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.

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 $$(-, +)$$.