UNO Katsuhiro Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (70176717)
NAGATOMO Kiyokazu Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (90172543)
KAWANAKA Noriaki Graduate School of Science, Osaka University Professor, 大学院・理学研究科, 教授 (10028219)
MIKI Kei Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (40212229)
SAKUMA Makoto Graduate School of Science, Osaka University Associate Prof., 大学院・理学研究科, 助教授 (30178602)
The aim of this research is to construct a new unifying method to study low dimensional topology from a view point of the quantum structure of Riemann surfaces. To do this, research was done for two areas. One is geometric aspect and the other is algebraic aspect. It is revealed that the volume of the complement of a hyperbolic knot is given by quantum invariants of the knot. Moreover, there is some indication that the volume of a hyperbolic three-manifold is also given by the quantum invariants of the manifold. These facts suggest that the quantum invariants of knots and three-manifolds include various geometric information, and efficiency of the method to study geometric properties from quantum invariants are pointed out.
We also study about the finite-type invariants and the web diagrams, which are related to certain expansion of quantum invariants, and get some new properties of them.
Quantum invariants are closely related to conformal field theory and theory of q-deformation, and some results for these theories are obtained. In the study of vertex operator algebras, the relation between modular forms and the correlation functions in conformal field theory is given. In the study of the theory of q-deformation, modular representations of finite groups are studied. The heart (quotient of the radical by the socle) of projective indecomposable modules are investigated, and the case that the heart is not indecomposable is determined. Moreover, q-deformation of a Frobenius-Schur character of complex reflection groups is defined and computed actually for the symmetric groups and imprimitive complex reflection groups.