$T\bar{T}$-flow effects on torus partition functions

In this paper, we investigate the partition functions of conformal field theories (CFTs) with the $T\bar{T}$ deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature.

1 Introduction The TT deformation of field theory has attracted much research interest in recent years both from viewpoint of field theory and in the context of holographic duality.
Among these progresses, the partition functions as well as correlation functions in deformed CFTs are of particular interest in our present study. The partition functions of the TT deformed CFTs have been computed in [6] by using the known deformed spectrum. Since the results in [6] are nonperturbative, the modular properties can be discussed, and it was shown that the partition functions are modular covariant. From other perspective, the deformed partitions were discussed from random metric point of view [29], and also in the context of holographic duality [36]. As for correlation functions, the deformed one-point functions of KdV charges operators were considered non-perturbatively based on the deformed spectrum [37]. Also the general deformed correlation functions in the UV were considered by J. Cardy in [38].
On the other hand, one can study the TT deformation in a perturbative way.
More concretely, suppose that one can expand L λ around λ = 0, where the first term L (0) corresponds to the un-deformed theory, the second term is the TT operator of un-deformed theory as appeared in the RHS of (4) with λ = 0, the third term and the terms omitted are presented since the stress tensor T λ is not fixed but also flow under the deformation. In other words, the stress tensor depends on λ.
A number of works were done in the framework of perturbation method, for example, in [1] the renormalization of free theory under the TT deformation is investigated by matching the S-matrix. Meanwhile, other physical quantities were also computed perturbatively, such as entanglement entropies, wilson loop and correlation functions [39][40][41]. In this work, we will continue to study the partition functions (which can be treated as zero-point functions) of deformed CFTs in a perturbative manner. The correlation functions of deformed theories were considered earlier in [42][43][44], where two-point functions and three-point functions were calculated, as well as the correlation functions of stress tensors. Later, these results were generalized to higher-point function cases [45,46], as well as including supersymmetry [47], torus CFTs [48], and especially the holographic dual of stress tensor correlation function in large c limit was considered in [49].
In these studies of correlation functions, it is worthwhile to note that computation is mainly performed in the first-order perturbation of CFT or in the case where the CFT is defined on the plane. Naturally, to make progress, the next step is that can we go beyond the first-order perturbation. However, this is a nontrivial question as can be seen as follow. As discussed above, in the first-order perturbation, the TT operator is known which is just constructed from the stress tensor of the undeformed CFT, while in higher-order perturbations, one must take the corrections of TT operator into consideration, namely, TT -flow effects. Unfortunately, in a general CFT, we do not have such an explicit expression on such kinds of corrections.
Nevertheless, as the first step towards higher-order perturbations, we can start with free theory, where the corrections of stress tensor and Lagrangian under the TT deformation can be constructed explicitly order by order. Based on this setup, we will study the corrections of deformed partition functions up to second-order by employing perturbation method. This also generalize our previous work [48], where the first-order partition functions of deformed CFTs on torus were computed.
Moreover, since we work in free theories, we will use Wick contraction rather than the Ward identity obtained in [48] to figure out the deformed correlation functions.
Finally, the two methods will lead to the same results.
The organization of this paper is as follows. In Section 2, we review the general method to obtain the deformed Lagrangian and stress tensor order by order, which can be used to expand the partition function up to the second-order that we are interested in. In Section 3, Section 4 and Section 5, we computed the first-and second-order corrections to the partition functions of free bosons, Dirac fermions and Majorana fermions respectively. We use Wick contraction to computed the deformed partition functions, also some proper regularization methods are chosen.
In Section 6 we continue to calculate the VEV of the first KdV charge in the deformed free theories up to the first-order, by using the perturbative QFT approach. We end in Section 7 with a conclusion and discussion. Our conventions, useful formulae, and some calculation details are presented in the appendices.

TT deformed partition function for generic 2d theory
In this section, we would like to compute the perturbation expansion of TT deformed partition function beyond the first-order. The procedure is based on the method first introduced in [2] (also see [50]), where deformed Lagrangian is obtained order by order. Let us first review this method below.
Consider a TT deformed QFT living in a two-dimensional Euclidean spacetime (M, g ab ) whose dynamics is governed by the local action Here L λ denotes the deformed Lagrangian parameterized by λ. The TT deformation can then be defined by the following flow equation where µν = g µρ g νσ ρσ is the volume element of the spacetime, and T λ µν is the stress tensor of the deformed theory, which is defined as Now expand of deformed Lagrangian and stress tensor in the power of λ In order to figure out L (n) , one can plugging (6) into both (4) and (5). By comparing each order in the resulting expressions, eventually, we obtain the following recursion where C i n ≡ n! i!(n−i)! . Note this recursion relations allow us to obtain L (n) and T (n) µν for arbitrary n, once L (0) , i.e. the un-deformed theory, is given.
With perturbations of L λ acquired, we continue to derive the corrections of the partition function to higher-order in perturbation theory in path integral language, which is where In what follows, we will focus on the TT deformed free theories on a torus, including free bosons, Dirac fermions, and Majorana fermions, where the deformed partition functions up to the second-order (11)(12) can be worked out analytically.

Free bosons
At first, what we would like to consider is the TT deformed free scalar on a torus T 2 . The corresponding action of the un-deformed theory reads 4 The identity g µν g ρσ − g ρν g µσ = µρ νσ is used.
where g is a normalization constant. According to the recursion relations (7-8) mentioned above, one could obtain the deformed Lagrangian and stress tensor starting from L (0) , Then the un-deformed stress tensor is 5 from which the first-order Lagrangian is given by and the corresponding first-order stress tensor is Reusing Eq. (7), we end up with the second-order Lagrangian We then could write out the corrections of the partition function (11) and (12) more concretely for bosonic fields Note that the expectation values in (19)(20) are defined in free theory, all of them could be evaluated directly by applying Wick contraction since the propagator is well-known for torus free scalar field [51], 5 In this paper, we use the conventional notation that T ≡ −2πT zz ,T ≡ −2πTzz, and Θ ≡ 2πT zz . The complex coordinates z := x + iy, where y is Euclidean time. ∂ := (∂ x − i∂ y )/2. The metric g zz = 1 2 .
Here ϑ 1 (z) is one of Jacobi ϑ-functions and η(τ ) is Dedekind η-function. Performing derivatives on (21) gives various two-point functions 6 where P (z) is Weierstrass elliptic function and we have applied the formula∂(z −1 ) = . For more details on elliptic functions please refer to Appendix A. The subsequent derivation of Wick contraction indicates that the expectation values of the composite operators ∂φ(z 1 ,z 1 ) 2 , ∂ φ(z 1 ,z 1 ) 2 , and |∂φ(z 1 ,z 1 )| 2 also make contributions. We regularize them by utilizing the pointsplitting method ∂ φ(z 1 ,z 1 )∂φ(z 1 ,z 1 ) = lim ∂φ(z 1 ,z 1 )∂φ(z 1 ,z 1 ) = lim With all ingredients in place, we next go on to investigate the corrections to the partition function of free bosons.

First-order
First, we note that the partition function of the free scalar on a torus is According to Eq. (19), at the first-order we shall just compute the value of T 2 d 2 x TT (0) (z,z) , which is consistent with [48]. Thus the first-order correction of the partition function is

Second-order
We next go on to consider the second-order correction to the partition function. We begin with calculating the first term of (20), whose integrand can be contracted as 8 where B ≡ ( π τ 2 − 2η 1 ),B ≡ ( π τ 2 − 2η 1 ), and A ≡ π τ 2 . Integrating the above expression amounts to compute the following integrals where g 2 is one of Weierstrass invariants whose definition can be found in Appendix A. We collect the detailed computation of the above integrals in Appendix B.2. Note some of the integrals are divergent, thus a proper regularization scheme is needed, which will be presented in Appendix B.1.
With the help of (31−35) and the following identity relating the quantity g 2 with the double integral of Eq.(31) is derived as Consequently, We next move to evaluate the second term in (20). Using Wick contraction, the integrand is After simple integration, one has Putting together (38) and (40), we obtain the second-order correction of the partition function under the TT deformation which is consistent with [6]. Note that we have minimally subtracted the divergent terms 9 when deriving the RHS of (37), and thus (41). It is possible to implement this minimal subtraction by adding the following counterterm 10 where stands for the radius of the infinitesimal disk regulator.

Free Dirac fermions
For the rest of the examples, we turn our attention to the fermionic fields defined on a torus. We first focus on a massless Dirac field whose action is Our convention for gamma matrices are Pauli matrices.
As before we make the expansion remarkably, the case of fermions will simplify a lot comparing with bosons by the fact that the higher-order terms of L (n) , n ≥ 2 are completely vanishing [50], due to the Grassmannian nature of fermionic fields. Following the derivation presented in [50], we obtain the full expression of L λ and T λ µν written in complex coordinates and It is well-known that the un-deformed partition function for Dirac fermions is given by 11 For the derivation, one can refer to Appendix D.
where ν = 1, 2, 3, 4 denotes the spin structures of fermions, corresponding to different boundary conditions 12 , ϑ ν are Jacobi ϑ-functions. The non-vanishing two-point functions for Dirac fermions with spin structure ν are where Performing derivatives on the propagators leads to the following correlation functions We need further to regularize these correlation functions when two points coincide with each other, in parallel with the bosonic case, we use the point-splitting method Now we have all the required ingredients to calculate the corrections to the partition function.

First-order
Using Wick contraction and the propagators and their derivatives listed above, we can compute the expectation value of T (0)T (0) and (Θ (0) ) 2 12 Z (0) 1 that corresponding to fermions with the double periodic boundary condition is zero, due to the property of Grassmann number [51].
Therefore the first-order correction of the partition function is Note that the first-order correction of free Dirac fermions shares the same structure with that of free bosons (30), which matches the conclusion in [6] obtained by the operator formalism. We're going to show that this is also true for the second-order correction.

Second-order
We now proceed to compute the second-order correction. Since there are no higherorder terms in Lagrangian (L (n) = 0 for n ≥ 2) for free massless Dirac fermions, (12) reduces to After using Wick contraction and discarding the purely divergent terms 13 , we obtain (70) 13 This is similar to the case of the free bosons in the previous section.
The integrals of the nontrivial integrands shown above are listed below For the detailed discussions of the above integrals please refer to Appendix B.3.
With the help of the above nontrivial integrals and identity involving g 2 , e ν−1 , and η 1 one can find that (70) equals Therefore the second-order corrections of the partition function with spin structure ν are which has the same structure with the bosonic case (41), and agrees with the result in [6]. Similar to the case of free bosons, for the deformed free Dirac fermions we can find the counterterm corresponding to the minimum subtraction scheme as follows 14 L DF,ct = λ 2 · 8g 2 π 2 ∂ψ * ψ∂ψ * ψ + 1 24π 3 6 . (79)

Free Majorana fermions
As the last example, we investigate the deformation of free massless Majorana fermions, whose un-deformed action is given by where Ψ=[ψψ] T , the gamma matrices are defined in the previous section.
Similar to the case of complex fermions, the TT flow of Lagrangian truncates at the first order, that is we have where and Note that one could obtain (82-84) by simply removing the " * " in (46)(47)(48)(49)(50)(51).
Taking derivatives on above propagators gives The regularized expectation value of the propagators and their derivatives when two points coincide are In analogy to the Dirac fermion case we now go on to compute the corrections to the partition function.

First-order
According to (82), the first-order correction of the partition function is which takes the same form that of in free massless bosons and free massless Dirac fermions.
Utilizing the nontrivial integrals and the identity (71)-(76) mentioned before, the double integral of (96) equals According to (97), we can obtain that the second-order correction of the partition function for deformed free Majorana fermions as was expected, the second-order corrections of Majorana fermions share the same structure as Dirac fermions (78) and free bosons (41), the conclusion of ref. [6] is confirmed again. Once again the counterterm can be found as 15 It's natural to ask whether the counterterms Eq.(79)(99) introduced in the first two orders are enough or not to cancel the divergences of the higher-order partition function in free fermionic theories. From the perspective of the TT deformation as a 15 The derivation is presented in Appendix C.
kind of irrelevant deformation, one can expect new divergent terms to appear in the higher-order, and there is no a priori reason that new counterterms added to canceling these divergences should be vanishing, although the higher-order deformations of the Lagrangians (48)(82) are truncated due to the Grassmannian structure of the fermion. It is an interesting future problem to perform higher-order calculations to determine the exact higher-order counterterms.

The first KdV charge
In the previous sections, the corrections of various TT deformed partition functions evaluated by the conformal perturbation theory based on Lagrangian path integral are in good agreement with results obtained by the non-perturbative approach [6].
In this section, we proceed with the perturbation method to study the TT -flow effects of the first quantum KdV charge 16 [52], for which there have been studies based on non-perturbation methods [4,37].
Let's first consider a generic CFT, for the sake of convenience we call it a seed later, on a cylinder with coordinate {z,z} and circumference L 17 . After the TT deformation, the deformed left-moving KdV charges P λ s in the resulting QFT take the form where the superscript λ represents the deformation parameter. For s = 1 where H λ = − L 0 dxT λ yy is deformed Hamiltonian and P λ = −i L 0 dxT λ xy is deformed momentum. The expectation value of P λ 1 in the deformed state |n λ thus reads where E λ n and P λ n represent the energy and momentum of the state |n λ respectively. From the TT -flow equations of E λ n , P λ n [1-3, 5] 16 We are grateful to the anonymous referee's suggestion to study the TT deformation of KdV charge. 17 where E n and P n are energy and momentum of the undeformed eigenstate |n in the seed, we could get the closed form for λ n|P λ 1 |n λ depended only on E n , P n , λ, and L From now on, we're going to focus on the case where n = 0 (i.e., the ground state) and the seed theory is free bosons or free Dirac fermions or free Majorana fermions.

Non-perturbative approach
respectively, which leads to λ 0|P λ 1 |0 λ equals 18 For the Dirac or Majorana fermions with periodic boundary condition ψ(z + L) = ψ(z) , the vacuum energy and momentum are which leads to P λ We next to reproduce the above results(106, 108) by utilizing conformal perturbative approach. 18 We denote λ 0|O λ |0 λ as O λ λ o for any flowing operator O λ on a cylinder.

Perturbative approach
According to (101), computing P λ 1 λ o amounts to compute the deformed one-point functions T λ λ o and Θ λ λ o . Thanks to our previous setup, we may obtain the onepoint functions on cylinder by taking the zero temperature limit of the corresponding one-point functions on torus, namely, In Lagrangian path integral formalism, O λ λ tor. equals to where We then make use of (110) to calculate the T λ λ tor. and Θ λ λ tor. of free bosons and free fermions respectively.
With the help of free propagators given in previous sections, after doing Wick contraction and simple integral on a torus, the final results are listed as follows. For free bosons, we get 19 For free Dirac fermions, And for free Majorana fermions, Take the zero temperature limits of (111-116) respectively, one obtains 20 Note that for the fermion cases, ν = 2, 3, 4 correspond to the periodic(space)antiperiodic(time), antiperiodic-periodic and antiperiodic-antiperiodic sectors respectively. It means that for fermions on a cylinder of circumference unity with periodic B. C.
It's easy to check that the perturbative results Eq.(127-129) (Eq. (130-131)) match the results come from non-perturbative method Eq.(106) (Eq.(108)) 21 . The discussions on whether these two approaches match each other at the second-order, which is technically involved, will be served as our future work.

Conclusion and Discussion
In this work, we perturbatively calculate the flow effects of TT deformation on the torus partition functions and the VEV of the first KdV charge P 1 under the Lagrangian path integral formalism. In previous cases [45,47,48], the authors have studied the correlation functions perturbatively up to the first-order deformation.
Generally speaking, to evaluate the correlation functions and higher ordered partition functions perturbatively, the flow of stress tensor must be taken into consideration. As a preliminary study, we focus on the discussions of free theories, including Although the results obtained from the Hamiltonian formalism are reproduced in the Lagrangian path integral formalism, in general, due to the emergence of higher derivative terms in the deformed Lagrangian (16,18), the equivalence between the Lagrangian path integral formalism and the Hamiltonian path integral formalism remains as a mystery. For instance, though Legendre transformation, it can be found 21 The center charges c F B = c DF = 2c M F = 1. 22 For the discussions of T andT flow in generic CFTs, please refer to [38,53].
that the Minkowski Hamiltonian of the deformed free bosons takes the form [43,54] H λ = 1 2λ − 1 + 1 + 2λ π 2 + φ 2 + 4λ 2 (πφ ) 2 where φ is the spatial derivative of φ and π the canonical momentum conjugate to φ. The higher power terms of π presented in (132) prevent us from getting the Lagrangian path integral directly from the corresponding Hamiltonian path integral, since how to deal with the generic integrals go beyond Gaussian integrals, for now, is still a major problem for mathematicians and physicists. It also leads to an open question of whether the Hamiltonian formalism is more fundamental than the Lagrangian formalism [55][56][57][58]. Fortunately, our results show, for the TT deformed theory, the use of disk regularization [59] together with minimum subtraction in Lagrangian formalism seems to be sufficient to match the Hamiltonian formalism.
For the instances considered in this paper, the second-order Lagrangian counterterms corresponding to the minimum subtraction are presented in (42,79,99) respectively.
To match the partition functions and correlation functions between the Lagrangian formalism and the Hamiltonian formalism up to the higher-order deformations will be interesting future work.
Further, it will be interesting to study the second-order deformation to the partition function in the interacting theories, e.g. massive fermions and bosons, Liouville field theory [60], and so on. The generic correlation functions with the TT -flow effects in SUSY extended CFTs will be also an interesting future direction with the following [47].

A Details of Weierstrass functions
In this Appendix, we give the definitions and properties of Weierstrass functions that appear in the calculations.
The first Weierstrass function P (z), called Weierstrass P-function, is defined as The Laurent series expansion of P (z) in the neighborhood of z = 0 is hence we have where g 2 and g 3 are called Weierstrass Invariants The second Weierstrass function ζ(z), called Weierstrass zeta-function, is a primitive function of −P (z) We then define and e 1 := P (w), e 2 := P (−w − w ), e 3 := P (w ), which are functions of the modular parameter τ . Note that there is an identity about η 1 (τ ) and Dedekind eta function η(τ ), which has been used in the bosonic calculations (29).

B Details of some integrations B.1 Prescription for regularization
Since the integrands over a torus we are interested in may contain singularities, in this Appendix we will discuss how to deal with these singularities based on the prescription given in [59].
Let us consider an integrand f (z,z) defined on a torus, which contains N number of singularities (r 1 , r 2 ...r N ). Following the prescription in [59], when performing the integrals, we integrate over not the whole torus T 2 , but over the regularized parallelogram-the parallelogram with small disks around the singularities removed (see Fig.1 for example). In the following, we denote the regularized torus by T 2 .
Suppose we find that then with the Stoke's theorem in 2D space 23 which can be applied to the regularized torus leading to where the contour integrals are anticlockwise. In this paper, we focus further on the case that F µ (z,z) can be written as F µ (z,z) = F µ 1 (z)F µ 2 (z), where F µ 1 is holomorphic function and F µ 2 is anti-holomorphic. For the j-th pole (r j ,r j ) of f (z,z) in T 2 , F µ (z,z) could be expanded around it as follows Therefore, on the grounds of the prescription in [59], we have where G( ) := −π j,n 2(n+1) C 1,z j,n C 2,z j,n+1 + C 1,z j,n+1 C 2,z j,n .
It is worth noting that for the case of F z holomorphic, meanwhile, Fz anti-holomorphic, it must have lim →0 G( ) = 0.

B.2 Integrals for bosonic fields
In this Appendix we record the details of integrals appearing in the calculations of free bosons part (32)(33)(34)(35).
Since all the integrands are double periodic, we can shift the variable of the integration to make life easier without changing the value of the integrals, i.e., We start with the integration of the P -function in a cell. Since P (z) = − ∂ζ(z) ∂z , with the integral strategy shown in Appendix B.1, we have 24 Finally, let us consider integration of |P (z) Similar to the case (154), we regularize the integral by simply discarding the divergent part, which gives According to the results of (148), (151), (154), (155) and (157), we have and

B.3 Integrals for fermionic fields
In this Appendix we present the details of integrals appearing in the calculations of free fermions part (71-75).
We first note that both ∂P ν (z) 2 and P ν (z)∂ 2 P ν (z) are elliptic functions with the modular parameter τ , where e 1 := P (w), e 2 := P (w + w ), e 3 := P (w ). Hence we can expand ∂P ν (z) 2 and P ν (z)∂ 2 P ν (z) in terms of ζ(z) and its derivatives, the results are Consequently, with the integral strategy shown in Appendix B.1, the first two inte- where we have utilized the integral 27 To compute the remaining three integrations, we need to work out the following integrals first In analogy with the bosonic case, in our regularization scheme, we simply drop out the divergent part to obtain the finite answer, that is, 27 For the definition of G( ), please refer to Appendix B.1.
clear that in our cases all the counterterms are proportional to λ 2 , thus from the expression of the second-order correction of the partition function, we know that T 2 L ct only need to cancel the divergent parts in λ 2 For free boson, we first rewrite the integrand (31) as where "..." stands for terms giving finite integral results. As shown in (B.2), under the disk regularization (B.1), we have 28 then the divergent part in λ 2 FB (x 2 ) is given by Finally, to implement the minimal subtraction, from (179) the following choice of counterterm is the simplest one In the following, we would like to determine the counterterms of the deformed free Dirac fermion. Though the integrals given in Appendix (B.3), we can find the divergent terms which we have omitted in the previous text (70), DF (z 2 ,z 2 ) = τ 2 12π 3 6 + τ 2 |e ν−1 | 2 π 3 2 + convergent part = τ 2 12π 3 6 + 16τ 2 g 2 π 2 ∂ψ * ψ∂ψ * ψ + convergent part.

D Derivation of the TT -flow for 2d fermions
In this Appendix, we reproduce the derivation of the TT -flow for 2d fermionic theories as shown in [50].
The action of the un-deformed fermionic theory living in a 2d Euclidean flat spacetime is given by One can rewrite it in a more general form, i.e., the form in curved spacetime, which is where X a µ := X a µ is independent of the metric. We then utilize the recursion relation (7)(8) to derive the expansion of L λ . First of all, the stress tensor of the un-deformed theory  29 The formula ∂e λ c ∂g µν = 1 4 e µc δ λ ν + e νc δ λ µ is used.
Let m = 0, the above results degenerate to the results in Section 4.