String Theory and M-Theory

162 Strings with space-time supersymmetry
vanishes for µ = i, + and is nonvanishing only for µ = −. Using the fermion
gauge choice (5.59), the first equation in (5.64) takes the form
(ΓµΠ µ α)P αβ
− ∂βΘ 1 = (Γ+Π + α + ΓiΠ i α)P αβ
− ∂βΘ 1 = 0. (5.68)
Multiplying this result by Γ + gives
Γ + (Γ+Π + α + ΓiΠ i α)P αβ
− ∂βΘ 1 = 2Π + α P αβ
− ∂βΘ 1 = 0. (5.69)
Using Π + α = p + δα,0 this gives
P 0β
− ∂βΘ 1 = 0. (5.70)
Using the definition of P αβ
− and the gauge choice hαβ = e φ ηαβ, this takes
the form
∂
∂τ
∂
+ Θ
∂σ
1 = 0. (5.71)
This is the equation of motion for Θ 1 in the light-cone gauge. It is considerably
simpler than the covariant equation of motion. Since this equation is
linear, it can be solved explicitly. In a similar way, the equations of motion
for X i and Θ 2 also become linear. One learns, in particular, that Θ 1 and Θ 2
describe waves that propagate in opposite directions along the string. This
fact can be traced back to the relative minus sign between the Θ 1 and Θ 2
dependence in S2.
The light-cone gauge action
The superstring theories considered here have ten-dimensional Lorentz invariance,
but in the light-cone gauge only an SO(8) transverse rotational
symmetry is manifest. The eight surviving components of each Θ form
an eight-dimensional spinor representation of this transverse SO(8) group
(or more precisely its Spin(8) covering group). There are two inequivalent
spinor representations of Spin(8), which are denoted by 8s and 8c. These
two representations describe spinors of opposite eight-dimensional chirality.
The ten-dimensional chirality of the spinors Θ 1,2 determines whether an 8s
or 8c representation survives in the light-cone gauge. Using the symbol S
for the surviving components of Θ, multiplied by a factor proportional to
p + , the choices are
IIA : p + Θ A → 8s + 8c = (S a 1 , S ˙a 2 ), (5.72)
IIB : p + Θ A → 8s + 8s = (S a 1 , S a 2 ). (5.73)