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Register a new userImproved Perturbation Theory and Four-Dimensional Space-Time in IIB Matrix Model
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We have analyzed IIB matrix model based on the improved mean field approximation (IMFA) and have obtained a clue that the four-dimensional space-time appears as its most stable vacuum. This method is a systematic way to give an improved perturbation series and was first applied to IIB matrix model by Nishimura and Sugino. In our previous paper we reformed this method and proposed a criterion for convergence of the improved series, that is, the appearance of the ``plateau. In this paper, we perform higher order calculations, and find that our improved free energy tends to have a plateau, which shows that IMFA works well in IIB matrix model.
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For the purpose of analyzing non-perturbative dynamics of string theory, Nishimura and Sugino have applied an improved mean field approximation (IMFA) to IIB matrix model. We have extracted the essence of the IMFA and obtained a general scheme, an improved Taylor expansion, that can be applied to a wide class of series which is not necessarily convergent. This approximation scheme with the help of the 2PI free energy enables us to perform higher order calculations. We have shown that the value of the free energy is stable at higher orders, which supports the validity of the approximation. Moreover, the ratio between the extent of ``our space-time and that of the internal space is found to increase rapidly as we take the higher orders into account. Our results suggest that the four dimensional space-time emerges spontaneously in IIB matrix model.
The spontaneous breakdown of SO(10) symmetry of the IIB matrix model has been studied by using the improved mean field approximation (IMFA). In this report, the eighth-order contribution to the improved perturbative series is obtained, which involves evaluation of 20410 planar two-particle irreducible vacuum diagrams. We consider SO(d)-preserving configurations as ansatz (d=4,7). The development of plateau, the solution of self-consistency condition, is seen in both ansatz. The large ratio of the space-time extent of d-dimensional part against the remaining (10-d)-dimensional part is obtained for SO(4) ansatz evaluated at the representative points of the plateau. It would be interpreted as the emergence of four-dimensional space-time in the IIB matrix model.
The origin of our four-dimensional space-time has been pursued through the dynamical aspects of the IIB matrix model via the improved mean field approximation. Former works have been focused on the specific choice of configurations as ansatz which preserve SO(d) rotational symmetry. In this report, an extended ansatz is proposed and examined up to 3rd order of approximation which includes both SO(4) ansatz and SO(7) ansatz in their respective limits. From the solutions of self-consistency condition represented by the extrema of free energy of the system, it is found that a part of solutions found in SO(4) or SO(7) ansatz disappear in the extended ansatz. It implies that the extension of ansatz works as a device to distinguish the stable solutions from the unstable ones. It is also found that there is a non-trivial accumulation of extrema including the SO(4)-preserving solution, which may lead to the formation of plateau.
We present a new scheme for extracting approximate values in ``the improved perturbation method, which is a sort of resummation technique capable of evaluating a series outside the radius of convergence. We employ the distribution profile of the series that is weighted by nth-order derivatives with respect to the artificially introduced parameters. By those weightings the distribution becomes more sensitive to the ``plateau structure in which the consistency condition of the method is satisfied. The scheme works effectively even in such cases that the system involves many parameters. We also propose that this scheme has to be applied to each observables separately and be analyzed comprehensively. We apply this scheme to the analysis of the IIB matrix model by the improved perturbation method obtained up to eighth order of perturbation in the former works. We consider here the possibility of spontaneous breakdown of Lorentz symmetry, and evaluate the free energy and the anisotropy of space-time extent. In the present analysis, we find an SO(10)-symmetric vacuum besides the SO(4)- and SO(7)-symmetric vacua that have been observed. It is also found that there are two distinct SO(4)-symmetric vacua that have almost the same value of free energy but the extent of space-time is different. From the approximate values of free energy, we conclude that the SO(4)-symmetric vacua are most preferred among those three types of vacua.
We derive a long distance effective action for space-time coordinates from a IIB matrix model. It provides us an effective tool to study the structures of space-time. We prove the finiteness of the theory for finite $N$ to all orders of the perturbation theory. Space-time is shown to be inseparable and its dimensionality is dynamically determined. The IIB matrix model contains a mechanism to ensure the vanishing cosmological constant which does not rely on the manifest supersymmetry. We discuss possible mechanisms to obtain realistic dimensionality and gauge groups from the IIB matrix model.
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