Worldsheet

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind as a direct generalization of the world line concept for a point particle in special and general relativity. The type of string, the geometry of the spacetime in which it propagates, and the presence of long-range background fields (such as gauge fields) are encoded in a two-dimensional conformal field theory defined on the worldsheet. For example, the bosonic string in 26 dimensions has a worldsheet conformal field theory consisting of 26 free scalar bosons. Meanwhile, a superstring worldsheet theory in 10 dimensions consists of 10 free scalar fields and their fermionic superpartners. 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 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 . In order to define the Polyakov action, the world-sheet is equipped with a world-sheet metric , which also has signature but is independent of the induced metric. Since Weyl transformations are considered a redundancy of the metric structure, the world-sheet is instead considered to be equipped with a conformal class of metrics . Then defines the data of a conformal manifold with signature .