Monte Carlo studies of dynamical compactification of extra dimensions in a model of nonperturbative string theory
The IIB matrix model has been proposed as a non-perturbative definition of superstring theory. In this work, we study the Euclidean version of this model in which extra dimensions can be dynamically compactified if a scenario of spontaneously breaking the SO(10) rotational symmetry is realized. Monte Carlo calculations of the Euclidean IIB matrix model suffer from a very strong complex action problem due to the large fluctuations of the complex phase of the Pfaffian which appears after integrating out the fermions. We employ the factorization method in order to achieve effective sampling. We report on preliminary results that can be compared with previous studies of the rotational symmetry breakdown using the Gaussian expansion method.
Dynamical compactification of extra dimensions in the Euclidean IIB matrix model
\FullConferenceThe 33rd International Symposium on Lattice Field Theory
14 -18 July 2015
Kobe International Conference Center, Kobe, Japan*
Large- reduced models have been proposed as the non-perturbative definition of superstring theory. Especially, the IIB matrix model  is one of the most successful proposals. The IIB matrix model is formally obtained by the dimensional reduction of ten-dimensional super-Yang-Mills theory to zero dimensions. In the IIB matrix model, spacetime is dynamically generated from the degrees of freedom of the bosonic matrices, despite the fact that it does not exist a priori in the model. Superstring theory is well-defined only in ten-dimensional spacetime, and it is an important question how our four-dimensional spacetime dynamically emerges. Monte Carlo studies of the IIB matrix model have a possibility to shed light on this question from a first principle calculation.
The Euclidean version of the IIB matrix model is obtained after a Wick rotation of the temporal direction. It has a manifest SO(10) rotational symmetry which, if spontaneously broken, yields a spacetime compactified to lower dimensions. However, its numerical simulation has been hindered by the “complex action problem”, because the Pfaffian obtained after integrating out the fermions is complex in general.
Apart from the matrix models of superstring theory, there are many interesting systems that are plagued by the “complex action problem”. Lattice gauge theories with a non-zero chemical potential are the ones that have attracted most of the attention in this context. In this work, we apply the “factorization method”, which was originally proposed in ref.  and generalized in ref. , to the Monte Carlo studies of the Euclidean version of the IIB matrix model. The IIB matrix model has also been studied analytically by the Gaussian Expansion Method (GEM) [4, 5]. Preliminary results of our Monte Carlo simulation are consistent with the GEM results and provide evidence that the factorization method is a successful approach to studying interesting systems that suffer from the complex action problem.
2 Factorization method
Generally, it is difficult to numerically simulate the complex action system
Since is not real positive, we cannot view it as a sampling probability in the Monte Carlo simulation. One way to calculate the vacuum expectation value (VEV) of an observable is to use the reweighting . Here, and are the VEV’s for the original partition function and the phase-quenched partition function , respectively. This is not an easy task since the phase may fluctuate wildly. In order to compute with given accuracy one needs O configurations, where is the system size. This is called the “sign problem” or the “complex action problem”.
Yet another problem is that the important configurations are different for different partition functions. This is called the “overlap problem”. We are plagued with this overlap problem in trying to obtain the VEV through the simulation of the phase-quenched partition .
which are strongly correlated with the phase . In the following, we define the normalized observables . We employ the factorization property of the density of states :
The constant is irrelevant in the following. is the density of states in the phase-quenched model. is the VEV in the constrained system
When the system size goes to infinity, the VEV’s are given by , where is the position of the peak of . This can be obtained by solving the saddle-point equation
When we properly choose the maximal set of the observables , we achieve effective importance sampling for the original partition function .
3 Euclidean version of the IIB matrix model
We study the IIB matrix model , which is defined by the following partition function:
where the bosonic part and the fermionic part are respectively
The bosons () and the Majorana-Weyl spinors () are traceless hermitian matrices. In the following, without loss of generality we set . The indices are contracted by the Euclidean metric after the Wick rotation. are the Gamma matrices after the Weyl projection, and is the charge conjugation matrix. This model has the SO(10) rotational symmetry. In ref. , it is shown that the partition function is positive definite without cutoffs.
This model is formally obtained by the dimensional reduction of ten-dimensional super Yang-Mills theory to zero dimensions. The IIB matrix model has the supersymmetry
For the linear combination and , we have
This leads to the interpretation of the eigenvalues of the bosonic matrices as the spacetime coordinates. Hence, the spontaneous symmetry breakdown (SSB) of the SO(10) rotational symmetry is identified with the dynamical compactification of the extra dimensions.
The order parameters of the SSB of the SO(10) rotational symmetry are the eigenvalues () of the “moment of inertia tensor”
which are ordered as before taking the expectation value. If grow and shrink in the large- limit, this suggests the SSB of the SO(10) rotational symmetry to SO and hence the dynamical compactification of ten-dimensional spacetime to dimensions. This scenario has been studied via GEM in ref. . The results of the studies of the SO symmetric vacua for are summarized as follows:
The extent of the shrunken directions () is , which does not depend on (universal compactification scale).
The ten-dimensional volume of the Euclidean spacetime does not depend on except (constant volume property). For the extent of the extended directions (), the volume is , with .
The free energy takes the minimum value at , which suggests the dynamical emergence of three-dimensional spacetime.
In ref. , the six-dimensional version of the Euclidean IIB matrix model was studied via GEM, and the six-dimensional version also turns out to have these three properties. The same model was studied numerically in ref. , and the results are consistent with the GEM results.
Next, we review the mechanism of the dynamical compactification of spacetime in the Euclidean IIB matrix model . Integrating out the fermions, we have
where is a anti-symmetric matrix. The indices run over , and are the structure constants of SU. are the coefficients in the expansion with respect to the SU generators . Under the transformation , becomes complex conjugate. We define the phase of the Pfaffian as . is real for the nine-dimensional configuration . When the configuration is -dimensional (), we find for , because the configuration is at most nine-dimensional up to the -th order of the perturbations. Thus, the phase of becomes more stationary for the lower dimensions. The numerical results in ref.  also suggest that there is no SSB of the rotational symmetry in the phase-quenched model. We calculate numerically, where is the VEV with respect to the phase-quenched partition function
We use the Rational Hybrid Monte Carlo (RHMC) algorithm, whose details are presented in Appendix A of ref. . The result in fig. 1 shows that converge to at large for all . This suggests that there is no SSB of the SO(10) rotational symmetry, and that the result is consistent with the constant volume property.
The model (6) suffers from a strong complex action problem, and we apply the factorization method to this system. It turns out to be sufficient to constrain only one eigenvalue ; namely the choice of the set in eq. (2) should be . This is because the larger eigenvalues do not affect much the fluctuation of the phase. This choice of is similar to that of the six-dimensional version of the IIB matrix model . When we constrain , the eigenvalues take the small value, which corresponds to the SO symmetric vacuum, with . This leads us to simulate the partition function of the constrained system
which is simulated via the RHMC algorithm. The ratio corresponds to the square of the ratio of the extents of the extended and shrunken directions , in the SO vacua with . The saddle-point equation (5) is now simplified as
in the large- limit. is the VEV of the partition function . We have , because under the transformation the Pfaffian becomes complex conjugate while the bosonic action (7) and the eigenvalues of the tensor (11) are invariant. The solution of the saddle-point equation (15) gives the VEV in the SO vacuum with . Solving this saddle-point equation amounts to finding the minimum of the free energy
in the SO vacuum with . The GEM result suggests that the free energy takes the minimum for the SO(3) vacuum. In order to reduce the CPU costs, we focus on the cases, which correspond to the SO(2), SO(3), SO(4) vacua, respectively.
In fig. 2 (LEFT) we plot for up to , where we observe a good scaling behavior at small
The coefficients and are obtained for each , by fitting the data. Then, we extrapolate the coefficients and obtain the large- limit, which corresponds to . This is represented by the solid line in fig. 2 (LEFT).
The function has a scaling behavior around
Subtracting this effect in order to reduce finite- effects, we plot for in fig. 2 (RIGHT). We find that the results scale reasonably well up to in the small- region . This implies the hard-core potential structure at small . In the six-dimensional version of the IIB matrix model, this effect is absent in the one-loop approximation , but is observed in the full model without one-loop approximation . The intersection of and represents the solution of the saddle-point equation (15). Fig. 2 (RIGHT) shows that the solution is close to for . For , too, we have obtained similar results, and the solution is close to 0.404. This is consistent with the “universal compactification scale” property.
Next, we compare the free energy (17) for the SO vacuum. The free energy at is
with . Due to the scaling behavior (19), the first term of the r.h.s of eq. (20) vanishes at large . Thus we compare . From fig. 3, we see that the free energy is much higher than and around . It is still difficult to determine whether the SO(3) or the SO(4) vacuum is energetically favored. More analysis will be reported elsewhere.
In this work, we have performed Monte Carlo simulations of the Euclidean version of the IIB matrix model using the factorization method, in order to study the dynamical compactification of the extra dimensions. The results turn out to be consistent with the GEM predictions. We have seen that in the phase-quenched model there is no SSB of the SO(10) rotational symmetry, and that the volume of spacetime is consistent with the GEM results. The function has a hard-core potential structure, and as a result of that, the computed shrunken dimensions are found to be consistent with the GEM results. Also, we have succeeded in finding that the SO(2) vacuum is energetically disfavored, compared to the SO(3) or SO(4) vacuum. The results of the Lorentzian version of the IIB matrix model, where (3+1)-dimensional spacetime is found to expand dynamically , and the scenario discussed in this work, suggest that the physical interpretation of the Euclidean IIB matrix model needs to be further investigated.
- Preprint number: KEK-TH-1860
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