Cocycle Properties of String Theories on Orbifolds

We study cocycle properties of vertex operators and present an operator representation of cocycle operators, which are attached to vertex operators to ensure the duality of amplitudes. It is shown that this analysis makes it possible to obtain the general class of consistent string theories on orbifolds.Comment: 40 pages (Talk given at Workshop on ``Recent Developments in String and Field theory", Kyoto, Japan on September 9-12, 1991.