this paper the bi-local theory of the Thermo field CFT corresponding to the O(N ) vector model. The bi-local composite operators in this case provide a simple set of invariant, 'single-trace' operators and their 1/N dynamics is faithfully given by the collective Hamiltonian. This was solved after linearization around the (thermal) background with the modes and states explicitly constructed. They are seen to be related to the primary higher spin operators whose Large N limit is evaluated and given in terms of collective modes. These modes are to represent the bulk fluctuating modes of the dual Higher spin theory. To summarize, a class of modes (coming from the off-diagonal) bi-locals are seen to have the features identical to 'evanescent' modes present in black hole and other nontrivial backgrounds. These modes are seen to participate in the bulk Higher spin fields that we reconstruct.
Another non-trivial feature of the bulk fields that have constructed is that (at finite temperature) they contain mixing of all spin operators of the CFT. This mixing is induced by the nontrivial thermal background.

A.1 Bi-local Operators
In TFD of O(N ) vector model, one can construct O(N ) invariant bi-local operators and their hermitian conjugates, A † θ ( p 1 , p 2 ), A † θ ( p 1 , p 2 ), C † θ ( p 1 , p 2 ) and D † θ ( p 1 , p 2 ). Moreover, total Hamiltonian and momentum are defined by To study algebra of these bi-local operators, it is convenient to define a matrix of bi-local operators. and Then, commutation relation of A θ and A † θ is given by In Large N limit, A θ and A † θ are annihilation and creation operators, respectively. One can evaluate other commutation relations.
Especially, the thermal vacuum is a specific coherent state with z( p) = θ( p).
Going back to the bi-local operators, we want to study Large N expansion of the bilocal operators and express them in terms of canonical conjugate pair of operators. At zero temperature, Large N expansion of bi-local operators was already found for the case of O(N ) vector model [46] and U (N ) vector model [27] using Holstein-Primakoff transformation. One can repeat the exactly same procedure for finite temperature O(N ) vector model because the only difference is that bi-local space of TFD is doubled. First, we consider O(N ) invariant Fock space consisting of A † θ ((I 1 , i 1 ), (J 1 , j 1 )) · · · A † θ ((I n , i n ), (J n , j n )) |0(θ) (A. 16) In this singlet sector, a Casimir constraint is given by Then, we present ansatz for B θ ((I, i), (K, k)).
One can easily confirm that these ansatz satisfy (A.10) and (A.12). Finally, one can solve the Casimir constraints for A θ ((I, i), (J, j)). For our purpose, this realization in terms of α's is enough to see (3.19) and (3.21). Also, we found other solutions for (A.17) to get other realizations [27,46] (And, one of them agrees with [28]). Those realizations give the same result because they satisfy (3.19) and (3.21) in Large N expansion. Note that the bi-local operators A and B can be expressed in terms of oscillators corresponding to canonical pair of bi-local field Ψ and Π, which can be expanded in 1/N [46]. Then, using the above realization, one can evaluate 1/N expansions among oscillators.