Worldsheet
String theory |
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Fundamental objects |
Perturbative theory |
Non-perturbative results |
Phenomenology |
Mathematics |
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
Mathematical formulation
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
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
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
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
References
- ISBN 978-1-4612-2256-9.
- ^ Susskind, Leonard (1970). "Dual-symmetric theory of hadrons, I.". Nuovo Cimento A. 69 (1): 457–496.
- ^ Tong, David. "Lectures on String Theory". Lectures on Theoretical Physics. Retrieved August 14, 2022.
- ^ Polchinski, Joseph (1998). String Theory, Volume 1: Introduction to the Bosonic string.