Numerical study of the $\mathcal{N}=2$ Landau--Ginzburg model

It is believed that the two-dimensional massless $\mathcal{N}=2$ Wess--Zumino model becomes the $\mathcal{N}=2$ superconformal field theory (SCFT) in the infrared (IR) limit. We examine this theoretical conjecture of the Landau--Ginzburg (LG) description of the $\mathcal{N}=2$ SCFT by numerical simulations on the basis of a supersymmetric-invariant momentum-cutoff regularization. We study a single supermultiplet with cubic and quartic superpotentials. From two-point correlation functions in the IR region, we measure the scaling dimension and the central charge, which are consistent with the conjectured LG description of the $A_2$ and $A_3$ minimal models, respectively. Our result supports the theoretical conjecture and, at the same time, indicates a possible computational method of correlation functions in the $\mathcal{N}=2$ SCFT from the LG description.


Introduction
In sufficiently low energies, any quantum field theory is expected to become scale invariant, all massive modes being decoupled. Such a scale-invariant theory would be described by a conformal field theory (CFT). If this low-energy theory gives rise to a nontrivial CFT, the original field theory is called the Landau-Ginzburg (LG) model or the LG description of the CFT [1]. The LG description thus provides a Lagrangian-level realization of CFT, although the existence of the Lagrangian of the latter is not always obvious.
A non-perturbative calculational method such as the lattice field theory may provide an alternative approach to this issue. In Ref. [25], the scaling dimension of the scalar field in the IR limit of the 2D N = 2 massless WZ model was measured by using a lattice formulation from Ref. [26]. The case of a single supermultiplet with a cubic superpotential W = Φ 3 , which is considered to become the A 2 minimal model in the IR limit, is studied. In Ref. [25], good agreement of the scaling dimension with that of the A 2 model was observed. As is wellrecognized, the lattice formulation is in general not compatible with the supersymmetry (SUSY) that must be a crucial element of the above LG correspondence. This is also the case for the lattice formulation of Ref. [26]. However, the formulation of Ref. [26] exactly preserves one nilpotent SUSY, utilizing the existence of the the Nicolai or Nicolai-Parisi-Sourlas map [27][28][29][30]. 1 Because of this exactly preserved SUSY, and because this 2D theory is super-renormalizable, it can be argued to all orders of perturbation theory that the full SUSY is automatically restored in the continuum limit. 2 The study of Ref. [25] thus paved the way for the numerical investigation of the N = 2 LG model, a triumph of the lattice field theory. 3 Somewhat later, in Ref. [41], the same W = Φ 3 model was analyzed by using the formulation in Ref. [42]; a similar result on the scaling dimension was obtained. A salient feature of the momentum cutoff formulation of Ref. [42] is that it preserves the full set of SUSY as well as the translational invariance even with a finite cutoff. The formulation is (almost) identical to the dimensional reduction of the lattice formulation [43] of the 4D WZ model on the basis of the SLAC derivative [44,45]. Although this formulation exactly preserves SUSY, it sacrifices the locality because of the SLAC derivative. See Ref. [46] for an analysis of the issue of the exact SUSY and the locality. Although the SLAC derivative generally suffers from some pathology [47][48][49], for the 2D N = 2 WZ model it can be argued [42] to all orders of perturbation theory that the locality is automatically restored in the continuum limit. This is precisely because of the exactly preserved SUSY and because this 2D theory is super-renormalizable. Since this formulation preserves the full SUSY, the construction of the associated Noether current, the supercurrent, is straightforward. Then, from the IR limit of the two-point function of the supercurrent, the central charge being fairly consistent with the A 2 model was observed. Thus, this study again supports the conjectured LG correspondence.
In this paper, following on from the study of Ref. [41], we carry out the numerical study of the N = 2 LG model on the basis of the formulation of Ref. [42]. In several aspects we extend and improve the analysis in Ref. [41]. First, we study a higher critical model W = Φ 4 , which would correspond to the A 3 minimal model, as well as W = Φ 3 to obtain further support for the LG correspondence and the validity of the formulation. For the scaling dimension, in this paper we use the two-point function in the momentum space instead of the susceptibility of Ref. [41]. Second, the numerical accuracy and the effective number of configurations in the Monte Carlo simulation are quite improved. Third, we also measure the central charge by using the two-point function of the energy-momentum tensor, not only by that of the supercurrent. In Ref. [41], it was reported that the former correlation function was too noisy for extracting the central charge; in the present paper, we avoid this problem by rewriting the correlation function of the energy-momentum tensor by that of the supercurrent by using SUSY Ward-Takahashi (WT) relations. It turns out that after this transformation, the correlation function of the energy-momentum tensor is rather useful to extract the central charge. We also repeat the calculation of the "effective central charge" in Ref. [41] that is an analogue of the Zamolodchikov c-function [50,51]. All our results below show a coherence picture being consistent with the conjectured LG correspondence.
In view of the LG/Calabi-Yau correspondence [17,[52][53][54], we hope that this kind of numerical method will eventually provide a computation method for scattering amplitudes in a superstring theory, whose world sheet theory is given by an N = 2 SCFT but not necessarily by the product of solvable minimal models.

The classical action
It is believed that the 2D N = 2 WZ model provides the LG description of the 2D N = 2 SCFT. 4 The action of the 2D WZ model can be obtained by the dimensional reduction of the 4D N = 1 WZ model [2] whose (Euclidean) action is given by (2.1) Here, θ andθ are Grassmann coordinates and Φ is the chiral superfield, consisting of a complex scalar A, a left-handed spinor ψ, and an auxiliary field F ; the coordinate y is given by where σ 0 is the unit matrix and σ 1,2,3 the Pauli matrices. The superpotential W (Φ) (W (Φ)) in Eq. (2.1) is assumed to be a polynomial of the superfield Φ (Φ). Under the dimensional reduction, we eliminate the dependence on the coordinates x 2 and x 3 . The coordinates x 0 and x 1 are identified with the 2D coordinates; in what follows, we use the complex coordinates quite often: The corresponding derivatives are given by Here, by N = 2, we mean N = (2, 2) and not N = (2, 0).

4
With these notations, 5 the Euclidean action of the 2D N = 2 WZ model is given by 6 The basic symmetries of this system, including SUSY, are summarized in Appendix A.

Momentum cutoff regularization
We quantize the system of Eq. (2.8) by employing a momentum cutoff regularization; this approach is studied in Ref. [42]. As emphasized in Ref. [42], this regularization exactly preserves important symmetries of the system, SUSY and the translational invariance. This is the good news. The bad news is that the regularization breaks the locality. In fact, this formulation is (when the integers L µ /a are odd implying a spacetime lattice with periodic boundary conditions; see below) nothing but the dimensional reduction of the SUSY-invariant lattice formulation of the 4D WZ model of Ref. [43] that is based on the SLAC derivative [44,45]. It is well recognized that the SLAC derivative generally suffers from some pathology [47][48][49].
On the other hand, for the 2D N = 2 WZ model, one can argue to all orders of perturbation theory that the locality is automatically restored when the UV cutoff is removed, thanks to the exactly preserved SUSY [42]. However, since this argument is based on perturbation theory, whose validity for the present massless WZ model is not clear due to the IR divergences, strictly speaking, the theoretical basis of our numerical simulation is not quite obvious. Nevertheless, our numerical results below (and those of Ref. [41]) show a coherent picture which strongly suggests the validity of the approach. We want to leave understanding the observed validity of our formulation as a future problem. Now, let us suppose that the system is defined in a box of physical size L 0 × L 1 . The Fourier transformation of each field ϕ(x) in Eq. (2.8) is then defined by where p µ = 2π L µ n µ , (n µ = 0, ±1, ±2, . . . ).
(2.10) 5 Defining a two-component Dirac fermion by ψ ≡ ψ 1 ψ2 andψγ 0 ≡ (ψ1, ψ 2 ), the 2D Dirac matrices are given by that is, The Euclidean action of the auxiliary field in the Wess-Zumino model has the "wrong sign", i.e, the sign is opposite to the Gaussian one. In this sense, the functional integral containing the Euclidean action of the auxiliary field is merely a formal expression. We understand that the auxiliary field is always expressed by using the equation of motion. The functional integral then becomes perfectly well defined under this understanding. Our computation below is based on such a well-defined functional integral.
Note that ϕ * (p) = ϕ(−p) * . (2.11) After eliminating the auxiliary field F by the equation of motion, the action in Eq. (2.8) in terms of the Fourier modes yields where and S B is the boson part of the action, 14) It is understood that the field product in W ′′ (A) and W ′′ (A) * is defined by the convolution of Eq. (2.13).
In order to define the functional integral, we then introduce the momentum cutoff Λ and restrict the momentum as |p µ | ≤ Λ ≡ π a for µ = 0 and 1.
We then define the partition function by (2.17) Equation (2.12) is the action in classical theory and thus is invariant under the SUSY transformation and the translation. Since these transformations act on field variables linearly (see Appendix A for their explicit form) and do not change the momentum label p, these transformations preserve the restriction on the Fourier modes in Eq. (2.16). As the consequence, our formulation in Eq. (2.17) manifestly preserves these symmetries [42].

Nicolai map
Our definition of the partition function in the regularized level, Eq. (2.17) of the 2D N = 2 WZ model allows the Nicolai or Nicolai-Parisi-Sourlas map [27][28][29][30], which renders the partition function Gaussian integrals [42]. 7 The point is that the Dirac determinant in Eq. (2.17) coincides with the Jacobian associated with the change of integration variables from (A, A * ) to (N, N * ), the variables defined in Eq. (2.14), up to the sign: Hence, after the integration over the fermion fields, the partition function is represented as where A i (i = 1, 2, . . . ) are solutions of the set of equations 20) and A * i are their complex conjugate. Note that, as Eq. (2.14) shows, e −SB is Gaussian in terms of the variables (N, N * ); this is, thus, a drastic simplification.
The representation in Eq. (2.19) thus presents the following simulation algorithm [35] (see also Ref. [55]): (1) Generate complex random numbers N (p) for each p µ in Eq. (2.16) whose real and imaginary parts obey the Gaussian distribution.
where O is an observable of interest. In Appendix B, we present a fast algorithm for the computation of sign det ∂(N,N * ) ∂(A,A * ) . (4) Repeat steps (1)-(3) and compute the averages over configurations of N : Here, ∆ is the normalized partition function, i.e., the Witten index [56,57]. 8 If the superpotential W is a polynomial of degree n, we should have ∆ = n − 1. 9 Since it is easy to generate Gaussian random numbers without any notable autocorrelation, the above algorithm is completely free from any undesired autocorrelation and the critical slowing down; this is a remarkable feature of this algorithm.
Unfortunately, in step (2) we cannot judge whether all the solutions of Eq. (2.20) are found or not because we cannot know a priori the total number of solutions A i for a given N . The best thing we can do is to collect as many solutions as possible. For this issue, the stability of the number of solutions under the increase of initial trial solutions in the solver algorithm, the agreement of ∆ with the expected Witten index and the observation of expected SUSY WT relations provide some consistency checks. In any case, the physical quantities we will compute in what follows, the scaling dimension and the central charge, cannot be free from the systematic error associated with the "undiscovered solutions." It is difficult to estimate the size of this systematic error at this time and the quoted values of the scaling dimension and the central charge should be taken with this reservation.

Simulation setup and classification of configurations
In this paper we consider the 2D WZ model of Eq. (2.8) with the superpotential with n = 3 and 4, which will be written in the abbreviated forms as W = Φ 3 and W = Φ 4 , respectively. We set the coupling constant in units of a = 1, as in Refs. [25,41]. To solve Eq. (2.20) with respect to A, we employ the Newton-Raphson (NR) method. 10 The quality of the obtained configuration A is estimated by the following norm of the residue: As we will see below, maximum values of this number are smaller than 10 −14 for all obtained configurations, and this is much smaller than the corresponding number in Ref. [41]. For a fixed configuration of N , we randomly generate 11 initial trial configurations of A so that we obtain 100 solutions for A allowing repetition of identical solutions; this is another improvement compared to the setup of Ref. [41]. A randomly generated initial configuration does not necessarily converge to a solution along the iteration in the NR method; sometimes it diverges and does not provide any solution. 12 Two solutions A 1 and A 2 are regarded as 10 For the generation of the configurations of N and for the computation of A and sign det ∂(N,N * ) we used a C++ library Eigen [58]. In particular, we extensively used the class PartialPivLU. 11 The initial value of the real and imaginary parts of A(p) is generated by the Gaussian random number with unit variance as in Ref. [41]. 12 In Ref. [41], the number of initial trial configurations is fixed to 100 but we found that this choice sometimes misses some solutions for A, especially for W = Φ 4 . 8 identical if the norm of the difference of the solutions, is smaller than 10 −13 .
For both cases W = Φ 3 and W = Φ 4 , for each box size L, we generate 640 configurations of N using the Gaussian random number. The box size L is taken as even integers from 8 to 36 for W = Φ 3 and from 8 to 30 for W = Φ 4 . We tabulate the classification of configurations we obtained in Tables 1-3 for W = Φ 3 and in Tables 4-6 for W = Φ 4 . The symbols such as (+++−) 2 , for example, imply the following: For a certain configuration of N , we found four solutions A i (i = 1, . . . , 4); sign det ∂(N,N * ) at three of those solutions is positive and negative at one solution. The subscript 2(= 1 + 1 + 1 − 1) stands for the contribution of that N configuration to ∆ in Eq. (2.23). Table 3, for example, shows that for L = 36 we had 13 such configurations of N out of 640 configurations.
In the tables, to indicate the quality of the configurations obtained we list ∆ from Eq. (2.23), which should reproduce 2 and 3 for W = Φ 3 and W = Φ 4 , respectively. For W = Φ 3 , our simulation gives ∆ = 2 exactly for all box sizes. For W = Φ 4 , ∆ deviates from 3 for L ≥ 26 but only slightly; from this, it might be possible to roughly estimate that the systematic error associated with the solution search (i.e., the possibility that some solutions are missed) is less than 0.5% even for W = Φ 4 .
For the same purpose, we also list the one-point function, where S B is defined in Eq. (2.14), which should identically vanish if the SUSY is exactly preserved [36,41]. 13 We also show the maximal value of the norm of the residue in Eq. (3.3) and the computation time in core · hour on an Intel Xeon E5 2.0 GHz.     (45)  The hot spot in our computation is the LU decomposition involved in the NR method whose computational time scales as ∝ N 3 for a matrix of size N . Thus, we expect that the computational time scales as ∝ L 6 as a function of the lattice size L. The actual computational time shown in Fig. 1 is fairly well explained by this theoretical expectation.

SUSY Ward-Takahashi relation
As mentioned above, our formulation exactly preserves SUSY even with a finite cutoff. Thus, barring the statistical error and the systematic error associated with the solution search, SUSY WT relations should hold exactly for any parameter. The observation of these relations  (25) 3 15 (16) where the explicit form of the SUSY transformation is given in Appendix A. First, Eq. (4.1) yields  whose real and imaginary parts are        and the real and imaginary parts are given by If the WT relations hold exactly, the "bosonic" points and the "fermionic" points in the plots should coincide with each other. Overall, we observe good agreements within 1σ, as expected. However, there still exist some deviations of order 2σ, especially in the real-part WT relations at the largest spatial momentum p 1 = π. To argue that these deviations are a result of statistical fluctuations and not due to the omission of some solutions in our solution search, we carried out the measurements corresponding to the left panels of Figs. 2 and 6, respectively but for L = 8, by changing the number of configurations by four times, i.e., 640 and 2560. The results are shown in Figs. 10 and 11. We see that although for 640 configurations there exist some discrepancies between the "bosonic" and "fermionic" ones of 15 A way to derive this relation is to introduce the source term for the auxiliary field: Then, after a (formal) Gaussian integration over the auxiliary field, this term changes to    order 2σ, when we increase the number of configurations by four times, the statistical error is halved and the discrepancies of the central values actually decrease. From this behavior, we think that the observed discrepancies in the WT relations are due to statistical fluctuations and they eventually disappear as the number of configurations is increased sufficiently.
Finally, we mention a general tendency of the statistical error in the correlation functions we found through the numerical simulation. Particularly in the high momentum region, the correlation functions of the scalar field suffer from larger statistical fluctuations than those of the fermion field (as seen in the left panel of Fig. 2). Actually, because of this problem we could not directly examine four-point SUSY WT relations including a four-point correlation function of A and A * . On the other hand, if we assume the validity of SUSY WT relations, we can use them to rewrite some noisy correlation functions into less noisy ones. This technique will be employed frequently in the following sections.

Scaling dimension
In this section, we measure the scaling dimension of the scalar field in the IR limit from the two-point correlation function. If the expected LG correspondence for the WZ model with W = Φ n holds, the chiral superfield is identified with the chiral primary field in the     Thus the two-point function of the scalar field is expected to behave as for large |z|. To obtain the value of the scaling dimension h +h, in Ref. [41], the authors computed the susceptibility To avoid the UV ambiguity at the contact point x ∼ 0, a small region around x = 0 was excised [25]. Then, for the scaling dimension, they obtained Here, we instead directly study the correlation function in the momentum space A(p)A * (−p) . The Fourier transformation of Eq. (5.2) reads (assuming h =h) for |p| small. Also since the SUSY WT relation of Eq. (4.3) shows that instead of the two-point function of the scalar field, we may use the two-point function of the fermion field, which is less noisy, as already mentioned. Figure 12 shows ln A(p)A * (−p) as a function of ln p 2 in the case of the maximal box size, i.e., L = 36 for W = Φ 3 and L = 30 for W = Φ 4 , respectively. We also show the fitting lines in the UV region π √ 2 ≤ |p| < π and in the IR region 2π L ≤ |p| < 4π L . Table 7 summarizes the scaling dimension obtained from the linear fit in the IR region, which is one of our main results in this paper. Recall, however, that those numbers may contain the systematic error associated with the solutions undiscovered by the NR method.
It may be of some interest to see how the values are changed if we do not include a few percent of "strange solutions" in Tables 1-6, such as (+++−) 2 in Table 1. Thus, we have computed the scaling dimension 1 − h −h by using only the (++) 2 -type solutions for W = Φ 3 (L = 36) and the (+++) 3 -type solutions for W = Φ 4 (L = 30). The result is:   We also plotted in Fig. 13  We clearly see the tendency that the measured scaling dimension approaches the expected value as L increases. The approach appears not quite smooth, however, so we do not try any fitting of this plot to extract the L → ∞ value; we suspect that this non-smoothness is due to statistical fluctuations as we observed for the SUSY WT relation in the previous section. From the 1 − h −h case presented in Fig. 13, we estimated the systematic error associated with the finite-volume effect. We estimate it by the maximum deviation of the central values at the three largest volumes; the values obtained in this way are presented in the second parentheses of 1 − h −h in Table 7. It is also interesting to see the "effective scaling dimension" that is obtained from the fitting in some restricted intermediate region of the momentum norm |p|. This is shown in Fig. 14. In both panels, the "effective scaling dimension" smoothly changes from that in the IR region (which is summarized in Table 7) and approaches 1 − h −h → 1 in the UV limit. This behavior is consistent with the expectation that the 2D N = 2 WZ models become the free N = 2 SCFT in the UV limit, in which the chiral multiplet should have the scaling dimension 1 − h −h = 1.

Central charge
In this section we consider the measurement of the central charge c, an important quantity that characterizes CFT. This appears, in the first place, in the operator product expansion (OPE) of the energy-momentum tensor, 16 Similarly, in N = 2 SCFT, the two-point functions of the supercurrent S ± and the U (1) current J are given by Thus, the central charge may also be obtained by computing these two-point functions.
To find the appropriate expression for the supercurrent, the energy-momentum tensor, and the U (1) current such that they form the superconformal multiplet in N = 2 SCFT is itself an intriguing problem, because in our system the N = 2 superconformal symmetry is expected to emerge only in the IR limit. As explained in Appendix A, we adopt the expressions of the former two which become (gamma-) traceless for the free massless WZ model, W ′ = 0. It appears that those expressions work as expected (see also Ref. [41]).
As in the previous section, we numerically compute the correlation function in the momentum space. We consider the two-point functions of the supercurrent, the energy-momentum tensor, and the U (1) current. As we will explain, these two-point functions are related to each other by SUSY, which is an exact symmetry of our formulation. Using this fact, the computation of the whole correlation function can be reduced to that for the supercurrent correlator.

Central charge from the supercurrent correlator
The argument in Appendix A gives the supercurrent in the momentum space, We thus compute the two-point function S + (p)S − (−p) . The Fourier transformation of Eq. (6.4) is, on the other hand, where we have introduced a regulator δ to tame the singularity at x = 0; K 2 is the modified Bessel function of the second kind. Since we are interested in the IR limit, taking the limit |p|δ → 0, we have We fit the measured two-point function S + (p)S − (−p) in the IR region by this function. We plot the two-point function S + (p)S − (−p) in Figs. 15 and 16 for the maximal box size, i.e., L = 36 for W = Φ 3 and L = 30 for W = Φ 4 . In each figure, the left panel is the real part of the correlation function and the right one is the imaginary part. The spatial momentum p 1 is fixed to the positive minimal value, p 1 = 2π/L. In these figures we also show the function on the right-hand side of Eq. (6.9) with the central charge c obtained from the fit in the IR region 2π L ≤ |p| < 4π L ; the central charges obtained in this way are tabulated in Table 8. Again, these numbers may contain the systematic error associated with the solutions undiscovered by the NR method.
Compared to the result of Ref. [41] for W = Φ 3 ,     In Fig. 17 we have plotted how the fitted central charge changes as a function of the box size L. From the c presented in Fig. 17, we estimated the systematic error associated with the finite-volume effect. We estimate it by the maximum deviation of central values at the largest three volumes; the values obtained in this way are presented in the second parentheses for c in Table 8. As for Fig. 14 in the previous section, it is interesting to see how the central charge obtained by the fit changes as a function of the fitted momentum region [41]. The result is shown in Fig. 18. This "effective central charge" depending on the momentum region is analogous to the supersymmetric version of the Zamolodchikov c-function [50,51]. As expected, the "effective central charge" changes from the IR value to c = 3 in the UV limit in which the system is expected to become a free N = 2 SCFT.   6.2. Central charge from the energy-momentum tensor correlator As discussed in Appendix A, the energy-momentum tensor T = T zz , which is expected to be consistent with the conformal symmetry, is given in the momentum space by It turns out that this expression as it stands leads to a very noisy two-point correlation function. Fortunately, noting the fact that the energy-momentum tensor of Eq. (6.11) is the SUSY transformation of the supercurrent in Eqs. (6.6) and (6.7), (6.12) where the SUSY transformation is given in Appendix A, we can express the two-point function of the energy-momentum tensor by a linear combination of two-point functions of the supercurrent which are less noisy: Note that this relation holds exactly in our formulation that preserves SUSY. The Fourier transformation of Eq. (6.3) is, by the same procedure as Eqs. (6.8) and (6.9), (6.14) We plot the two-point function T (p)T (−p) of Eq. (6.13) in Figs. 19 and 20 for the maximal box size, i.e., L = 36 for W = Φ 3 and L = 30 for W = Φ 4 . In each figure, the left panel is the real part of the correlation function and the right one is the imaginary part. The spatial momentum p 1 is fixed to the positive minimal value, p 1 = 2π/L. In these figures we also show the function in Eq. (6.14) with the central charge c obtained from the fit in the IR region 2π L ≤ |p| < 4π L . The central charges obtained in this way are tabulated in Table 9; this is another main result of this paper. Recall again, however, that these numbers may contain the systematic error associated with the solutions undiscovered by the NR method. We repeated the computation of the central charge c by using only the (++) 2 -type solutions for W = Φ 3 (L = 36) and the (+++) 3 -type solutions for W = Φ 4 (L = 30), to see how the   One may note that the fit in Table 9 is better than that in Table 8, in the sense that χ 2 /d.o.f. is very close to 1 in the former. This is due to the fact that the real and imaginary parts of the two-point correlation function of Eq. (6.13) are exactly (anti-)symmetric under p → −p, while the numerical data of S + (p)S − (−p) itself does not possess this property. 17 The number of data points is thus effectively doubled.
In Fig. 21, we plotted how the fitted central charge changes as a function of the box size L. From c presented in Fig. 21, we again estimated the systematic error associated with the finite-volume effect. The values obtained in this way are presented in the second parentheses for c in Table 9.
Also, in Fig. 22 the "effective central charge" obtained from the fit in various momentum regions is depicted; from IR to UV, it again shows the expected behavior analogously to the Zamolodchikov c-function.     Finally, we consider the U (1) current correlator. As discussed in Appendix A, the U (1) current is given by The two-point function of this current is expected to behave in the IR limit as We note that the supercurrent S ± can be rewritten as the SUSY transformation of J, Therefore, This shows that the computation of the U (1) current correlator is identical to the energymomentum tensor correlator of Eq. (6.13) up to a proportionality factor. We expect that we would obtain almost the same results as the previous subsection, so we do not carry out the analysis on this correlator.

Conclusion
In this paper, following on from the study of Ref. [41], we numerically studied the IR behavior of the 2D N = 2 WZ model with the superpotentials W = Φ 3 and W = Φ 4 . We used the SUSY-invariant momentum-cutoff formulation which allows, because of exact symmetries, a straightforward construction of the Noether currents, i.e., the supercurrent and the energymomentum tensor. The simulation algorithm is free from autocorrelation because it utilizes the Nicolai map. From two-point correlation functions in the momentum space, we determined the scaling dimension of the scalar field (Table 7) and the central charge (Table 9) in the IR region. It appears that these results, with the flow of the "effective central charge" in Fig. 22, are consistent with the conjectured LG correspondence to the A 2 and A 3 minimal SCFT. 18 As future prospects, we may further extend the present study to WZ models with multiple superfields and more complicated superpotentials such as the ADE-type theories in Table 10 [16] (the results of the present paper apply to the A 2 , A 3 , and E 6 models).
For a possible application of the present calculational method to the superstring compactification to the Calabi-Yau quintic threefold, the simulation of the W = Φ 5 model will be an important starting point. We are now considering various possible extensions of the present study.
Qα is, on the other hand, defined bȳ andQ2 We see that these transformations fulfill simple anti-commutation relations, The supercurrent, the Noether current associated with SUSY can be read off by considering the localized SUSY transformation in the action. That is, under where ϕ stands for a generic field and ξ α (x) andξα(x) are localized Grassmann parameters, the action changes as (A22) Here, superscripts ± denote the U (1) charge ±1, which will be defined in Sect. A.3 below. The definition of the supercurrent S ± µ is still ambiguous because of the freedom to add a divergence-free term and/or a term that is proportional to the equation of motion. We can remove the ambiguity [41] by imposing the gamma-traceless condition, that is, for the massless free WZ model, W ′ = 0.
where we have assumed that both integers L 0 and L 1 are even, We write the matrix in Eq. (B2) as It should be noted that the diagonal matrix P , whose p, q element is 2p z δ p,q , does not have an inverse because it has zero at p = 0; what we want to do is to remove this zero. In an analogous way, we can write, for the general case, where W 0,0 is the component at (p, q) = (0, 0), det ′ is the determinant in the subspace in which the components with p = 0 or q = 0 are omitted, and Note that this is simply the determinant of a 2 × 2 matrix. Since the right-hand side of Eq. (B7) refers to the subspace in which P has an inverse, the Jacobian can be expressed as Here, the inverse of P is given by P −1 p,q = 1 2p z δ p,q = pz 2|p z | 2 δ p,q . (B11) Thus, substituting the matrix elements in Eq. (B2), we have