worldsheet

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime.{{cite book|last1=Di Francesco|first1=Philippe|last2=Mathieu|first2=Pierre|last3=Sénéchal|first3=David|year=1997|isbn=978-1-4612-2256-9|title=Conformal Field Theory |doi=10.1007/978-1-4612-2256-9|page=8}} The term was coined by Leonard Susskind{{cite journal |first=Leonard |last=Susskind |title=Dual-symmetric theory of hadrons, I. |journal=Nuovo Cimento A |volume=69 |issue=1 |pages=457–496 |year=1970}} 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{{cite web |url=http://www.damtp.cam.ac.uk/user/tong/string.html |title=Lectures on String Theory |last=Tong |first=David |website=Lectures on Theoretical Physics |access-date=August 14, 2022}} 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{{cite book|last1=Polchinski|first1=Joseph|year=1998|title=String Theory, Volume 1: Introduction to the Bosonic string}} $\mathbf{g}$ , 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

Category:String theory

Category:Leonard Susskind