New Types of Matrix Models

Today particle physics except for gravity is well described by the standard model. However, gravity cannot be quantized in the same method because we cannot renormalize it. Therefore the main problem of current particle physics is to establish a consistent quantum theory which contains both the standard model and gravity. Under these circumstances, the most hopeful and popular candidate is the string theory. The reason to favor the string theory is its wonderful nature. We can give as concrete examples that the theory has no ultraviolet divergence and includes gravitational field as well as matter and gauge fields automatically. However, due to the infinite ground states, this theory has no capability to predict; therefore we cannot answer why the standard model emerges. On the other hand it is possible to consider that this problem is the problem in the framework of perturbative formulation of the theory, because the completed region of the string theory is only the perturbative region. So if the non-perturbative formulation of the theory is accomplished, it is quite likely that this problem is resolved. Of course, it is pure speculation, but it seems quite probable that the non-perturbative effects turn infinite ground states into single one. What must not be forgotten is that one theory never finish before the non-perturbative formulation is completed. One of candidates for the non-perturbative formulation of the string theory at present is the string field theory. Although a considerable number of studies have been conducted on these theories, the only successful string field theories so far are the ones formulated in the light-cone gauge. So it is not clear whether we can extract some essential information of the non-perturbative effects. Another candidate is what is called the matrix model. With the advent of the BFSS model as a starter, many proposals have been being made since. The common idea of these models is that they reproduce sting or membrane theory in the large-N limit. In a sense the matrix model is similar to the lattice gauge theory, which is the non-perturbative formulation of the field theory, in that they can be analyzed using numerical simulation. Therefore it is reasonable to suppose that we will develop current matrix models a little further and find the true model. A virtue of the matrix model is that it has a possibility of putting an interpretation on the space-time itself. However, some important questions such as "what would be the real mechanism to realize the 4-dimensional world from the 10(or 11)-dimensional universe" and "how is the diffeomorphism introduced into the theory" remain unsettled. One of them is the question of background independence. Consider the IKKT model for example. This model has an SO(10) × SU(N) symmetry, and this is just a symmetry like some theory was expanded around the flat background. Therefore we cannot deny the existence of different matrix model whose expansion around a special background gets the IKKT model. On this point Smolin proposed a new type of matrix model in which the action is cubic in matrices. Matrices are built from the super Lie algebra osp( 1|32; Ｒ), and one multiplet is pushed into a single supermatrix. Smolin\u27s conjecture is that the expansions around different backgrounds of the osp(1|32; Ｒ) matrix model will reduce to the BFSS or IKKT model. However, as far as the IKKT model is concerned, the theory made from Smolin\u27s way dose not reproduce the supersymmetry of the IKKT model. That is, indeed the 10-dimensionality is realized, but the half of supersymmetry required by the IKKT model cannot be held. Anyway, the model described by a single matrix alone is very attractive, and Smolin\u27s courageous attempt demonstrated one concrete possibility. Moreover, as Smolin\u27s u(1|16, 16) model has demonstrated, the matrix models are not irrelevant to the loop quantum gravity which is another approach to the Theory of Everything. Furthermore, it was pointed out that the matrix string theory has a connection with the matrix model based on the exceptional Jordan algebra Ｊ, while B.Kim and A. Schwarz have discussed a tie-in between the IKKT model and the Jordan algebra ｊ with its spinor representation. For these reasons, doing research on extended matrix model is very interesting and important. Over and above, we should not overlook the fact that several approaches which are very similar to the matrix model have been pursued by other fields. It might be inferred from these circumstantial evidence that the attempt to renounce the space-time as a continuum holds one important key to the future progress of physics. It seems at least that there is no need to relate the matrix model to the string theory alone. For these purposes, we investigate new types of matrix models based on the complex exceptional Jordan algebra and the super Lie algebras. In the former case, a matrix Chern-Simons theory is directly derived from the invariant on E6. It is stated that the same argument as Smolin which derives an effective action similar to the matrix string theory can also be held in our model. The only difference is that our model has twice as many degrees of freedom as Smolin\u27s model has. One way to introduce the cosmological term is the compactification on directions. It is of great interest that the properties of the product space Ｊc × ｇ, in which the degrees of freedom of our model live, are very similar to those of the physical Hilbert space. In the latter case, we investigate three super Lie algebras, osp(1|32;Ｒ), u(1|16,16), and gl(1|32;Ｒ). In paticular, we study the supersymmetry structures of these models and discuss possible reductions to the IKKT model. In addition to those, a different u(1|16, 16) model from Smolin\u27s, and some kind of topological effective action derived using Wigner-Inönü contraction are also discusse