Unified Theories of Space and Matter

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Our achievements under the support of this Grant are summarized into following three categories.(1)Quantum Gravity and StringsSince the theory of quantum gravity is unrenormalizable the perturbational approach is not permitted. To understand the dynamics of gravity, therefore, one frequently studies various model theories that reflect some aspects of the problem. The two dimensional conformal model and the matrix model are most popular approaches. In the latter one can sum up over all topologies of two dimensional manifolds and find non-perturbative solutions for correlation functions. Many of studies were made along this line under our project, among which here we quote the study of loop correlation amplitudes in a two dimensional universe. The one-loop correlation amplitude is nothing but the two-dimensional version of Hawking's no boundary universe amplitude.Some people studied the topological conformal field and gave a new geometrical interpretation to the topology sum over manifol … More ds. In the string theory existence of "the universal string theory" was studied. All the superstrings and the bosonic strings are able to be deduced as a result of successive spontaneous symmetry breakdown of the universal string theory.(2)Origin of Gauge SymmetryIn most of cases one assume a high gauge symmetry in an extremely high energy region, then introduces a spontaneous symmetry breakdown to get the standard or GUT theory. In this project we found a new mechanism that even if a theory had a low gauge symmetry in high energy, some higher gauge symmetry is generated in the low energy region. This happens when the Kaluza-Klein compactification of space is taken place. The basic mechanism working there is shown to be the Berry phase effect.(3)Applications of Mathematical Techniques developed in String Theory Conformal field theory, w-algebra and Bethe anzatz method were applied to various mathematical problems. One of them is the thorough study of the XXZ-Spin Chain Model. The model had been studied in case of attractive force, now the repulsive spin-chain was analyzed in detail. Another interesting example is an application of w-algebra to the study of the motion of micro-organisms. Less