In
string theory, a worldsheet is a two-dimensional
manifold which describes the embedding of a string in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
.
The term was coined by
Leonard Susskind as a direct generalization of the
world line
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from c ...
concept for a point particle in
special
Special or specials may refer to:
Policing
* Specials, Ulster Special Constabulary, the Northern Ireland police force
* Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer
Literature
* ''Specia ...
and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
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
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the co ...
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
superpartner
In particle physics, a superpartner (also sparticle) is a class of hypothetical elementary particles predicted by supersymmetry, which, among other applications, is one of the well-studied ways to extend the standard model of high-energy physics.
...
s.
Mathematical formulation
Bosonic string
We begin with the classical formulation of the bosonic string.
First fix a
-dimensional
flat spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
(
-dimensional
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
),
, which serves as the
ambient space
An ambient space or ambient configuration space is the space surrounding an object.
While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latt ...
for the string.
A world-sheet
is then an
embedded surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, that is, an embedded 2-manifold
, such that the
induced metric has signature
everywhere. Consequently it is possible to locally define coordinates
where
is
time-like while
is
space-like.
Strings are further classified into open and closed. The topology of the worldsheet of an open string is
, where