Worldsheet 📖 Wikipedia

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.

We begin with the classical formulation of the bosonic string.

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

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

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

Meanwhile the topology of the worldsheet of a closed string^[3] is ${\displaystyle \mathbb {R} \times S^{1}}$, and admits 'coordinates' ${\displaystyle (\tau ,\sigma )}$ with ${\displaystyle -\infty <\tau <\infty }$ and ${\displaystyle \sigma \in \mathbb {R} /2\pi \mathbb {Z} }$. That is, ${\displaystyle \sigma }$ is a periodic coordinate with the identification ${\displaystyle \sigma \sim \sigma +2\pi }$. The redundant description (using quotients) can be removed by choosing a representative ${\displaystyle 0\leq \sigma <2\pi }$.

In order to define the Polyakov action, the world-sheet is equipped with a world-sheet metric^[4] ${\displaystyle \mathbf {g} }$, which also has signature ${\displaystyle (-,+)}$ 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 ${\displaystyle [\mathbf {g} ]}$. Then ${\displaystyle (\Sigma ,[\mathbf {g} ])}$ defines the data of a conformal manifold with signature ${\displaystyle (-,+)}$.