Lectures on String Theory

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E. Exercises
Exercise 1. Show that the Hessian matrix associated to the Nambu-Goto Lagrangian
has for each σ two zero eigenvalues corresponding to Ẋµ and X ′µ .
Exercise 2. How reparametrization invariance can be used to bring the equation
⎛
⎞ ⎛
⎞
∂
⎝ (ẊX′ )X ′µ − (X ′ ) 2 Ẋ µ
√
⎠ + ∂ ⎝ (ẊX′)Ẋµ − (Ẋ)2 X ′µ
√
⎠ = 0 .
∂τ
(ẊX′ ) 2 − Ẋ2 X ′2 ∂σ
(ẊX′ ) 2 − Ẋ2 X ′2
to the simplest form?
Exercise 3. The Polyakov string. Prove that equations of motion for the fields
X µ imply conservation of the two-dimensional stress-energy tensor
∇ µ T µν = 0
Exercise 4. Show that T αβ = 0 implies that the end points of the open string
move with the speed of light.
Exercise 5. Non-relativistic string.
• Consider a string in equilibrium on the x-axis between (0, 0) and (L, 0) and
suppose that the infinitesimal parts of the string can move only in the y-
direction. Derive the Lagrangian with µ the mass density and T the string
tension.
• Derive the equation of motion from this Lagrangian and keep explicit attention
to the boundary conditions.
• Analyze the boundary terms. What must you impose in order to have a stationary
action?
• Construct the momentum function P .
• Calculate the time derivative of the momentum (consider the boundary conditions).
What do you conclude?
• Fourier transform the x-coordinate and solve the eom. Do this for both boundary
conditions.