Worldsheet

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

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

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

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

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

• String theory
• Leonard Susskind
• String theory stubs

• Articles with short description
• Short description is different from Wikidata
• All stub articles