We study the properties of the space-time that emerges dynamically from the matrix model for type IIB superstrings in ten dimensions. We calculate the free energy and the extent of space-time using the Gaussian expansion method up to the third order. Unlike previous works, we study the SO(d) symmetric vacua with all possible values of d within the range 2 ≤ d ≤ 7, and observe clear indication of plateaus in the parameter space of the Gaussian action, which is crucial for the results to be reliable. The obtained results indeed exhibit systematic dependence on d, which turns out to be surprisingly similar to what was observed recently in an analogous work on the six-dimensional version of the model. In particular, we find the following properties: i) the extent in the shrunken directions is given by a constant, which does not depend on d; ii) the ten-dimensional volume of the Euclidean space-time is given by a constant, which does not depend on d except for d = 2;iii) The free energy takes the minimum value at d = 3. Intuitive understanding of these results is given by using the low-energy effective theory and some Monte Carlo results.

Recently, the Gaussian expansion method has been applied to investigate the dynamical generation of 4d space-time in the IIB matrix model, which is a conjectured nonperturbative definition of type IIB superstring theory in 10 dimensions. Evidence for such a phenomenon, which is associated with the spontaneous breaking of the SO(10) symmetry down to SO(4), has been obtained up to 7-th order calculations. Here we apply the same method to a simplified model, which is considered to exhibit an analogous spontaneous symmetry breaking via the same mechanism as conjectured for the IIB matrix model. The results up to 9-th order demonstrate a clear convergence, which allows us to unambiguously identify the actual symmetry breaking pattern by comparing the free energy of possible vacua and to calculate the extent of "space-time" in each direction. §1. IntroductionIt has long been believed that matrix models may be useful as a nonperturbative formulation of string theory, and hence play an important role similar to the lattice formulation of quantum field theory. Indeed, matrix models have been quite successful in formulating non-critical string theory, and after the development of such notions as string duality and D-branes, the idea has been extended also to critical strings. The IIB matrix model 1) is one such proposal, which is conjectured to be a nonperturbative definition of type IIB superstring theory in 10 dimensions. It is a supersymmetric matrix model, which can be formally obtained as the zero-volume limit of 10d SU (N ) super Yang-Mills theory.In this model space-time is represented by the eigenvalue distribution of ten bosonic matrices. 2) If the distribution collapses dynamically to a four-dimensional hypersurface, which, in particular, requires the SO(10) symmetry of the model to be spontaneously broken, we may naturally understand the dimensionality of our space-time as a result of the nonperturbative dynamics of superstring theory. In Ref.3) the first evidence for the above scenario was obtained by calculating the free * ) 488 J. Nishimura, T. Okubo and F. Sugino energy of space-times with various dimensionalities using the Gaussian expansion method up to 3rd order. Higher-order calculations, 4), 5) as well as tests of the method itself in simpler models, 6), 7) have strengthened the conclusion considerably. Further evidence for the emergence of a four-dimensional space-time based on perturbative calculations around fuzzy-sphere like solutions is provided in Ref. 8).While these results are certainly encouraging, it is desirable to understand the mechanism responsible for the spontaneous symmetry breaking (SSB) of rotational symmetry. In Ref. 9) it is pointed out that the phase of the fermion determinant favors lower-dimensional configurations, since the phase becomes stationary around such configurations. * ) Indeed, Monte Carlo simulations show that SSB does not occur in various models without such a phase factor. 12)-14) Unfortunately, including the effects of the phase in Monte Carlo simulations is techni...

The IKKT matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, spacetime is conjectured to emerge dynamically from the microscopic matrix degrees of freedom in the large-N limit. Indeed in the Lorentzian version, Monte Carlo studies suggested the emergence of (3+1)-dimensional expanding spacetime. Here we study the Euclidean version instead, and investigate an alternative scenario for dynamical compactification of extra dimensions via the spontaneous symmetry breaking (SSB) of 10D rotational symmetry. We perform numerical simulations based on the complex Langevin method (CLM) in order to avoid a severe sign problem. Furthermore, in order to avoid the singular-drift problem in the CLM, we deform the model and determine the SSB pattern as we vary the deformation parameter. From these results, we conclude that the original model has an SO(3) symmetric vacuum, which is consistent with previous results obtained by the Gaussian expansion method (GEM). We also apply the GEM to the deformed matrix model and find consistency with the results obtained by the CLM.

We investigate the spontaneous breaking of the SO(D) symmetry in matrix models, which can be obtained by the zero-volume limit of pure SU(N ) super Yang-Mills theory in D = 6, 10 dimensions. The D = 10 case corresponds to the IIB matrix model, which was proposed as a non-perturbative formulation of type IIB superstring theory, and the spontaneous breaking corresponds to the dynamical compactification of space-time suggested in that model. First we study the D = 6 case by the Gaussian expansion method, which turns out to yield clearer results than the previous results for the D = 10 case for certain technical reasons. By comparing the free energy of the SO(d) symmetric vacua for d = 2, 3, 4, 5, we conclude that the breaking SO(6) → SO(3) actually occurs. We find that the extent of space-time in the shrunken directions is almost independent of d. In units of this universal scale, the extended directions seem to have large but still finite extents depending on d. We show that these results for the extent of space-time can be explained quantitatively by an argument based on the low-energy effective theory. With these new insights, we reconsider the previous results for the IIB matrix model, and find that they are also consistent with our argument based on the low-energy effective theory. Thus we arrive at comprehensive understanding and some quantitative predictions concerning the nature of the spontaneous symmetry breaking taking place in these models. The space-time picture that emerges from the IIB matrix model and its implication on possible interpretations of the model are also discussed.Subject Index: 125 §1. Introduction It has long been considered that matrix models provide a non-perturbative formulation of string theories analogous to lattice gauge theory for QCD. After the revolution triggered by the discovery of D-branes, there appeared concrete proposals for critical superstring theories and M theory, 1)-3) which take the form of large-N reduced models. For instance, the IIB matrix model, 2) which is proposed as a non-perturbative formulation of type IIB superstring theory, can be obtained by the zero-volume limit of pure SU (N ) super Yang-Mills theory in 10 dimensions.As a possible non-perturbative phenomenon analogous to quark confinement in * )

We advocate a method to improve systematically the self-consistent harmonic approximation (or the Gaussian approximation), which has been employed extensively in condensed matter physics and statistical mechanics. Such a method was previously applied to the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory in ten dimensions. Remarkably the dominance of four-dimensional space-time in the partition function was suggested from calculations up to the 3rd order. Recently this calculation has been extended to the 5th order, and the same conclusion has been obtained. Here we apply this Gaussian expansion method to the bosonic version of the IIB matrix model, where Monte Carlo results are available, and demonstrate the convergence of the method by explicit calculations up to the 7th order. More generally we study matrix models obtained from dimensional reduction of SU(N ) Yang-Mills theory in D dimensions, where the D = 10 case corresponds to the bosonic IIB matrix model. Convergence becomes faster as D increases, and for D 10 it is already achieved at the 3rd order.