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内容記述 |
In this doctoral thesis, we study the three-dimensional supersymmetric Yang-Mills theory from a viewpoint of matrix models. Particularly, we use the model which is called the IIB matrix model.<br /> First of all, we give a brief review of the IIB matrix model which was proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya in 1997. We can describe the following points as several properties of the IIB matrix model. This model can be regarded as a large <i>N</i> reduced model of the ten-dimensional N=1 supersymmetric <i>SU(N)</i> Yang-Mills theory. It was shown that a large <i>N</i> gauge theory can be equivalently described by its reduced model on 0-dimensional spacetime. In this dimensional reduction, a spacetime translation is represented in the color of <i>SU(N)</i> space, and the eigenvalues of bosonic matrices in the IIB matrix model are interpreted as the momenta of fields. Therefore, the basic assumption in this identification is that the eigenvalues are uniformly distributed. On the other hand, this model was proposed as the non-perturbative formulation of the type-IIB superstring theory. The following is several reasons that the IIB matrix model is a non-perturbative formulation of the type-IIB superstring theory. First, the action can be related to the Green-Schwarz action of a type-IIB superstring theory by taking a large <i>N</i> limit. In fact, we can describe an arbitrary number on interacting D-strings and anti-D-strings in the type-IIB superstring as diagonal blocks of bosonic matrices in the IIB matrix model. And the off-diagonal blocks of it represent interactions among these strings. The IIB matrix model describes not only the single-body system, but also the multi-body system of D-strings in the type-IIB superstring theory. Therefore, it must be clear that the IIB matrix model is not the first quantized theory, but a full second quantized theory of the type-IIB superstring. Then, we can point out a second evidence for the conjecture that the IIB matrix model is a non-perturbative formulation of the type-IIB superstring theory. The evidence is that Wilson loops of the IIB matrix model satisfy the string field equations of motion for the type-IIB superstring in the light-cone gauge. The IIB matrix model can describe joining and splitting interactions of fundamental strings created by the Wilson loops. Thus, the IIB matrix model could become a non-perturbative formulation of the type-IIB superstring theory. Finally, the dynamics of eigenvalues of bosonic matrices in the IIB matrix model represent the dynamical generation of spacetime in our universe. It can be interpreted that the spacetime consists of discretized points, and that eigenvalues of matrices represent their spacetime coordinates in the IIB matrix model. Thus, the dynamics of the IIB matrix model is such that the resulting eigenvalue distributions can be interpreted as any Riemannian geometry. For example, if the diagonal elements distribute within a manifold which extends in four dimensions but shrinks in six dimensions, then a natural interpretation is that the spacetime is four-dimensional.<br /> The final property is a very characteristic and exciting topic in several properties of the IIB matrix model. There are considerable amount of investigations toward understanding the four-dimensional spacetime by using the IIB matrix model. For example, I may list the following studies: branched polymer picture, complex phase effects and mean-field approximations. These studies seem to suggest that the IIB matrix model predicts the four-dimensionality of spacetime. But it is difficult to analyze dynamics of the IIB matrix model in a generic spacetime. So it is considered that we would like to understand general mechanisms to single out the four-dimensionality of spacetime through the studies of concrete examples. In 2002, it was proved that fuzzy homogeneous spaces are constructed using the IIB matrix model. The homogeneous spaces are constructed as <i>G/H</i> where <i>G</i> is a Lie group and H is a closed subgroup of <i>G</i>. When a background field is given to bosonic matrices in the IIB matrix model, the stability of this matrix configurations can be examined by investigating the behavior of the effective action under the change of some parameters of the background. The stabilities of fuzzy <i>S</i><sup>2</sup>, fuzzy <i>S</i><sup>2</sup> × <i>S</i><sup>2</sup>, fuzzy <i>S</i><sup>2</sup> × <i>S</i><sup>2</sup> × <i>S</i><sup>2</sup> and fuzzy <i>CP</i><sup>2</sup> have been investigated in the past. By the above researches, we have found that the IIB matrix model favors the configurations of four-dimensionality and more symmetric manifolds.<br /> Recently, there were interesting developments about constructions of curved spacetimes by matrix models. Hanada, Kawai and Kimura have introduced a new interpretation on the IIB matrix model in which covariant derivatives on any <i>d</i>-dimensional spacetimes can be described in terms of <i>d</i> large <i>N</i>bosonic matrices in the IIB matrix model. In this interpretation, the Einstein equation follows from the equation of the IIB matrix model, and symmetries under local Lorentz transformation and diffeomorphism are included in the unitary symmetry of the IIB matrix model. On the other hand, the relations between supersymmetric Yang-Mills theories on curved spacetimes and a matrix model have been proposed by Ishiki, Shimasaki, Takayama and Tsuchiya. The relations is as follows: the relation between the supersymmetric Yang-Mills theory on <i>R</i> × <i>S<sup>3</sup></i> and the supersymmetric Yang-Mills theory on <i>R</i> × <i>S<sup>2</sup></i>, and the relation between the supersymmetric Yang-Mills theory on <i>R</i> × <i>S</i><sup>2</sup> and the plane-wave matrix model. They have made a connection between the supersymmetric Yang-Mills theory on <i>R</i> × <i>S</i><sup>3</sup> and the plane-wave matrix model up to showing the above relations. They also have showed that <i>S</i><sup>3</sup> can be described in terms of three matrices which are configured by aligning representation matrices of <i>SU(2)</i> on a diagonal.<br /> We investigate the effective action of a deformed IIB matrix model with a Myers term on <i>S</i><sup>3</sup>background describing by covariant derivatives on <i>S</i><sup>3</sup> and irreducible representation matrices of <i>SU(2)</i>. In he both cases, we find that the highly divergent contributions at the tree and one-loop level are sensitive to the UV cutoff. However the two-loop level contributions are universal since they are only logarithmically divergent. We expect that the higher loop contributions are insensitive to the UV cutoff since three-dimensional gauge theory is super renormalizable. We can thus conclude that the effective action of the deformed IIB matrix model on <i>S</i><sup>3</sup> is stable against the quantum corrections as it is dominated by the tree level contribution. Therefore, we find that the <i>S</i><sup>3</sup> background is one of the non-trivial solutions of the IIB matrix model. We recall here that we have obtained the identical conclusions for the <i>S</i><sup>2</sup> case.We thus believe that the two- and three-dimensional spheres are classical objects in the IIB matrix model since the tree level effective action dominates. We can in turn conclude that they are not the solutions of the IIB matrix model without a Myers term. We still expect that the IIB matrix model favors the configurations of four-dimensional spacetime. |
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