A matrix model which has the manifest ten-dimensional N = 2 super Poincare invariance is proposed. Interactions between BPS-saturated states are analyzed to show that massless spectrum is the same as that of type IIB string theory. It is conjectured that the large-N reduced model of ten-dimensional super Yang-Mills theory can be regarded as a constructive definition of this model and therefore is equivalent to superstring theory.

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.Comment: 37 pages, LaTex with PTPTex.sty, 2 epsf figures. Proceedings of the 13th Nishinomiya Yukawa Memorial Symposium (November, 1998

We study the large N reduced model of D-dimensional Yang-Mills theory with special attention to dynamical aspects related to the eigenvalues of the N × N matrices, which correspond to the space-time coordinates in the IIB matrix model. We first put an upper bound on the extent of space time by perturbative arguments. We perform a Monte Carlo simulation and show that the upper bound is actually saturated. The relation of our result to the SSB of the U(1) D symmetry in the Eguchi-Kawai model is clarified. We define a quantity which represents the uncertainty of the space-time coordinates and show that it is of the same order as the extent of space time, which means that a classical space-time picture is maximally broken. We develop a 1/D expansion, which enables us to calculate correlation functions of the model analytically. The absence of an SSB of the Lorentz invariance is shown by the Monte Carlo simulation as well as by the 1/D expansion.

We reconsider the matrix model formulation of type IIB superstring theory in (9+1)-dimensional space-time. Unlike the previous works in which the Wick rotation was used to make the model well defined, we regularize the Lorentzian model by introducing infrared cutoffs in both the spatial and temporal directions. Monte Carlo studies reveal that the two cutoffs can be removed in the large-N limit and that the theory thus obtained has no parameters other than one scale parameter. Moreover, we find that three out of nine spatial directions start to expand at some "critical time," after which the space has SO(3) symmetry instead of SO(9).

We study theories with SU (2|4) symmetry, which include the plane wave matrix model, 2 + 1 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k . All these theories possess many vacua. From Lin-Maldacena's method which gives the gravity dual of each vacuum, it is predicted that the theory around each vacuum of 2 + 1 SYM on R × S 2 and N = 4 SYM on R × S 3 /Z k is embedded in the plane wave matrix model. We show this directly on the gauge theory side. We clearly reveal relationships among the spherical harmonics on S 3 , the monopole harmonics and the harmonics on fuzzy spheres. We extend the compactification (the T-duality) in matrix models a la Taylor to that on One can see from (D.8) that in the N 0 → ∞ limit, this formula reduces to N 0 tr(

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We investigate relationship between a gauge theory on a principal bundle and that on its base space. In the case where the principal bundle is itself a group manifold, we also study relations of those gauge theories with a matrix model obtained by dimensionally reducing them to zero dimensions. First, we develop the dimensional reduction of YangMills (YM) on the total space to YM-higgs on the base space for a general principal bundle. Second, we show a relationship that YM on an SU (2) bundle is equivalent to the theory around a certain background of YM-higgs on its base space. This is an extension of our previous work [29], in which the same relationship concerning a U (1) bundle is shown. We apply these results to the case of SU (n + 1) as the total space. By dimensionally reducing YM on SU (n + 1), we obtain YM-higgs on SU (n + 1)/SU (n) ≃ S 2n+1 and on SU (n + 1)/(SU (n) × U (1)) ≃ CP n and a matrix model. We show that the theory around each monopole vacuum of YM-higgs on CP n is equivalent to the theory around a certain vacuum of the matrix model in the commutative limit. By combining this with the relationship concerning a U (1) bundle, we realize YM-higgs on SU (n + 1)/SU (n) ≃ S 2n+1 in the matrix model. We see that the relationship concerning a U (1) bundle can be interpreted as Buscher's