Mathematical Structure and its Physical Interpritation of 3-dimonsional Quantum Gravity as Topological Field Theory

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In Gravity theory based on 2+1 dimensional Chern-Simons action, which is topological, scattering amplitudes between 2 particles are given in terms of vacuum expectation values of Wilson loop operators which are extended to include 3-point vertices. Especially, tetrahedron-type operators play an essential role. Unfortunately, explisit expressions of such operators in field theory are not yet known up to now. 2+1 dimensional Chern-Simons theories have, however, intimate connection with 1+1 dimensional rational conformal field theory. Such a relationship makes it possible for us to evaluate vacuum expectation valuses of extened Wilson loop operators exactly. General method to caluculate vacuum expectation values of the tetrahedron-type oerators are almost established by now.One of the most important reason why 2+1 dimensional Chern-Simons theories are exactly soluble is rich mathematical structures contained there. Indeed, vacuum expectation values of standard Wilson loop operators are nothing but knot polynomials and relation with quantum group is also made clear. A conection between such mathematical structures and physics is, however, not yet distinct. Also, how to interprit the mathematical structure interms of field theoretical language is stil open question. On the otherhand, 2+1 dimensional Chern-Simons theories are 2+1 dimensional gravity under the apropriate choice of gauge group. So, it is very important to continue to study relateionships between mathematical structures such as knot polyniomials and quantum group and gravity in order to undeastand 3+ldimensional gravity theories.