Lectures on String Theory

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provided the parameter ɛ satisfies the following equation
ρ β ρ α ∂ β ɛ = 0 .
Exercise 55. If in the previous exercise the parameter ɛ is constant then we
deal with global supersymmetry transformations. Derive the Noether current which
corresponds to this global symmetry of the action.
Exercise 56*. Express the spin connection ω via vielbein e by using the condition
of vanishing of the torsion
T a αβ = D α e a β − D β e a α = 0 .
Exercise 57. Derive the on-shell algebra of supersymmetry transformations
[δ 1 , δ 2 ] =? .
Exercise 58. Using the Noether method derive the currents corresponding to
the Poincaré symmetry for the fermionic string. Express the corresponding Noether
charges via oscillators for both Neveuw-Schwarz and Ramond sectors.