Lectures on String Theory

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are constraints on the dynamics of our system. Using the action we find
T αβ = 1 2 ∂ αX µ ∂ β X µ − 1 4 η αβ∂ γ X µ ∂ γ X µ + i 4 ¯ψρ α ∂ β ψ µ + i 4 ¯ψρ β ∂ α ψ µ
G α = 1 4 ρβ ρ α ψ µ ∂ β X µ
Note that G α is ρ-traceless, i.e.
ρ α G α = 0 .
This equation is an analog of T α α = 0. Finally, by using equations of motion one can
show that the stress tensor and the supercurrent are conserved
∂ α T αβ = 0 , ∂ α G α = 0 .
These conservation laws lead to existence of infinite number of conserved charges. To
analyze the algebra of contraints in more detail it is convenient to use the world-sheet
light-cone coordinates σ ± . In the light-cone coordinates the action becomes
S = 1
2π
Equations of motion are
∫
(
)
d 2 σ ∂ + X∂ − X + i(ψ + ∂ − ψ + + ψ − ∂ + ψ − ) .
∂ + ∂ − X µ = 0 , ∂ − ψ µ + = ∂ + ψ µ − = 0 .
Solving equations of motion for fermions we get
ψ µ + = ψ µ +(σ + ) , ψ µ − = ψ µ −(σ − ) .
This, it appears that two components of the Majorana fermion are left- and rightmoving
fields on the world-sheet.
are
The components of the stress-tensor T +− = 0 = T −+ , while the other components
T ++ = 1 2 ∂ +X∂ + X + i 2 ψ +∂ + ψ + ,
T −− = 1 2 ∂ −X∂ − X + i 2 ψ −∂ − ψ − .
The components of the supercurrent are
G + = 1 2 ψ +∂ + X ,
G − = 1 2 ψ −∂ − X .
The conservation laws look in the light-cone coordinates as
∂ − G + = ∂ + G − = 0 , ∂ − T ++ = ∂ + T −− = 0 .