Correlation functions of the CFTs on torus with $T\bar{T}$ deformation

In this paper, we investigate the correlation functions of the conformal field theory (CFT) with the $T\bar{T}$ deformation on torus in terms of perturbative CFT approach, which is the extension of the previous investigations on correlation functions defined on a plane. We systematically obtain the first order correction to the correlation functions of the CFTs with $T\bar{T}$ deformation in both operator formalism and path integral language, and later generalize it to the higher order perturbations which are involved in the multiple $T$ and $\bar{T}$ insertion. As consistenty checks, we compute the deformed partition function, namely zero-point correlation function, up-to leading order and check the results in the free field theories as examples. Further, we also get the second order formula of the partition function which is consistent with previous result in literature.


Introduction
Recently a class of exactly solvable deformation of 2D QFTs with rotational and translational symmetries called TT deformation [1][2][3] attracts a lot of research interest. With TT deformation, the deformed Lagrangian L(λ) can be written as where the composite operator TT (z) constructed from stress tensor within the theory L(λ) was first introduced in [1]. Although such kind of irrelevant deformation is usually hard to handle, it still has numerous intriguing properties. A remarkable property is integrability [2,4,5]. If the un-deformed theory is integrable, there exists a set infinite of commuting conserved charges or KdV charges. After TT deformation, these charges can be adjusted such that they still commute with each other [2,4]. Hence in some sense the deformed theory is solvable. Furthermore, such deformation is well under control by the fact that it is possible to compute many quantities in the deformed theory especially when the un-deformed theory is a CFT, such as S-matrix, energy spectrum, correlation functions, entanglement entropy and so on [6][7][8][9][10][11]. The TT deformation is a special one among a infinite set of deformations constructed from bilinear combinations of KdV currents [2,4]. These deformations also preserve the integrability of the un-deformed theory. Besides TT deformation, other deformations in this set including the so-called JT deformation also receive much attention from both field theory and holographic points of view [12][13][14][15][16][17][18][19][20]. In addition, the TT deformation can also be understood from some other perspectives and generalizations [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].
In particular, within λ < 0, the TT -deformed CFT is suggested to be holographically dual to AdS space with Dirichlet boundary condition imposed at finite radius [38,39].
Correlation functions are fundamental observables in QFTs, so it is of great importance to study the correlation functions in its own right. The correlation functions have many important applications, e.g. quantum chaos, quantum entanglement, and so on. One example is the four-point functions which are related to out of time order correlator (OTOC), a quantity that can be used to diagnose the chaotic behavior in field theory with/without the TT deformation [50][51][52][53]. To measure the quantum entanglement, the computation of entanglement (or Rényi) entropies involves the correlation functions [54][55][56][57]. In the present work we are interested in studying the correlation functions in the TT deformed CFT. In particular, the TT deformed partition function, namely zero-point correlation function, on torus could be computed and was shown to be modular invariant [58,59]. Furthermore the partition function with chemical potentials for KdV charges turning on was also analyzed [60]. The correlation functions with TT deformation in the deep UV theory were investigated in a non-perturbtive way by J. Cardy [11].
Meanwhile, one can also proceed with conformal perturbation theory. Here we have to emphasize that we focus on the deformation region nearby the un-deformed CFTs, where the CFT Ward identity still holds and the effect of the renormalization group flow of the operator with the irrelevant deformation is not taken into account in the current setup. The conformal symmetry can be regarded as an approximate symmetry up to the lowest orders of the TT deformation and the correlation functions can be also obtained nearby the original theory. The total Lagrangian is expanded near the critical point for small coupling constant λ The first order of deformed correlation functions take the following form where the expectation value in the integrand is calculated in the underformed CFTs by Ward identity, and the integration domain is the torus T 2 . In the perturbative CFTs approach, the deformed two-point functions and three-point functions were consider in [41,61] up to the first order in coupling constant. Subsequently, the present authors have considered the four-point functions [51]. Also we generalized this study to the case with supersymmetric extension [62]. Note that in the previous studies, these theories were defined on plane. In the present work, we would like to consider the theories defined on torus which will be very important to understand the boundary theory which is the holographic dual to the BTZ black hole [63]. The other motivation to study the correlation functions in the deformed theory on the torus is associated with reading the information about multiple entanglement entropy of the multi-interval [64][65][66]. To obtain the deformed correlation functions, one has to calculate the integrand in eq. (3) by Ward identity and do the integral over the torus T 2 with the help of a proper regularization scheme. The Ward identity on torus associated with the energy momentum tensor, e.g. T orT , has different structure compared with that on the plane [51] [62].
In terms of perturbative approach, we obtain the correlation functions with TT deformation systematically by using both operator formalism and path integral language following the analysis in [67][68][69]. Further, the correlation functions with multiple T and T insertion can be also obtained, for example, the case with a TT TT insertion, which is associated with the second order correction to the correlation functions. With these results in hand, as applications, we also obtained the deformed partition function up to second order, which is in good agreement with the results offered by [58] where the partition function is directly obtained by the counting of the known deformed spectrum.
The plan of this paper is as follows. In section 2, we discuss the Ward identity associated with T andT insertion on torus and apply it to study the first order perturbation of partition function. Then we check the partition functions in the deformed free bosonic and fermionic field theories. In section 3, we compute the generic Ward identity associated with multi-T andT insertion, and apply it to the second order perturbation of the partition function with TT deformation. In section 4, we offer the Ward identity on torus by using path integral method. Conclusions and discussions are given in the final section. In appendices, we would like to list some relevant techniques and notations which are very useful in our analysis.

TT -deformation
In this section we will calculate the first order TT correction to the correlation functions of the CFTs on torus. As examples, the results are applied to the first order corrections to the partition function in free field theories with TT deformation.

Correlation functions in the TT -deformed CFTs
To obtain the correlation functions of the CFTs with TT insertion on torus, the procedure is similar with the case in which there is only a single T -insertion as examined in [68,69], where the correlation functions were derived in the operator formalism. Interestingly, the same results were also obtained in path integral language [67]. Let us recall the well-known result about the T inserted correlation functions on torus in where X ≡ φ 1 (w 1 ,w 1 )...φ n (w n ,w n ), a string of primary operators, P (z), ζ(z) are the Weierstrass P -function and zeta function respectively, η 1 = ζ(1/2), and τ is modular parameter of the torus (For our conventions, please refer to appendix A). Note that though the prefactor (ζ(w − w i ) + 2η 1 w i ) is not doubly periodic on coordinate w, the correlation function T (w)X is doubly periodic on w by translation symmetry. In fact, eq.(4) can be regarded as a generalization of Ward identity on plane. As w → w i , the usual OPE on the plane is reproduced where we used the expansion of functions P (w) ∼ 1/w 2 , ζ(w) ∼ 1/w in the neighbor- In what follows we will review how to derived eq.(4) in operator formalism as in [68].
At first, the partition function on torus is defined by the following trace over the Hilbert Then the correlation functions of X({w i ,w i }) = φ 1 (w 1 ,w 1 )...φ n (w n ,w n ) (We will suppress the anti-holomorphic coordinatesw i dependence in X for simplicity hereafter) takes the form To obtain the T inserted correlator T (w)X({w i }) , we started with the coordinate z on plane which is related to standard coordinate w on cylinder via the exponential map z = e 2πiw . With plane coordinate z, one can expand the stress tensor as Now consider the quantity tr(T pl (z)X({z i })q L 0 −c/24 ), 3 using (8), which equals where X pl ({z i }) are primary operators defined on plane. The first term can be converted to the derivative with respect to the modular parameter τ while the second term equals 4 Note the commutator on the RHS can be further expressed as a contour integral Here the contour γ encircles the operators located at z i , i = 1, ..., n. Then eq.(9) is where the following formula [68] is used with z 0 = e i2πw 0 , z = e 2πiw . Note the contour γ does not encircle z.
Next we transform all the quantities above on plane to coordinate w on torus by exponential map. For stress tensor on torus T (w), one has and the primary fields X pl ({z i }) transform accordingly to X({w i }) on torus. It follows that eq. (14) can be written as where the contour on torus γ ′ transformed from γ on plane encloses w i not w. It can be shown that the above equation is also valid when X contains component of the stress tensor T . The second term on the RHS can be further evaluated by substituting into the OPE which leads to where in the last step the translation symmetry is used ( i ∂ w i X = 0). Finally we obtain tr(T (w) After dividing both side of eq.(20) by Z, the result eq.(4) is produced.
Based on the derivation above, we can next consider TT insertion, which can be done by replacing X in eq.(17) withT (v)X. Since OPE T withT is regular, only the OPE T φ i will contribute to the contour integral. Following the same line as eq.(18)-eq.(20), the TT inserted correlation function is given by Here we have implicitly included the factorqL 0 −c/24 inside the trace. Equivalently, eq.(21) can be expressed as where the last two terms being proportional to i ∂ w i T (v)X can be computed as follows. Using translation symmetry, one has Substituting the anti-holomorphic counterpart of eq.(4) into the RHS, then one can see that ∂ v T (v)X is analytic on torus except at the contact points v ∼ w i . Explicitly, one can get which means the last two terms in the last line of eq.(22) are contact terms vanishing on torus except at contact points. Following the prescription in [72], when computing the integral in the first order perturbation of TT deformed correlation functions, we excise these singular points v = w i from the integral domain where D(w i ) is a small disk centered at v = w i . Therefore the last two terms in the last line of eq. (22) make no contribution to the first order TT deformed correlation functions.
It is interesting to apply the TT inserted formula to the case without primary operator φ i , i.e., X is identity operator, which is where we have used T (v) = −2πi∂τ ln Z. The above result indicates the expectation value of operator TT on torus does not dependent on the position w, v, this is reasonable since the holomorphic and anti-holomorphic stress does not effect each other in CFTs. Note the same phenomenon also presents in the cylinder case [1].
Without operators φ i , eq. (27) can be derived in a more direct way. To see this we start with the trace of a single insertion of stress tensor on plane where we used eq.(11) such that the terms with n = 0 vanish. Next transform that to torus by the map (16) tr The expectation value of T is then obtained Now consider T (z 1 )T (z 2 ) insertion, which is Noting [L n ,L n ] = 0 and using eq.(11), one has which indicates only the term with n = m = 0 will contribute to the summation in eq. (31). Further making transformation to torus and using eq.(30), we finally obtain which reproduced eq. (27).
It is interesting to note that the expectation value TT is related to the first order perturbation of partition function under TT deformation. The deformed partition function is with the un-deformed partition function Z = Dφe −S . After performing integral and using eq. (27), the first order perturbation of partition function is which is in good agreement with the result in [58], where the partition function with TT deformation is computed by using the deformed spectrum and also the modular properties of partition function is investigated in [58]. In section 3 we will compute the second order perturbation where the TT (u 1 )TT (u 2 ) is obtained. Before doing that, we would like to apply the first order results to free field examples as consistent checks.

Free field theories
Now we apply the formula eq.(27) to free field theories, and show that eq.(27) is consistent with the results obtained by Wick contraction.
Let us first consider the free boson on torus. The corresponding un-deformed partition function is where η(τ ) is the Dedekind η function. The two-point function of scalar fields is wellknown, which takes the form [70] φ(z 1 ,z 1 )φ(z 2 ,z 2 ) = − log Here the last term is non-holomorphic and comes from the zero mode. Performing derivatives on above two-point function gives The holomorphic and anti-holomorphic stress tensor for boson are T = − 1 2 (∂φ) 2 , T = − 1 2 (∂φ) 2 respectively. The expectation value can be calculated by point-splitting method where eq.(39) is used. Note this result is consistent with eq.(30). 6 Using Wick contraction and eq.(40), we can further compute the expectation value of TT operator T (z 1 )T (z 2 ) = 1 4 : (∂φ(z 1z1 )) 2 :: (∂φ(z 2 ,z 2 )) 2 : which is equal to eq.(27) as Note that the result for TT is more complicated than T T (see for example [72]), this is because in the latter case, the two holomorphic stress tensor T can interaction with each other while not for T andT . 6 which can be verified with the help of the identity for Dedekind η function Next we will go on to the first order correction to the partition function of free fermions. There are four kinds of spin structures denoted as ν = (1, 2, 3, 4) for free fermions. The two-point function for fermion with spin struction ν is [70] The partition function Z ν is product of holomorphic and antiholomorphic part The holomorphic stress tensor is given by and similar for the anti-holomorphic part. By subtracting the divergent part, the expectation value is Using wick theorem which indicates that eq. (27) is valid for free fermions. 7 Here the function P ν (z) is defined by [71]

Higher order deformations
In this section we would like to calculate the correlation functions with higher order TT insertion, which follows closely to the multi-T insertion studied in [68]. At first, let us review how to obtain the correlation functions with multiple T operators. we will take T T insertion as an example in the following.
We begin by replacing X in eq.(17) with T (v)X, which is where the contour γ ′ encloses w i as well as v. To perform the contour integral the following OPE beside eq.(18) is needed After computing the integral and using translation symmetry we obtain T T inserted correlation functions [68] tr(T (w)T (v) With eq.(54) in hand, it is straightforward to write down the following expression for multiple-T case which is a recursion relation for multiple-T correlation functions [68]. Next we will consider the cases where multiple-T andT are presented. For example, adding oneT to eq.(56), one can obtain where the contour encloses u, v, w i , however the OPE T (w)T (v) has no singular term again, thus the contour integral around v makes no contribution. This implies the computation of contour integral in the last line is similar with multiple-T cases. Finally, we obtain a recursion relation for multiple-T andT inserted correlation functions If we replace T (w) withT (w) in the first line, then the anti-holomorphic counterpart formula of eq.(57) can also be derived which is expressed in terms of anti-holomorphic quantities.
As mentioned in the last section, we will apply formula eq.(57) as well as its antiholomorphic counterpart to study the second order perturbation of the TT deformed partition function, which involves the integral Here the expectation value in the integrand has two T and twoT insertion.
One can note that the RHS does not dependent onv 1 .
Next consider T (v 1 )T (v 2 )T (u 2 )T (u 1 ) which can be expressed in terms of eq. (59) and eq.(60) by using the recursion relation wherev 12 =v 1 −v 2 . Note the last term equals zero since T (u 2 )T (u 1 )T (v 1 ) is independent ofv 1 . Using (59) and (60), the above equation can be further expressed 4πi(P ′′ (v 12 )(P (u 12 ) + 2η)∂ τ ln Z − P ′′ (u 12 )(P (v 12 ) + 2η)∂τ ln Z) Let v 1 = u 1 , v 2 = u 2 in eq.(62), we obtain the integrand in eq. (58), and the integrals needed to calculate are 8 We used (see the appendix C) In computing these integrals, following the prescription for regularization in [72], we have removed the singular points out the integration domain. Here we only listed the results, for the detailed discussions please refer to appendix B. After putting together the integrals, we obtain which is equal to the second order partition function computed in [58].

Deformed correlation functions in path integral formalism
In this section we will derive the correlation functions with TT insertion following the line of [67] where the T T insertion was obtained in path integral formalism.
We start with the definition of stress tensor, assuming there is a Lagrangian description for the theory where S is the action of the theory, then the expectation value of stress tensor is given by with partition function More generally the correlation functions is defined by The Ward identity corresponding to three types of local symmetries: reparametrization, local rotation and Weyl scaling in CFT can be given by [67] where e a µ is the zweibein field coupled with CFT and ω ν is the spin connection. The vector fields ξ µ parameterize the transformation of zweibein: e µ a → e µ a −ξ ν ∂ ν e µ a +∂ ν ξ µ e ν a . s k , d k are the spin and dimension of the field φ k . R is the scalar curvature of the surface, which is equal to zero for torus. And Note that eq.(76) contains correlation function with a single stress tensor inserted. In order to obtain double stress tensors insertion, one can further vary eq.(76) with respect to metric. The resulting expression is If we let ρ = σ = z and ξz = 0 in above equation, the correlation function with T T insertion can be obtained as presented in [67]. Similarly, the TT insertion can be obtained by setting ρ = σ =z and ξz = 0, as what will be shown in the following.
Setting ρ = σ =z in eq.(78), we obtain (∇wξ ν Tw ν (w)X + ∇wξ µ T µw (w)X + ξ λ ∇ λ Tww(w)X ) Setting ξz = 0 in above equation leads to where h k = 1 2 (d+s) and we omitted the term Tz z ... = 0. To extract the T zz (z)Tww(w)X outside the integral on the RHS of eq.(80), the Green function G z vv for operator ∇ z on Riemann surface with genus g is employed [67] where h j vv (v) are holomorphic quadratic differentials on the Riemann surface, and η zz ,i are Beltrami differentials dual to holomorphic quadratic differentials, i.e., d 2 z √ gg zz h j zz η zz ,i = δ j i . Let ξ z (z) = G z vv (z, v), then eq.(80) can be written as where the last term on LHS is called Teichmuller term. All the formulae derived so far are valid for general Riemann surface. Here we are interested in the case g = 1, i.e., the torus, in which case the metric are flat (R = 0), y j = −τ , and the corresponding Beltrami differential and quadratic differential for torus are The explicit expressions for G z vv (z, v) on torus is With these parameters in hand, let us first consider the Teichmuller term which can be computed explicitly similar to [67] h where the last term can be evaluated by substituting eq.(82). The derivative in the last term does not vanish, since the correlator can be non-analytical in z as T zz (z) approaches other operators. As for the first term, it turns out to be 9 dz T zz (z)Tww(w)X = i∂ τ Tww(w)X + i∂ τ ln Z Tww(w)X .
Finally the Teichmuller term is (89) 9 In this section, in order to compare our results to that of [67], we follow the convention in that paper, where the stress tensor on torus is related to previous section upto a factor 2π, and the stress tensor on plane T pl is the same with previous definition, thus eq.(16) become Here T pl (w ′ ) = L n /w ′n+2 , T (w) = (−2π) e −2πiwn (L cy ) n , with (L cy ) n = L n − δ n,0 c/24, then Combine with the remaining terms in eq.(82) which can be computed straightforwardly, the TT inserted correlation function is given by where the term ∂ w Tww(w)X in last line does not vanish since Tww(w)X is not analytic in w as Tww goes to X, as mentioned before. In fact, ∂ w Tww(w)X is proportional to delta functions such as δ (2) (w − w k ) (which can be seen by substituting the expression of oneT inserted function TwwX ). Therefore the terms in the last line of eq. (90) are contact terms. In addition, the term k z∂ w k Tww (w) X is also contact term (see eq.(23)). As discussed around eq.(26), when we consider the first order of TT deformed correlation functions, the contact points is dropped out from the integral. Upon ignoring the contact terms eq.(90) is consistent with the result in section 2. Therefore the operator formalism and path integral method are consistent with each other when we consider the first order TT deformed correlation functions.

Conclusions and discussions
Motivated by studying the quantum chaos and the entanglement of multiple partite subsystem, one has to know the correlation functions on torus with the TT deformation. In this work, to study the correlation functions of the CFTs on torus with TT deformation, we apply the Ward identity on torus and do a proper regularization procedure to figure out the correlation functions with TT deformation in terms of perturbative field theory approach. It can be regarded as a direct generalization of previous studies [51] [62] on correlation functions in the TT deformed bosonic and supersymmetric CFTs defined on plane. It is well known that the the correlation functions on plane with T andT can be obtained straightforwardly by using the Ward identity, while the Ward identity on the torus is very complicated and Ward identity associated with the T andT is unknown in the literature. In this work, we obtained the TT deformed correlation functions perturbatively in both operator formalism and in path integral language. As a consistent check, the first order correction to the partition function agrees with that obtained by different approach [58] in literature. We explicitly calculate the first order correction to partition function in the free field theories and we confirm the validity by comparing with the results obtained by Wick contraction. Moreover, the higher order correction to the correlation functions have been obtained systematically. As a check, the resulting second order correction to the partition function is consistent with the results in [58] obtained by the counting the full deformed energy spectrum.
Since resulting correlation functions are applicable for generic CFTs with the deformation, they are useful to study the holographic aspects of the dual boundary CFTs with finite size, finite temperature effects. In addition, it is interesting to investigate the correlation functions of the supersymmetric theories on the torus, as we did in [62].

Appendices A Conventions
In our convention the torus denoted as T 2 is defined by the identification of complex number w ∼ w + 2w 1 + 2w 2 with 2w 1 = 1, 2w 2 = τ .
In the following we collect some formulae regarding elliptic functions which are useful in this work. The Weierstrass P -function is defined by [71] P (z) = 1 The Weierstrass P -function is an elliptic function (doubly periodic on complex plane) with periods 2w 1 and 2w 2 . P (z) is even and has only one second order pole at z = 0 on torus. The Laurent series expansion in the neighborhood of z = 0 can be expressed as where c 2n are constants.
The Weierstrass ζ(z) function is defined by which is related with P (z) as Note ζ(z) is odd and has a simple pole at z = 0 around which the Laurent expansion takes the form Since an elliptic function can not have only one simple pole on torus, ζ(z) is not doubly periodic. Instead, ζ(z) satisfies the quasi-doubly periodic conditions with ζ(w 1 ) equals the Dedekind η function (also denoting η 1 ≡ ζ(w 1 )) and ζ(w 2 ) ≡ η ′ .
These quantities satisfy the following identity
Since P (z) is doubly periodic and the translation does not change the integral, the T 2 d 2 zP (z − y)(= T 2 d 2 zP (z)) is where in the last step eq.(97) is used to eliminate η ′ . One has to be careful to valuate this integral, since there is a singular point at z = y in the integrand. In fact, following the prescription for the regularization [72] (see also [73]), we choose the domain of integral on the torus excluding the singular point as T 2 −D(y), where D(y) denotes a small disk centered at the point z = y and the corresponding boundary is ∂T 2 − ∂D(y). Further, one can check that the integral above along the contour ∂D(y) makes no contribution to the final answer. By the same reason, we can handle the integral (103),(105) and (106) below in the similar manners.
From eq.(99), we obtain d 2 z(P (z) + 2η) = π, (100) whose complex conjugate is This integral is exactly equal to the one obtained by using the different method in [72].
With transforming the stress tensor on plane into cylinder, we finally obtain T (u 1 )T (u 2 ) in eq.(59).
Following the steps deriving T (u 1 )T (u 2 ) , we will finally obtain the same express as presented in eq. (60). Similarly, the deriving of four-point function T (u 1 )T (u 2 )T (v 1 )T (v 2 ) in eq.(62) can be proceeded.