Bosonic string
We begin with the classical formulation of the bosonic string.
First fix a -dimensional flat spacetime (-dimensional Minkowski space), , which serves as the ambient space for the string.
A world-sheet is then an embedded surface, that is, an embedded 2-manifold , such that the induced metric has signature everywhere. Consequently it is possible to locally define coordinates where is time-like while is space-like.
Strings are further classified into open and closed. The topology of the worldsheet of an open string is , where , a closed interval, and admits a global coordinate chart with and .
Meanwhile the topology of the worldsheet of a closed string[3] is , and admits 'coordinates' with and . That is, is a periodic coordinate with the identification . The redundant description (using quotients) can be removed by choosing a representative .