Lectures on String Theory

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Exercise 16. Prove that that (closed) string has an infinite set of integrals of
motion: for any function f the quantities
are conserved.
L f =
∫ 2π
0
dσf(σ − )T −− , ¯Lf =
∫ 2π
0
dσ f(σ + )T ++ ,
Exercise 17. Obtain an expression for the Virasoro generators L m and ¯L m in
terms of string oscillators.
Exercise 18. Show that the operators D n = −ie inθ d dθ
obey the Virasoro algebra.
Exercise 19. Consider an open string solution 0 ≤ σ ≤ π:
X 0 = t = Lτ
X 1 = L cos σ cos τ ,
X 2 = L cos σ sin τ ,
X i = 0, i = 3, . . . , d − 1 .
• Show that this solution satisfies the Virasoro constraints and open string boundary
conditions.
• Compute the mass of string.
• Compute the angular momentum J ≡ J 12 of string.
• Show that J = α ′ M 2 , where α ′ = 1
2πT
is called a slope of the Regge trajectory.
Exercise 20.
Poincare group
By using the Poisson brackets between the generators of the
{P µ , P ν } = 0
{P µ , J ρσ } = η µσ P ρ − η µρ P σ
{J µν , J ρσ } = η µρ J νσ + η νσ J µρ − η νρ J µσ − η µσ J νρ
show that for a certain choice of a function f the following expression
J 2 = 1 ( )
Jαβ J αβ + f(P 2 )P α J αλ P β J βλ , P 2 ≡ P µ P µ .
2