2008-05-152024-05-18https://scholars.lib.ntu.edu.tw/handle/123456789/697904Abstract: String theories in critical dimensions (10 for superstrings and 26 for bosonic strings) are of great interest to physicists because they are the only candidates of the theory of everything. Furthermore, they are also the only candidates of the theory of quantum gravity which admits a well-defined perturbative approach. However, string theories in critical dimensions are extremely complicated and they present a lot of puzzles and miracles. On the other hand, string theories in lower dimensions (e.g. D = 0, 1, 2) are much easier to understand and in many ways they possess the essential features of critical string theories, although by themselves they have no real physical significance. Matrix models have been constructed as dual descriptions of non-critical string theories, and scattering amplitudes to all orders in perturbation theory have been found. The most important development in string theory in the past 10 years was the discovery of AdS/CFT duality by Maldacena. This duality is an explicit example of the holographic principle of quantum gravity. The most famous AdS/CFT duality is the duality between type IIB string theory in AdS_5xS^5 space and the N=4 super Yang-Mills theory in 3+1 dimensions. Although this duality has passes numerous checks, it has never been really understood with a logical reasoning based on fundamental principles. In fact, even the specific content of the duality is not totally clear. A lot of it remains to be understood, and we believe that this will be the core to understand the many puzzles in quantum gravity. Recently, it was found by Maldacena and his collaborators that the 1/2 BPS sector of the AdS/CFT duality mentioned above can be described by the matrix model dual to a non-critical string theory. This opens up the gate for us to utilize the techniques developed for non-critical string theory to understand the AdS/CFT duality. This is one of the most important and popular research topics in string theory at this moment. The most salient feature of the AdS/CFT duality is that the two equivalent theories live on spacetime of different dimensions. The same feature also exists for the duality between non-critical string theories and matrix models. In the latter case, there are infinite symmetries in the higher dimensional theory (string theory) that can be used to uniquely determine scattering amplitudes. We believe that this is at least the mathematical reason for its equivalence to a lower dimensional theory. Is there also an infinite dimensional symmetry in the critical string theory? Is it responsible for the AdS/CFT duality? These are questions we would like to answer.