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In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime.[1] The term was coined by Leonard Susskind[2] 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.

Mathematical formulation[edit]

Bosonic string[edit]

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 .

World-sheet metric[edit]

In order to define the Polyakov action, the world-sheet is equipped with a world-sheet metric[4] , 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 .


  1. ^ Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997). Conformal Field Theory. p. 8. doi:10.1007/978-1-4612-2256-9. ISBN 978-1-4612-2256-9.
  2. ^ Susskind, Leonard (1970). "Dual-symmetric theory of hadrons, I.". Nuovo Cimento A. 69 (1): 457–496.
  3. ^ Tong, David. "Lectures on String Theory". Lectures on Theoretical Physics. Retrieved August 14, 2022.
  4. ^ Polchinski, Joseph (1998). String Theory, Volume 1: Introduction to the Bosonic string.