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.

-dimensional flat spacetime-dimensional Minkowski space, which serves as the ambient space for the string.

A world-sheet is then an embedded surface, such that the induced metric is time-like is space-like.

.

Meanwhile the topology of the worldsheet of a closed string.

World-sheet metric

In order to define the Polyakov action, the world-sheet is equipped with a world-sheet metric 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 defines the data of a conformal manifold.

References