Worldsheet

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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

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Bosonic string

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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

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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  .

References

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  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.