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

在基于2+1维Chern-Simons作用量的引力理论中，粒子间的散射振幅是由Wilson环算子的真空期望值给出的，并将其推广到包含3点顶点。特别是四面体型算子起着至关重要的作用。不幸的是，这种算子在场论中的显式表达式至今还不清楚。2+1维Chern-Simons理论与1+1维有理共形场论有着密切的联系。这种关系使我们有可能准确地估计广义Wilson环算子的真空期望值。计算四面体型算子真空期望值的一般方法已基本建立，2+1维Chern-Simons理论精确可解的一个重要原因是其中包含了丰富的数学结构。实际上，标准Wilson圈算符的真空期望值是纽结多项式，并且与量子群的关系也很清楚。然而，这种数学结构和物理学之间的联系还不明显。如何用场论语言解释数学结构也是一个有待解决的问题。而2+1维Chern-Simons理论在适当选择规范群的情况下是2+1维引力。因此，继续研究纽结多项式、量子群等数学结构与引力的关系，对于理解三维引力理论是十分重要的。

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.