6
1 Introduction
(a)
(b)
Fig. 1.3 (a) Open and (b) closed strings
− −−−−−−− →
Fig. 1.4 Trajectory x
μ
c (τ, σ ) of a closed string in spacetime (worldsheet). It begins and ends at
the circles parametrized by x i (σ ) and x f (σ ). The worldsheet is topologically a cylinder and is
parametrized by (τ, σ ) ∈ [τ i , τ f ] × [0, 2π)
and 1.5. To each topology is associated different boundary conditions and types of
strings:
• closed: periodic and anti-periodic boundary conditions;
• open: Dirichlet and Neumann boundary conditions.
While a closed string theory is consistent by itself, an open string theory is not and
requires closed strings.
Spectrum
In order to gain some intuition for the states described by a closed string, one can
write the Fourier expansion of the fields X μ (in the gauge g ab = η ab and after
imposing the equations of motion)
X
μ (τ, σ ) ∼ x
μ
+ p
μ τ +
i
√
2
n∈Z ∗
1
n
α
μ
n e
−in(τ −σ )
+ ¯
α
μ
n e
−in(τ +σ )
,
(1.1)
1 Introduction
(a)
(b)
Fig. 1.3 (a) Open and (b) closed strings
− −−−−−−− →
Fig. 1.4 Trajectory x
μ
c (τ, σ ) of a closed string in spacetime (worldsheet). It begins and ends at
the circles parametrized by x i (σ ) and x f (σ ). The worldsheet is topologically a cylinder and is
parametrized by (τ, σ ) ∈ [τ i , τ f ] × [0, 2π)
and 1.5. To each topology is associated different boundary conditions and types of
strings:
• closed: periodic and anti-periodic boundary conditions;
• open: Dirichlet and Neumann boundary conditions.
While a closed string theory is consistent by itself, an open string theory is not and
requires closed strings.
Spectrum
In order to gain some intuition for the states described by a closed string, one can
write the Fourier expansion of the fields X μ (in the gauge g ab = η ab and after
imposing the equations of motion)
X
μ (τ, σ ) ∼ x
μ
+ p
μ τ +
i
√
2
n∈Z ∗
1
n
α
μ
n e
−in(τ −σ )
+ ¯
α
μ
n e
−in(τ +σ )
,
(1.1)