Mikhail Orionovich Katanaev (Steklov Mathematical Institute, Moscow, Russia)

Local and global integrability of two-dimensional gravity

Two-dimensional Gravity is proved to be an integrable model. A general solution of the equations of motion is explicitly found without gauge fixing. It consists of two classes of solutions: (i) constant curvature and zero torsion surfaces (the Liouville theory) and (ii) nonconstant curvature and nonzero torsion surfaces (new solutions). Solutions are found locally and then extended along geodesics providing global structure of surfaces. As an example, we consider General Relativity assuming the space-time to be a warped product of two surfaces. In this case, General Relativity reduces to two-dimensional gravity. The vacuum solutions contain solutions describing many wormholes, cosmic strings, domain walls of curvature singularities, cosmic strings surrounded by domain walls, solutions with closed timelike curves and other solutions along with the Schwarzschild and (anti-) de Sitter solutions.