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We show that Yang-Mills matrix integrals remain convergent when a Myers term is added, and stay in the same topological class as the original model. It is possible to add a supersymmetric Myers term and this leaves the partition function invariant.
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We review our proposal for a constructive definition of superstring, type IIB matrix model. The IIB matrix model is a manifestly covariant model for space-time and matter which possesses N=2 supersymmetry in ten dimensions. We refine our arguments to reproduce string perturbation theory based on the loop equations. We emphasize that the space-time is dynamically determined from the eigenvalue distributions of the matrices. We also explain how matter, gauge fields and gravitation appear as fluctuations around dynamically determined space-time.
Assuming that the large-$N$ master field of the Lorentzian IIB matrix model has been obtained, we go through the procedure of how the coordinates of emerging spacetime points can be extracted. Explicit calculations with test master fields suggest that the genuine IIB-matrix-model master field may have a fine-structure that is essential for producing the spacetime points of an expanding universe.
The large-$N$ master field of the Lorentzian IIB matrix model is of course not known, but we can assume that we already have it and investigate how the emerging spacetime metric could be extracted. We show that, in principle, it is possible to obtain both the Minkowski metric and the spatially flat Robertson-Walker metric.
The vertex operators for the supergravity multiplet can be constructed through the Wilson lines in IIB matrix model. We investigate the structure of the vertex operators and the symmetry of their correlation functions. For this purpose, we perturb the theory by the Wilson lines dual to the supergravity multiplet. The structure of the Wilson lines can be determined by requiring $cal{N}$=2 SUSY under the low energy approximation. We argue that the generating functional of the correlators is invariant under local SUSY transformations of IIB supergravity.
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.
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