• Source: 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.




Mathematical formulation






= Bosonic string

=
We begin with the classical formulation of the bosonic string.
First fix a



d


{\displaystyle d}

-dimensional flat spacetime (



d


{\displaystyle d}

-dimensional Minkowski space),



M


{\displaystyle M}

, which serves as the ambient space for the string.
A world-sheet



Σ


{\displaystyle \Sigma }

is then an embedded surface, that is, an embedded 2-manifold



Σ
↪
M


{\displaystyle \Sigma \hookrightarrow M}

, such that the induced metric has signature



(
−
,
+
)


{\displaystyle (-,+)}

everywhere. Consequently it is possible to locally define coordinates



(
τ
,
σ
)


{\displaystyle (\tau ,\sigma )}

where



τ


{\displaystyle \tau }

is time-like while



σ


{\displaystyle \sigma }

is space-like.
Strings are further classified into open and closed. The topology of the worldsheet of an open string is




R

×
I


{\displaystyle \mathbb {R} \times I}

, where



I
:=
[
0
,
1
]


{\displaystyle I:=[0,1]}

, a closed interval, and admits a global coordinate chart



(
τ
,
σ
)


{\displaystyle (\tau ,\sigma )}

with



−
∞
<
τ
<
∞


{\displaystyle -\infty <\tau <\infty }

and



0
≤
σ
≤
1


{\displaystyle 0\leq \sigma \leq 1}

.
Meanwhile the topology of the worldsheet of a closed string is




R

×

S

1




{\displaystyle \mathbb {R} \times S^{1}}

, and admits 'coordinates'



(
τ
,
σ
)


{\displaystyle (\tau ,\sigma )}

with



−
∞
<
τ
<
∞


{\displaystyle -\infty <\tau <\infty }

and



σ
∈

R


/

2
π

Z



{\displaystyle \sigma \in \mathbb {R} /2\pi \mathbb {Z} }

. That is,



σ


{\displaystyle \sigma }

is a periodic coordinate with the identification



σ
∼
σ
+
2
π


{\displaystyle \sigma \sim \sigma +2\pi }

. The redundant description (using quotients) can be removed by choosing a representative



0
≤
σ
<
2
π


{\displaystyle 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




g



{\displaystyle \mathbf {g} }

, which also has signature



(
−
,
+
)


{\displaystyle (-,+)}

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



[

g

]


{\displaystyle [\mathbf {g} ]}

. Then



(
Σ
,
[

g

]
)


{\displaystyle (\Sigma ,[\mathbf {g} ])}

defines the data of a conformal manifold with signature



(
−
,
+
)


{\displaystyle (-,+)}

.




References

Kata Kunci Pencarian:

• Worldsheet
• Polyakov action
• String (physics)
• Relationship between string theory and quantum field theory
• Bosonic string theory
• Type II string theory
• Black hole
• GSO projection
• Non-critical string theory
• Critical dimension