Numerical study of the Landau–Ginzburg model with two superfields
In the low energy limit, the two-dimensional massless 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 , , and minimal models. In this paper, by using a supersymmetric-invariant non-perturbative formulation, we simulate the WZ model with two superfields corresponding to the , , and 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.
Numerical study of the Landau–Ginzburg model with two superfields
\@hangfrom Subject Index B16, B24, B34
At an extremely low-energy scale, the two-dimensional massless Wess–Zumino (WZ) model  with a quasi-homogeneous superpotential is believed to become an superconformal field theory (SCFT). This conjecture of the Landau–Ginzburg (LG) description has been studied from various aspects [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. 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 and 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 . These studies are based on either a lattice formulation in Ref. , or a supersymmetry-preserving formulation with a momentum cutoff in Ref. ; both non-perturbative formulations make essential use of the existence of the Nicolai map [29, 30, 31, 32].111For some related works, see Refs. [26, 28] and references therein. It appears that the two-dimensional massless WZ model is numerically studied in Ref. . In above numerical studies, their results of the scaling dimension and the central charge are consistent with the expected values in the and minimal models within numerical errors. Therefore, we have now numerical evidences for the LG/SCFT correspondence in the case of the , , and () minimal models.
In this paper, on the basis of the momentum cutoff regularization  and the analysis in Ref. , we simulate the two-dimensional WZ model corresponding to - and -type theories. The method in Ref.  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 , , and 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 -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. . The detailed discussions for the formulation are given in Ref. . These preceding studies treat the two-dimensional 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 . In what follows, we work in the momentum space with an ultraviolet (UV) cutoff,
where the Greek index runs over and , and repeated indices are not summed over; is taken as even integers, and is a unit of dimensionful quantities. A limit removes the UV cutoff, being similar to the continuum limit. In fact, when we take as odd integers, the unit itself is the lattice spacing in the dimensional reduction of the four-dimensional lattice formulation  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 . 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 . We basically use the complex coordinates for the momentum, and . In general, the two-dimensional WZ model contains superfields, . A supermultiplet consists of a complex scalar , left- and right-handed spinors , and a complex auxiliary field . Then, the action of the two-dimensional WZ model with a quasi-homogeneous superpotential is given by
where denotes the convolution
The field products in and are understood as this convolution. Integrating over the auxiliary fields , we obtain the action in terms of the physical component fields,
where is the bosonic part of the total action
The new variables (2.5) specify the so-called Nicolai map [29, 30, 31, 32], the change of variables from to . This mapping simplifies the path-integral weight drastically; the partition function where the fermion fields are integrated is given by
where (, , …) is a set of solutions of the equation
The weight is a Gaussian function of the variables . All we should do to obtain configurations of and is the following: We generate complex random numbers from the Gaussian distribution for each momentum, and then, solve numerically the algebraic equation (2.7) with respect to . Note that, however, our numerical root-finding analysis may suffer from the systematic error associated with some undiscovered solutions to Eq. (2.7). Ref.  addresses the difficulty of the algorithm.
3 Central charge from the EMT correlator
In a two-dimensional SCFT, the central charge appears in the two-point function of EMT; in terms of the Fourier mode , we have 222In the spacetime with the complex coordinates and , the EMT correlator is given by . This convention is identical to that of Refs. [26, 49, 50].
The two-dimensional WZ model, which itself is not superconformal invariant and hence does not behave as Eq. (3.1), is believed to give an LG description of SCFT. Thus, the IR behavior of the WZ model would be governed by relations as Eq. (3.1) in SCFT. The central charge can be computed from the fit function (3.1) in the IR region.
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 in the free-field limit  (see also Refs. [51, 52]). Following the corresponding computation in Ref. , 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. . Thus we can obtain a less noisy form of the EMT correlator
where is the supercurrent defined by
Since the formulation exactly preserves the supersymmetry, this relation between EMT and the supercurrent holds.
4 Numerical setup and sampling configurations
In what follows, we consider the WZ model corresponding to the , , and minimal models. The superpotential is defined by
We set the couplings and to in the unit of . The box size is taken as even integers , , , , , and for ; , , , , , and for ; , , and for . Other numerical setups are identical to those in Ref. , and hence see Ref.  for more detailed information of our program.
and should be reproduced numerically, and indicate the quality of our configurations. The computational time is given in corehour on the Intel Xeon E5 2.0 GHz for with , …, and with , …, , and the Intel Xeon Gold 3.0 GHz for with , and with , …, .
5 Numerical determination of the central charge
Let us show the main result of this paper, the numerical determination of the central charge for the , , and models.
We plot the correlation function (3.3) in Figs. 1–3 for the maximal box size with the fitting curve (3.1); the central charge is obtained from the fit in the IR region . The left panel in each figure is devoted to the real part of the two-point function and the right one is the imaginary part. The horizontal axis indicates the momentum , and the momentum is fixed to the positive minimal value . In Table 7, we tabulate the numerical results of the central charge for all box sizes in the , , and models.
The central charge for the maximal box size in Table 7 reads
This is the main result in this paper. Here, a number in the second parentheses indicates the systematic error associated with the finite-volume effect. In Eqs. (5.1) and (5.3), we estimate this as follows: We pick out the largest three volumes for each minimal model; from the central values at two smaller ones, we extrapolate to the larger regime as a linear function of the inverse volume , and then, obtain an extrapolated value at the maximal volume (see Fig. 4); the systematic error is identified with the deviation between this and the central value in Eqs. (5.1) and (5.3).333The fit function is a possible choice, but there would be no theoretical evidence to support this choice. Because the behavior of the limit appears not quite smooth as in Fig. 4, we do not attempt to extrapolate to the infinite volume limit. In Eq. (5.2), since we have more than two would-be convergent results at large , the systematic error is estimated by the maximum deviation of the central values at the three largest volumes as in Ref. . Eqs. (5.1)–(5.3) are consistent with the expected values, for within , for within , and for within the numerical errors; the standard deviations are evaluated by the sum of the statistical and systematic errors.
As mentioned in Refs. [25, 26], it is interesting to plot the “effective central charge,” which is analogous to the Zamolodchikov’s -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 for . Then Fig. 5 shows that the “effective central charge” connects the IR central charge to an UV value . This number is consistent with the central charge in the expected free SCFT. Recall that is the number of supermultiplets in the free WZ model, and we have set in this paper.
In this paper, we numerically studied the two-dimensional WZ model corresponding to the , , and minimal models. Utilizing the supersymmetry-preserving formulation with a momentum cutoff , 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 , , , , , and models in Table 1; the or () 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 minimal model, or simpler theories with several supermultiplets which is not a Gepner model.
We would like to thank Katsumasa Nakayama, Hiroshi Suzuki, and Hisao Suzuki for helpful discussions. We are grateful to Hiromasa Takaura for a careful reading of the manuscript. Our numerical computations were partially carried out by using the supercomputer system ITO of Research Institute for Information Technology (RIIT) at Kyushu University. This work was supported by JSPS KAKENHI Grant Number JP18J20935.
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