Worldsheet

 ( String Theory ) 

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 ) 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 .

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Mathematical formulation

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Bosonic string We begin with the classical formulation of the bosonic string.

First fix a $d$-dimensional flat spacetime ($d$-dimensional Minkowski space), $M$, which serves as the ambient space for the string.

A world-sheet $\backslash Sigma$ is then an embedding surface, that is, an embedded 2-manifold $\backslash Sigma\; \backslash hookrightarrow\; M$, such that the induced metric has signature $(-,+)$ everywhere. Consequently it is possible to locally define coordinates $(\backslash tau,\backslash sigma)$ where $\backslash tau$ is time-like while $\backslash sigma$ is space-like.

Strings are further classified into open and closed. The topology of the worldsheet of an open string is $\backslash mathbb\{R\}\backslash times\; I$, where $I\; :=\; 0,1$, a closed interval, and admits a global coordinate chart $(\backslash tau,\; \backslash sigma)$ with and $0\; \backslash leq\; \backslash sigma\; \backslash leq\; 1$.

Meanwhile the topology of the worldsheet of a closed string is $\backslash mathbb\{R\}\backslash times\; S^1$, and admits 'coordinates' $(\backslash tau,\; \backslash sigma)$ with and $\backslash sigma\; \backslash in\; \backslash mathbb\{R\}/2\backslash pi\backslash mathbb\{Z\}$. That is, $\backslash sigma$ is a periodic coordinate with the identification $\backslash sigma\; \backslash sim\; \backslash sigma\; +\; 2\backslash pi$. The redundant description (using quotients) can be removed by choosing a representative .

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World-sheet metric In order to define the Polyakov action, the world-sheet is equipped with a world-sheet metric $\backslash mathbf\{g\}$, 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 $\backslash mathbf\{g\}$. Then $(\backslash Sigma,\; \backslash mathbf\{g\})$ defines the data of a conformal manifold with signature $(-,\; +)$.

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