标题： 同伦不变量与(2+1)维引力中的时间演化 显示英文标题

标题： Homotopy Invariants and Time Evolution in (2+1)-Dimensional Gravity

摘要： 我们建立ISO(2,1)同伦不变量与(2+1)维引力的多边形表示之间的关系。 多边形闭合条件以及SO(2,1)环路条件，对于基本群在ISO(2,1)中的表示来说，等价于ISO(2,1)环路条件。 此外，不变量空间上的辛结构与多边形表示的辛结构密切相关。 我们选择一个多边形变量作为内部时间并计算哈密顿量，然后显式地进行哈密顿-雅可比变换。 我们与其他作者关于g = 1和g = 2 (N = 0)的结果进行对比。 显示英文摘要

摘要： We establish the relation between the ISO(2,1) homotopy invariants and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2,1) cycle conditions, are equivalent to the ISO(2,1) cycle conditions for the representa- tions of the fundamental group in ISO(2,1). Also, the symplectic structure on the space of invariants is closely related to that of the polygon representation. We choose one of the polygon variables as internal time and compute the Hamiltonian, then perform the Hamilton-Jacobi transformation explicitly. We make contact with other authors' results for g = 1 and g = 2 (N = 0).