4
Open and Closed Strings
4.1
World-Sheets with Boundaries
In addition to the theory of closed strings just described, we may construct another
consistent theory by considering world-sheets Σ with boundaries. Such a theory
necessarily contains open strings. In this case, we can insert open string states to the
boundaries and closed string states to the bulk of Σ as in Fig. 4.1.
Let us first characterize the open string Hilbert space V o . For this, we need to
fix the boundary conditions for the various fields in the world-sheet CFT. Let us
focus on Σ of genus 0 with one boundary and no closed string insertions. Hence,
the world-sheet of an open string is topologically the infinite strip [0, π] × R. By the
conformal mapping z = −e −iw (w = σ + iτ with (σ, τ ) ∈ [0, π] × R), the strip
is mapped to the upper half plane H. For concreteness, we take Neumann boundary
conditions ∂ n φ(z, ¯
z)| ∂Σ = 0, and similarly for c or b in place of φ. Here, ∂ n is
the derivative in the direction normal to the boundary. Alternate (e.g. Dirichlet)
boundary conditions will not affect our treatment below as long as they preserve
the conformal symmetry of the world-sheet action, that is, admissible boundary
conditions must satisfy the Cardy condition (T − ¯
T )| ∂Σ = 0. The fields living
on H can be separated into holomorphic and anti-holomorphic parts, but due to
the boundary conditions, these two parts combine into a single holomorphic field
defined on the whole complex plane C. This is the doubling trick; the holomorphic
field in the lower half plane is expressed in terms of the anti-holomorphic field
in the upper half plane through the reflection of the coordinate, ∂ z φ(z)| Im(z<0) =
∂ ¯
z φ(¯ z)| Im(z>0) , etc. We then expand each field on C in a Laurent series (mode
expansion)
i∂ z φ(z) =
n∈Z
α n
z n+1 , c(z) =
n∈Z
c n
z n−1 , b(z) =
n∈Z
b n
z n+2 ,
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_4
55
Open and Closed Strings
4.1
World-Sheets with Boundaries
In addition to the theory of closed strings just described, we may construct another
consistent theory by considering world-sheets Σ with boundaries. Such a theory
necessarily contains open strings. In this case, we can insert open string states to the
boundaries and closed string states to the bulk of Σ as in Fig. 4.1.
Let us first characterize the open string Hilbert space V o . For this, we need to
fix the boundary conditions for the various fields in the world-sheet CFT. Let us
focus on Σ of genus 0 with one boundary and no closed string insertions. Hence,
the world-sheet of an open string is topologically the infinite strip [0, π] × R. By the
conformal mapping z = −e −iw (w = σ + iτ with (σ, τ ) ∈ [0, π] × R), the strip
is mapped to the upper half plane H. For concreteness, we take Neumann boundary
conditions ∂ n φ(z, ¯
z)| ∂Σ = 0, and similarly for c or b in place of φ. Here, ∂ n is
the derivative in the direction normal to the boundary. Alternate (e.g. Dirichlet)
boundary conditions will not affect our treatment below as long as they preserve
the conformal symmetry of the world-sheet action, that is, admissible boundary
conditions must satisfy the Cardy condition (T − ¯
T )| ∂Σ = 0. The fields living
on H can be separated into holomorphic and anti-holomorphic parts, but due to
the boundary conditions, these two parts combine into a single holomorphic field
defined on the whole complex plane C. This is the doubling trick; the holomorphic
field in the lower half plane is expressed in terms of the anti-holomorphic field
in the upper half plane through the reflection of the coordinate, ∂ z φ(z)| Im(z<0) =
∂ ¯
z φ(¯ z)| Im(z>0) , etc. We then expand each field on C in a Laurent series (mode
expansion)
i∂ z φ(z) =
n∈Z
α n
z n+1 , c(z) =
n∈Z
c n
z n−1 , b(z) =
n∈Z
b n
z n+2 ,
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_4
55