Worldsheet

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 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 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 .

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 (-, +).

References

References

1. (1997). "Conformal Field Theory".
2. Susskind, Leonard. (1970). "Dual-symmetric theory of hadrons, I.". Nuovo Cimento A.
3. Tong, David. "Lectures on String Theory".
4. (1998). "String Theory, Volume 1: Introduction to the Bosonic string".