String Theory and M-Theory

158 Strings with space-time supersymmetry
The crucial minus sign in this formula is determined from the requirement
that Ω3 should be closed, that is, dΩ3 = 0. To see this substitute the explicit
formula dΠ µ = −(d ¯ Θ 1 Γ µ dΘ 1 + dΘ 2 Γ µ dΘ 2 ) into
dΩ3 = c(d ¯ Θ 1 ΓµdΘ 1 − d ¯ Θ 2 ΓµdΘ 2 )dΠ µ . (5.45)
The minus sign ensures the cancellation of the cross terms that have two
powers of dΘ 1 and two powers of dΘ 2 . The terms that are quartic in dΘ 1
or dΘ 2 , on the other hand, vanish due to Eq. (5.43).
Let us now compute the kappa symmetry variation of Ω3,
δΩ3 = 2c(dδ ¯ Θ 1 ΓµdΘ 1 − dδ ¯ Θ 2 ΓµdΘ 2 )Π µ
−2c(d ¯ Θ 1 ΓµdΘ 1 − d ¯ Θ 2 ΓµdΘ 2 )δ ¯ Θ A Γ µ dΘ A . (5.46)
Using Eq. (5.43) again, the second line of this expression can be recast in
the form
Therefore,
and thus
−2c(δ ¯ Θ 1 ΓµdΘ 1 − δ ¯ Θ 2 ΓµdΘ 2 )dΠ µ . (5.47)
δΩ3 = d 2c(δ ¯ Θ 1 ΓµdΘ 1 − δ ¯ Θ 2 ΓµdΘ 2 )Π µ
, (5.48)
δΩ2 = 2c(δ ¯Θ 1 ΓµdΘ 1 − δ ¯Θ 2 ΓµdΘ 2 )Π µ . (5.49)
To be explicit, setting c = 1/π gives
δS2 = 2
π
d 2 σε αβ (δ ¯ Θ 1 Γµ∂αΘ 1 − δ ¯ Θ 2 Γµ∂αΘ 2 )Π µ
β . (5.50)
The term S2 is required to have this variation, since then the variation of
the entire action under κ transformations takes the form
δS = 4
d
π
2 σε αβ (δ ¯ Θ 1 P+Γµ∂αΘ 1 − δ ¯ Θ 2 P−Γµ∂αΘ 2 )Π µ
β . (5.51)
The orthogonal projection operators P± are defined by
with
P± = 1
(1 ± γ) (5.52)
2
γ = − εαβΠ µ αΠν βΓµν 2 √ . (5.53)
−G
It now follows that the action is invariant under the transformations
δ ¯ Θ 1 = ¯κ 1 P− and δ ¯ Θ 2 = ¯κ 2 P+ (5.54)