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

In the low energy limit, the two-dimensional massless $\mathcal{N}=2$ Wess--Zumino (WZ) model with a quasi-homogeneous superpotential is believed to become a superconformal field theory. This conjecture of the Landau--Ginzburg (LG) description has been studied numerically in the case of the $A_2$, $A_3$, and $E_6$ minimal models. In this paper, by using a supersymmetric-invariant non-perturbative formulation, we simulate the WZ model with two superfields corresponding to the $D_3$, $D_4$, and $E_7$ models. Then, we numerically determine the central charge, and obtain the results that are consistent with the conjectured correspondence. We hope that this numerical approach, when further developed, will be useful to investigate superstring theory via the LG/Calabi--Yau correspondence.


Introduction
At an extremely low-energy scale, the two-dimensional massless N = 2 Wess-Zumino (WZ) model [1] with a quasi-homogeneous superpotential is believed to become an N = 2 superconformal field theory (SCFT). This conjecture of the Landau-Ginzburg (LG) description has been studied from various aspects . However, we have no complete proof of this conjecture. It is difficult to prove because the coupling constant becomes strong at the low-energy region and the perturbative theory possesses infrared (IR) divergences; the LG description is a truly non-perturbative phenomenon. An interesting approach to this issue may be a numerical and non-perturbative technique on the basis of the lattice field theory.
By using such numerical approaches, the A 2 and A 3 minimal models were simulated in Refs. [24][25][26], where superpotentials in the corresponding WZ model are given as the cubic and quartic ones containing a single superfield; see Table 1 [15]. These studies are based on either a lattice formulation in Ref. [27], or a supersymmetry-preserving formulation with a momentum cutoff in Ref. [28]; both non-perturbative formulations make essential use of the existence of the Nicolai map [29][30][31][32]. 1 In above numerical studies, their results of the scaling dimension and the central charge are consistent with the expected values in the A 2 and A 3 minimal models within numerical errors. Therefore, we have now numerical evidences for the LG/SCFT correspondence in the case of the A 2 , A 3 , and E 6 ( ∼ = A 2 ⊗ A 3 ) minimal models.
In this paper, on the basis of the momentum cutoff regularization [28] and the analysis in Ref. [26], we simulate the two-dimensional N = 2 WZ model corresponding to Dand E-type theories. The method in Ref. [26] is generalized to the WZ model with multiple superfields and more complicated superpotentials. Then, from an IR behavior of the energy-momentum tensor (EMT), we numerically determine the central charge of the D 3 , D 4 , and E 7 models; we obtain the results that are consistent with the conjectured correspondence. We also measure the "effective central charge" [25,26], which is analogous to the Zamolodchikov's c-function [34,35]. Although the theoretical background of the formulation is not completely obvious so far, our computational results support the validity of the formulation even if we consider multi-superfield theories. We hope to apply this approach to some models which is neither a minimal model nor a product of minimal models (Gepner model [10,13]), and then develop a numerical method to investigate superstring theory via the LG/Calabi-Yau correspondence [16,[36][37][38]. 2 Nicolai mapping for the multi-superfield WZ model Our numerical simulation is based on the formulation in Ref. [28]. The detailed discussions for the formulation are given in Ref. [26]. These preceding studies treat the two-dimensional N = 2 WZ model with a single superfield. In this section, we summarize basic formulas of the formulation for the WZ model with multiple superfields. Suppose that the system is defined in a two-dimensional Euclidean physical box L 0 × L 1 .
In what follows, we work in the momentum space with an ultraviolet (UV) cutoff, where the Greek index µ runs over 0 and 1, and repeated indices are not summed over; L µ /a is taken as even integers, and a is a unit of dimensionful quantities. A limit a → 0 removes the UV cutoff, being similar to the continuum limit. In fact, when we take L µ /a as odd integers, the unit a itself is the lattice spacing in the dimensional reduction of the four-dimensional lattice formulation [39] based on the SLAC derivative [40,41]. It is well recognized that the regularization based on the SLAC derivative violates the locality. The four-dimensional SLAC derivative is plagued by the pathology that the locality of the theory is not automatically restored in the continuum limit [42][43][44][45]. In the two-or three-dimensional case, on the other hand, one can argue the restoration of the locality in the continuum limit within perturbation theory for massive WZ models [28]. For massless models, it is not clear whether the restoration is automatically accomplished because perturbation techniques are hindered by the IR divergences. We believe that the numerical results in Refs. [25,26] and ours below support the validity of the formulation.
For simplicity, we set a = 1. We basically use the complex coordinates for the momentum, p z = (p 0 − ip 1 )/2 and pz = (p 0 + ip 1 )/2. In general, the two-dimensional N = 2 WZ model contains N Φ superfields, {Φ I } I=1,...,N Φ . A supermultiplet Φ I consists of a complex scalar A I , left-and right-handed spinors (ψ I ,ψ I ), and a complex auxiliary field F I . Then, the action of the two-dimensional N = 2 WZ model with a quasi-homogeneous superpotential W ({A}) is given by where * denotes the convolution The field products in ∂W ({A})/∂A I and ∂W ({A})/∂A I ∂A J are understood as this convolution. Integrating over the auxiliary fields {F }, we obtain the action in terms of the physical component fields, where S B is the bosonic part of the total action The new variables {N} (2.5) specify the so-called Nicolai map [29][30][31][32], the change of variables from {A} to {N}. This mapping simplifies the path-integral weight drastically; the partition function where the fermion fields are integrated is given by where {A} k (k = 1, 2, . . . ) is a set of solutions of the equation The 3 Central charge from the EMT correlator In a two-dimensional SCFT, the central charge c appears in the two-point function of EMT; in terms of the Fourier mode The two-dimensional N = 2 WZ model, which itself is not superconformal invariant and hence does not behave as Eq. Let us write down explicit expressions of EMT and its correlator in the WZ model (2.2). Since our formulation preserves some spacetime symmetries exactly, we can straightforwardly construct Noether currents associated with the symmetries, for example, the supercurrent and EMT. To remove the ambiguity of EMT, we require the traceless condition T zz = Tz z = 0 in the free-field limit [26] (see also Refs. [51,52]). Following the corresponding computation in Ref. [26], EMT is given in the momentum space by Like as the single-supermultiplet case, it can turn out that this expression of EMT is the super-transformation of the supercurrent; for the definition of the super-transformation, see Appendix A in Ref. [26]. Thus we can obtain a less noisy form of the EMT correlator where S ± z is the supercurrent defined by Since the formulation exactly preserves the supersymmetry, this relation between EMT and the supercurrent holds.

Numerical setup and sampling configurations
In what follows, we consider the N Φ = 2 WZ model corresponding to the D 3 , D 4 , and E 7 minimal models. The superpotential is defined by            Table 7, we tabulate the numerical results of the central charge for all box sizes in the D 3 , D 4 , and E 7 models.
The central charge for the maximal box size in Table 7 reads c = 1.595(31) (41) for D 3 , As mentioned in Refs. [25,26], it is interesting to plot the "effective central charge," which is analogous to the Zamolodchikov's c-function [34,35]. This is obtained from the fit in a variety of momentum regions from IR to UV; we take the fitted momentum regions as 2π L n ≤ |p| < 2π L (n + 1) for n ∈ Z + . Then Fig. 5 shows that the "effective central charge"        L n ≤ |p| < 2π L (n + 1), for n ∈ Z + .

Conclusion
In this paper, we numerically studied the two-dimensional N = 2 WZ model corresponding to the D 3 , D 4 , and E 7 minimal models. Utilizing the supersymmetry-preserving formulation with a momentum cutoff [28], we numerically determined the central charge from the IR behavior of the WZ model. Although the theoretical background of our computational approach is not clear so far, our results for the theories with two superfields are consistent with the conjectured correspondence between the LG model and the minimal series of SCFT. In the paper and the preceding studies [24][25][26], we have the numerical evidences of typical minimal models: the A 2 , A 3 , D 3 , D 4 , E 6 , and E 7 models in Table 1; the A 4 or E 8 ( ∼ = A 2 ⊗ A 4 ) minimal model is left to be simulated.
To investigate superstring theory by using our numerical approach, we may start from the numerical simulation of the A 4 minimal model, or simpler theories with several supermultiplets which is not a Gepner model.