Lectures on String Theory

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where A(m) is a function of m. The aim of this exercise is to find the constraints on
A(m) that follow from the condition that the relations above define the Lie algebra.
• That does the antisymmetry requirement on a Lie algebra tells you about
A(m)? What is A(0)?
• Consider the Jacobi identity for the generators L m , L n and L k with m+n+k =
0. Show that
(m − n)A(k) + (n − k)A(m) + (k − m)A(n) = 0.
• Use the last equation to show that A(m) = αm and A(m) = βm 3 , for constants
α and β, yield consistent central extensions.
• Consider the last equation with k = 1. Show that A(1) and A(2) determine all
A(n)
Exercise 29.
• Use the Virasoro algebra to show that if a state is annihilated by L 1 and L 2
then it is annihilated by all L n with n ≥ 1.
• Consider the Virasoro generators L 0 , L 1 and L −1 . Write out the relevant
commutators. Do these operators form a subalgebra of the Virasoro algebra?
Is there a central term here?
Exercise 30. Consider open string. The fundamental commutation relation is
[X µ (σ, τ), P ν (σ ′ , τ)] = iη µν δ(σ − σ ′ ) , σ ∈ [0, π] .
• Show that consistency with the oscillator expansion implies that
δ(σ − σ ′ ) = 1 ∞∑
cos nσ cos nσ ′
π
n=−∞
• Why the fundamental commutation relation compatible with open string boundary
conditions?
• Prove this representation for the δ-function by using the fact that any function
f(σ) with σ ∈ [0, π] and vanishing derivative at σ = 0, π can be expanded as
f(σ) =
∞∑
A n cos nσ .
n=0