Application of string theory to localization problem

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Anderson localization has been studied by field theoretical method. In this research, the relation to string theory is investigated and by the use of the string scattering amplitude, the renormalization group equation for the Anderson localization is determined up to five-loop order. The multi-moment of conductivity fluctuation is also shown to be related to the dilaton and the massive field renormalizations. The string theory is closely related to the two-dimensional gravity coupled to matter fields. Matrix model for such case is investigated especially for the central charge c greater than one. The string susceptibility is evaluated from the perturbative series expansion. In a strong magnetic field, two dimensional electron within the lowest Landau level shows the quantum Hall effect. In a two-subband model, the state at the band center is shown to be delocalized and the exact value of the conductivity is obtained. Two dimensional electron in a strong magnetic field is described by Laughlin wave function. Using a complex matrix, which represents the two dimensional coordinate of electrons, we evaluate the correlation function at the edge state and confirmed the result of the conformal field theory. Also the relation to the two dimensional gravity is investigated by adding the interaction term.