# Worldsheet

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 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

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

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

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

#### World-sheet metric

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