The correlation function of (1,1) and (2,2) supersymmetric theories with T T̄ deformation

Center for Theoretical Physics and College of Physics, Jilin University, Changchun 130012, People’s Republic of China Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Golm, Germany c School of Physics and Astronomy, Sun Yat-Sen University, Guangzhou 510275, China d Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, Sun Yat-Sen University, Guangzhou 510275, China


Introduction
Studying exactly solvable models in 2D QFT can help us get a deep understanding of general field theory. The next step is naturally taken to consider the deviation from these exactly solvable models. In the language of renormalization group flow, in general the study of deformations by turning on relevant operators is under more controllable than irrelevant deformations which may introduce infinite divergences in the UV. However, a special kind of irrelevant deformation of 2D QFT was shown to have a number of remarkable properties even in the UV [1][2][3]. Such deformation preserves the integrability if the undeformed theory is integrable, also the spectrum and the S-matrix can be calculated. In addition, the deformed theory can be renormalized perturbatively systematically [4].
Among these deformations there is a special one, referred to as TT deformation, have attracted much attention recently . Here T is related to stress tensor of the theory. The deformed Lagrangian S(λ) can be written as where the operator TT (z) was first introduced in [1]. For conformal field theory, it was found that the partition function of deformed theory can be computed and remains modular invariant, and one can even obtain Cardy-like formula in deformed CFT.
There are many directions to generalize the TT deformation, then an interesting question to ask is that what will happen when additional symmetry is presented in the theory, for example, conformal symmetry discussed above. In [56][57][58][59] (see also [60,61]), the authors have taken into account the supersymmetry, more specific, N = (0, 1) and extend SUSY with N = (1, 1), (2, 0), (2,2) was considered. In these studies, the supersymmetric version of TT operator appeared in eq.(1) was constructed based on the supercurrent multiplet [62], and the deformed Lagrangian is also given for free theory with or without potential. Taking N = (1, 1) for example [56], the deformed action takes the form with Here O(ξ) = J +++ (ζ)J −−− (ζ)−J − (ζ)J + (ζ), (J +++ , J − ) and (J −−− , J + ) are two pairs of superfields, which include stress energy tensor (For more details for this construct, please refer to [56]). Moreover, it was shown that the deformation constructed in this way preserves solvability and supersymmetry. Furthermore, the operator O in eq. (2) is equal to bosonic TT as appeared in eq.(1) up to total derivative terms vanished on shell O = TT + EOM ′ s + total derivatives.
Similar relationships between bosonic TT and its supersymmetric counterparts are also hold in other extend SUSY mentioned above.
In this work, we are interested in studying the correlation functions in TT deformation of superconformal field theory perturbatively. Correlation functions are fundamental observables in QFT, thus it is of great importance to study the correlation functions in its own right. The behavior of correlation functions was studied in both TT [19,45] and JT [17] perturbatively, and unperturbatively in deep UV region by J.
Cardy [18]. Inspired by these progress, here we would like to add supersymmetry to the undeformed theory. Since we will work with Euclidean signature, we would like to focus on our attention to the superconformal field theory with N = (1, 1) and N = (2, 2) supersymmetry. As discussed above the operators O and TT are equal on shell up to some total derivative terms, thus we will employ the latter as the definition for TT deformation in the process of computing correlation functions. Here we have to emphasize that we only focus on the deformation region nearby the undeformed CFTs, where the CFT Ward identity still holds and it is not necessary to take account the effect of the renormalization group flow of the operator with the irrelevant deformation. Therefore, the conformal symmetry can be considered as an approximate symmetry up to the first order of the TT deformation and the correlation functions can also be obtained nearby the original theory. Moreover, both in holography and quantum field theory, these correlation functions can be also applied to obtain various interesting quantum information quantities in the deformed field theory, e.g. the Rényi entanglement entropy of local quench in various situations [63][64][65], entanglement negativity [66], entanglement purification [67], information metric [68,69], etc.
The remaining parts of the paper are organized as follows. In section 2, we first briefly review the Ward identity in (1,1) superconformal and also the correlation functions in undeformed theory, then formulate the 2-,3-, and n-point (n-pt) correlation functions with TT inserted, the last step is to perform the integral in conformal perturbation theory using dimensional regularization. In section 3, we first discuss the Ward identity and undeformed correlators in (2,2) superconformal field theory. Then following the same line as section 2, we compute the 2-,3-, and n-point deformed correlation function. In section 4. We discuss the dimensional regularization methods used in section 2 and section 3. In the final section, conclusions and discussions will be given.

N =(1,1) superconformal symmetry
In this section we review (1,1) superconformal symmetry and the corresponding Ward identity. The coordinates on superspace are analytic coordinates Z = (z, θ) and antianalytic coordinatesZ = (z,θ) where z,z are two complex coordinates and θ,θ are Grassmannian coordinates. The (1,1) superconformal algebra is the direct sum of (1,0) and (0,1) algebra, thus for simplicity we may subsequently only write out the analytic part. For (1,1) theory the superderivative is [70][71][72][73] The superfield A superfield Φ(Z,Z) is called primary superfield if it transforms as under conformal transformation Here ∆,∆ are the anomalous dimensions of Φ(Z,Z). The infinitesimal version of eq. (9) is where only the analytic part of the transformation is considered. Furthermore, one can obtain the OPE between the superfield J(Z) containing stress tensor T (z) and primary superfield Φ with dimension ∆, which is the generalization of OPE between stress tensor and primary field T (z)φ(z ′ ) in CFT. This can be done by substituting eq.(11) back to eq.(7) and using super-Cauchy theorem 4 which implies where the SUSY invariant distance Z 12 = z 1 − z 2 − θ 1 θ 2 and θ 12 = θ 1 − θ 2 . We then obtain the following OPE [72] From this OPE, the N = (1, 1) superconformal Ward identity can be written as and similar expressions forJ(Z).
It is important to apply Ward identity to global superconformal transformation whose algebra osp(2|1) is a subalgebra of superconformal algebra. By employing Ward identity and the fact that correlator of primary superfields is invariant under global superconformal transformation since it is a true symmetry of the theory, these correlators will be highly constrained. And similar to the cases in bosonic CFT, it is possible to completely fix 2-and 3-point correlators up to to some constant factors. The 2-pt correlator is with c 12 a constant and 3-pt correlator is where the second factor in the right hand side can also be written as Here c 123 , c ′ 123 are constants, ∆ ij = ∆ i + ∆ j − ǫ ijk ∆ k , and θ 123 is defined as which is invariant under global conformal transformation. By definition θ 123 is Grassmannodd, thus θ 2 123 = 0 and eq.(20) follows.
As for n-pt correlators with n ≥ 4, they depend on 2n coordinates z i , θ i , i = 1, ..., n, and 5 constraints corresponding to 5 generators of osp(2|1). Thus there are 2n − 5 independent variables in n-pt correlators. Actually, there exists the same number of independent osp(2|1) invariants, i.e. 2n − 5, which are [72] w j ≡ θ 12j , j = 3, ..., n, U k ≡ Z 123k , k = 4, ..., n, where θ 12j is defined in eq.(21) and Z ijkl is an analogue of cross ratio in CFT In terms of these variables the n-pt function can be determined as with i =j ∆ ij = 2∆ j , ∆ ij = ∆ ji and similar for∆ ij . Here f is a function can not be fixed by global superconformal symmetry, and it depends on the theory under consideration.
With the results discussed above, we can compute the TT deformed correlators.
The variation of action under TT deformation can be constructed as where the minus sign comes from the anti-commutation nature of θ. Thus to first order in λ the variation of n-pt correlator is Note that the correlator inside the integral can be evaluated via Ward identity. In the following section we will compute eq.(26) for n = 2, 3 and n ≥ 4.

2-pt correlators
In this section we will consider the 2-pt correlators with TT deformation. The undeformed correlator takes the form as eq. (18) Having obtained J (Z)Φ 1 Φ 2 we are in position to consider J(Z)J(Z)Φ 1 Φ 2 which follows as where in the second step for later convenience we name the terms involving derivatives as G, and the remaining terms as F with n = 2 in the present case. To evaluate the right hand side of eq.(34), first consider the anticommutator between P and J = F + G, noting P, G, F are all Grassmannian In principle by setting θ 1,2 → 0, one can get the results for bosonic CFT, which is 7 Also T 11 can be evaluated in an alternatively way as which is equal to result in eq. (44). The integral in the last step was computed in [19].
Comparing this with the CFT results given in eq. (8) in [19] as One can find that the last constant is different in eq.(48) and eq.(49). This difference can be understood from the way we performing the integrals. On one hand, we can use dimensional regularization to evaluate the integral directly which will result in eq. (49). On the other hand, we can compute the above integral in an indirect way as we did at the beginning, i.e., Firstly, expanding the integrand into several terms as below, then using dimensional regularization to compute each integral, finally adding up the contribution of individual term which leads to eq. (48). The difference between eq.(48) and eq.(49) can be eliminated by redefine ǫ.

3-pt correlators
The general form of 3pt correlators can be written as where a, c are two undetermined constants and for later convenience we denote As discussed in 2-pt correlators in the previous section, we first consider the correlator JΦ 1 Φ 2 Φ 3 which can be calculated by using the definition of G, F in eq.(35) as follows where P (defined by GO 3 ≡ P O 3 ) turns out to be In the last step of eq.(54) we have omitted the "crossing" terms such as GŌ 3 , Gθ 123 (By crossing terms we mean the terms with holomorphic derivative ∂ z acting on antiholomorphic coordinates, or ∂z acting on holomorphic coordinates, which will result in a δ-function as ∂ z (1/z) =δ(z). Note that we have encountered crossing term as in eq.(32) in the 2pt correlator case), since these terms will vanish when integrating over θ. To be concrete, taking the term GŌ 3 for example Let us first focus on the last two terms which are crossing terms. After some computation the last term is For the same reason as discussed below eq.(41), this term should be dropped out. As for the term G(Ḡθ 123 ), after employing the anti-commutator G(Ḡθ 123 ) can be written as where the term Gθ 123 is omitted in the first step, and also for ∂ z jθ 123 in the second step since they do not contain θ. Thus finally we get which is also singular and should be dropped. This can be seen by noting that if we interchange the position in J(Z)J(Z)Φ 1 ... , and to consider J (Z)J(Z)Φ 1 ... we will obtain a term different with eq.(61) as Finally we obtain the 3pt correlator as Let us first consider the terms containing no a Next evaluating the a 1 -terms which contains two parts, the first part is and the second part is As for the a 2 -term denoted as V 2 , by observing eq.(63) we find V 2 = −aV 12 θ 123θ123 , thus V 12 + V 2 = V 12 e −aθ 123θ123 . In summary, the result for 3-pt correlators with TT perturbation to first order is where the identity k,k =i∆ ik = 2∆ i . is used to simplify the final expression.

n-pt correlators
For n point with n ≥ 4, the undeformed correlator functions take the form as with Assuming all Φ i have the same dimension (∆,∆), we have Again the crossing termsḠZ ijkl ,Ḡθ ijk ,ḠO n do not depend on θ and we will not consider these terms below. Now evaluate where P takes the same form as eq.(55) with summation from 1 to n, and where we introduced the notation ∂ R z j , D R j , ∂ R θ which act on z i , θ i but not onz i ,θ i , and similarly let ∂ L z j ,D L j , ∂ L θ act onz i ,θ i but not on z i , θ i (thus ∂ R z j (1/z j ) = 0). When inserting JJ , yields Naively the last term in eq.(73) looks like a crossing term, but this is not the case as can be see below where for example one has with G R acting only on U j , w j but not onŪ j ,w j . Eventually one can get In summary the TT deformed correlator is of the form Hence using the results for integrals in section 4, the final result is Setting n = 4, the above results can be used to investigate, for example, the OTOC.
The superfield can be written as and its conjugate To consider the OTOC involving two fields φ, ψ 1 , from (45) in [19], at first order one of the 4pt functions needed to compute is where in the integrand, we can replace Z ij → z ij ,Z ij →z ij . In the bosonic CFT, 4-pt correlators can be expressed as conformal blocks whose universal properties are known in some cases, thus the OTOC can be computed [74], while in eq.(82) the function f is unknown in general. Thus it is more difficult to compute OTOC here.
Super-analytic transformation can be defined via the transformation law of covariant derivatives as Superconformal primary fields are defined such that under super-analytic transformation they transform as where ∆, J are the dimension and charge of Φ respectively. The OPE between energy momentum superfield J(Z) and primary superfield have been considered in [75,77] where Z 12 = z 12 − θ 1θ2 −θ 1 θ 2 (alsoZ 12 =z 12 −θ 1θ2 −θ 1θ2 ). In analogy with (1,1) case in the previous section, from this OPE, we can get the Ward identity as 8 In NS sector the n-pt correlators on the right hand side of eq.(88) are constrained by Ward identity corresponding to global superconformal Osp(2|2) transformation [77].
When n = 2, the correlator is fixed as where ∆ 1 = ∆ 2 , Q 1 + Q 2 = 0 and similar for∆,Q. Note here we have written the antiholomorphic part explicitly. 8 For the N = 2 Super-Cauchy theorem see [75] For n = 3 the correlators take the form with A ij = −A ji , 3 j=1,j =i A ij = −Q i , and similar for theĀ ij ,Q i . Note that not all A ij are fixed, this is because for 3-pt case there are nine coordinates (z i , θ i ,θ i ), i = 1, 2, 3, and eight generators for osp(2|2), thus there remains one degree of freedom which corresponds to the invariant quantity with R 2 123 = 0. The n-pt correlators can be fixed by Ward identity up to an undetermined function where x i is Osp(2|2) invariant variables which may be either R ijk or Z ijkl It should be point out that only 3n − 8 variables R ijk , Z ijkl are independent.
The variation of action under TT deformation can be constructed as Also to first order the n-pt correlators is In the following section we will consider eq.(95)) with n = 2, 3 and n ≥ 4.

2-pt correlators
Up to a constant prefactor, the 2-pt correlators take the form as To obtain TT deformed correlators, first consider only the correlators with holomorphic component of stress tensor inserted, from eq.(88), this is where for later convenience we introduced G, F such that G contains derivatives and F does not To evaluate eq.(97), firstly, let us consider the crossing terms (holomorphic derivatives ∂ z acting on antiholomorphic coordinates or vice versa) in eq.(97). In analogy with the (1, 1) case, it can be shown that this kind of terms vanish when integrating over θ,θ, thus it will not contribute to the final results eq.(95). Explicitly, consider the crossing where in the last step we have used (103) 9 Some useful expressions In the same manner one has dθdθGeQ 2θ 12θ12 Z 12 = 0. Therefore we can derive eq.(97) in the following without considering crossing terms, which is and P is defined as (similar forP ,F ) Further Performing the integral over z using dimensional regularization, yields

3-pt correlators
Using Ward identity, the 3pt correlators take the general form as Following the same line as 2-pt correlators, we first consider It can be shown that the crossing terms do not contribute, i.e.
Therefore we only need to consider where x i can be either of the following invariant variables Note that only 3n − 8 of R ijk and Z ijkl are independent.
Let us first consider only holomorphic component J(Z) inserted where we will encounter new crossing terms GR ijk , GZ ijkl in addition to these appeared in eq.(112). By using eq.(103) it can be checked that they will vanish, i.e.
Thus we will not consider crossing terms in eq.(121), then where P 1 , P 2 be of the same form as defined in eq.(113) and eq.(114). Here Q equals Gf , which is where for simplicity we have abbreviated R ijk as R, Z ijkl as Z and suppressed the summation R ijk , Z ijkl . Note in the first step in eq.(124) we omit the terms vanishing after integration over θ,θ. Following the same way we introduceQ as Next consider JJΦ 1 ...Φ n , which is where the last term should be dropped as discussed in previous sections. And the term (GQ) Φ 1 ...Φ n /f is very similar to the (1,1) case as discussed in eq.(75), which is not a crossing term. Actually, Gathering all the results together, we then have integration the final result is Thus the following operator appeared in first order perturbation of OTOC can be computed by utilizing eq.(130).

Dimensional regularization
Using Feynman parametrization and dimensional regularization one can obtain the following basic integral [17] (Let z 1 = z 2 ) 10 with ǫ being a infinitesimal constant. Next consider I 12 (z 1 ,z 2 ) with z 1 = z 2 where in the last step d = 2 is set directly since there is no divergence in the integral, and analytical continuation of the dimension is not required. Here V d = 2π d/2 /Γ(d/2) 10 The notation of integrals is taken the same form as [19] I a1,··· ,am,b1,··· ,bn (z i1 , · · · , z im ,z j1 , · · · ,z jn ) ≡ is the area of (d − 1)-sphere with unit radius, also we denote A 2 = (1 − u)u|z 12 | 2 and use the coordinates transformation Let us mention that the result in eq.(136) is consistent with eq.(135), i.e. they satisfy For I 22 (z 1 ,z 2 ) with z 1 = z 2 , similarly we can obtain In summary, by using dimensional regularization we can obtain the following basic integrals which appear in N =(1,1) case I 11 (z i ,z j ) = −π(− 2 ǫ + ln |z ij | 2 + γ + ln π + O(ǫ)), where in the last line the integrals with two points coincide are listed. For these integrals by translation symmetry, we can set z i = 0, thus there is no scale in the integrals and we can set these integrals equal zero in dimensional regularization. Note that the integral I 22 (z i ,z j ) is proportional to a delta function δ (2) (z ij ) in (B.7) of [17]. However, we will omit this delta function here due to the fact that once we let z i = z j in I 22 (z i ,z j ), as mentioned above, by translation symmetry there is no scale in the integral. Thus the term δ (2) (z ij ) in (B.7) of [17] is simply replaced by zero in eq.(139).
By using Feynman parametrization, following the same line as above, we can also obtain the integrals needed in the N = (2, 2) case, which are I 13 (z i ,z j ) = π (z ij ) 2 , I 31 (z i ,z j ) = π (z ij ) 2 , I 23 (z i ,z j ) = I 32 (z i ,z j ) = I 33 (z i ,z j ) = 0, where we also let the integrals with two points coinciding with each other vanish.

Conclusions
In the paper we investigated the correlation functions with TT deformation for N = (1, 1) and N = (2, 2) superconformal field theory perturbatively to the first order in the coupling constant. This extends previous work on the correlation function from bosonic CFT to supersymmetry case. Much like the bosonic CFT, the undeformed 2and 3-point function almost fixed by global superconformal symmetry, while the n-pt (n ≥ 4) functions dependent on a undetermined function f . By using superconformal Ward identities, we work out the correlation functions with TT operator inserted. It is shown that all the integral in first order perturbation can be decomposed into several basic integral as listed in the last section. As a consequence, we only need to evaluate these integrals, which have been done with dimensional regularization. As a possible application, we briefly mentioned the OTOC under deformation. Unlike the bosonic CFT, where the conformal blocks in 4-pt functions can be used to evaluate the OTOC, for superconformal field theory, there is an unknown function f in 4-pt functions. Thus more information about the function f is required to study the OTOC of superconformal field theory.
In the present paper we only considered the effect of TT deformation on correlation functions perturbatively near the IR conformal fixed point. Since TT deformation is believed to have good behavior in the UV, it is interesting to study the correlation functions of superconformal theory in the UV as what has been done for the bosonic CFT in [18]. Another interesting problem is to study the correlation functions in N = (1, 0) and N = (2, 0) theories, which exist for Lorentz signature. Possibly, one can also consider the JT deformation in supersymmetry theory recently studied in [61].

A Integrals in 2pt-correlators
There are nine terms in eq.(42), the first one have been considered in eq. (44). Below by using the integrals in section 4 we list the remaining eight terms in the integral eq.(42).
The second term